Introduction: Cardano equations for the solubility of third-degree equations, rings and fields, the characteristics of a field, known facts about rings of polynomials, field extensions, construction of some field extensions, the sub-ring generated by a subset, the subfield generated by a subset, algebraic and transcendent elements, algebraically closed fields.

Splitting fields: Simple extensions and maps between simple extensions, splitting fields, existence of the splitting field, uniqueness up to isomorphisms of the splitting field, multiple roots, formal derivatives, separable polynomials and perfect fields, minimal polynomials and their characterizations.

The fundamental Theorem of Galois Theory: Group of the automorphisms of a field, normal, separable and Galois extensions, characterizations of separable extensions, Fundamental Theorem of Galois Correspondence, examples, Galois group of a polynomial, Radical extensions, solvable groups and Galois's Theorem on solving equations , Theorem of the existence of primitive elements.

The computation of Galois group: Galois groups as subgroups of $ S_n $, transitive subgroups of $ S_n $, characterization of irreducibility in terms of transitivity, polynomials with Galois groups in $ A_n $, Theory of discriminants, Galois groups of polynomials with degree up to $4$, examples of polynomials with Galois group $S_p$.

Cyclotomic fields: Definitions, Galois group, maximal real subfields, quadratic subfields, Galois groups, cyclotomic polynomials and their properties, Theorem of the inverse Galois theory for abelian groups.

Finite fields: Existence and uniqueness of finite fields, Galois group of a finite field, subfields of a finite field, enumeration of irreducible polynomials on finite fields. Construction of the algebraic closure of a finite field with $ p $ elements.

Constructions with ruler and compass: Definition of constructible points of the plane, constructible real numbers, characterization of constructible points in terms of fields, constructible subfields and construction of constructible numbers, cube duplication, trisection of angles, quadrature of the circle and Gauss's theorem for the construction of regular polygons with ruler and compass.

J. S. Milne.Fields and Galois Theory. Course Notes v4.22 (March 30, 2011).
S. Gabelli. Teoria delle Equazioni e Teoria di Galois. Springer UNITEXT (La Matematica per il 3+2) 2008, XVII, 410 pagg., ISBN: 978-88-470-0618-8
E. Artin.Galois Theory. NOTRE DAME MATHEMATICAL LECTURES Number 2. 1942.
C. Procesi.Elementi di Teoria di Galois. Decibel, Zanichelli, (Seconda ristampa, 1991).

Graphs: basic definitions. Simple and non simple graphs, planarity, connectivity, degree, regularity, incidence and adjacency matrices. Examples of families of graphs. The '' handshaking lemma ''.
Graphs obtained from others: complement, subgraph, cancellation and contraction. Isomorphisms and automorphisms of graphs. Connectivity: paths, cycles. A graph is bipartite if and only if each cycle has equal length.
Connetivity and connected component. Connectivity for sides and vertices. Eulerian and semi-Eulerian graphs.
Euler's theorem: a connected graph is Eulerian if and only if every vertex has an even degree. Hamiltonian graphs. Sufficient conditions to guarantee that a graph is Hamiltonian: the theorems of Ore and Dirac.
the Ore theorem. Trees and forests. The cyclomatic number and the "cutset" rank of a graph.
Fundamental system of cycles and cuts associated with a generating forest. Enumeration of generating forests. The Cayley theorem.
Generating trees: the "greedy" algorithm for the "connector problem". Planar graphs. K3.3 and K5 are not planar. Statement of the theorem of Kuratovski and variations.
Euler's formula for planar graphs. The dual of a planar graph.
Correspondence between cycles and cuts for planar graphs and their dual. Dual abstract. A graph that admits a dual abstract is planar. Colorings: initial considerations and some properties.
Colorings: the 5 colors theorem. Graphs on surfaces: classification of topological surfaces.
Coloring of faces and duality between this problem and the coloring of vertices. Reduction of the proof of the 4-color theorem to the coloring of cubic graph faces. '' The marriage problem ": Hall's theorem.
Hall's theorem in the language of transversals. Criteria of existence of transversal and partial transversivities. Application to the construction of Latin squares. Direct graphs: basic notions and orientability.
The Max-Flow Min-Cut theorem and Menger's theorem.
Complexity of algorithms and applications to the theory of graphs.
Introduction to the theory of matroids: definitions using bases and independent elements. Graphical and cographic matroids, vector matroids and the problem of representability.
Definition of matroid using the cycles and the rank function. Minors of a matroid. Transverse matroids and the Rado Theorem for matroids.
Union of matroids and applications: existence of disjoint bases in a matroid. Duality for matroids and applications to graphic and cographic matroids. Planar matroids and the generalization of Kuratovski's theorem for matroids.
Elements of algebraic graph theory: the incidence matrix and the Laplacian matrix of an oriented graph. The vertex space and the space of edges of a graph. Subspaces of the cycles and subspace of the cuts of a defined oriented graph of the incidence matrix.
Basis for the space of the cycles and for the space of the cuts of a graph. The Riemann-Roch theorem for graphs.
Proof of the "generalized Matrix Tree theorem". The contraction / restriction algorithm for matroids. Examples. The number of acyclical orientations of a graph.
Graph polynomials: the chromatic polynomial, the "reliabiliaty" polynomial. Examples. The polynomial rank of a matroid. Properties and first applications.
Proof of the structure theorem for functions on matroids that satisfy contraction/restriction properties. Their incarnation through the polynomial rank.
Whitey's moves and two isomorphisms for graphs. Isomorphism between graphical matroids implies isomorphism between graphs in case the graphs are 3 connected.
Whitney's Theorem for graphic matroids: sketch of the demonstration. Characterization for minors excluded from binary and regular matroids. The theorem of Seymour.

R. Diestel: Graph theory, Spriger GTM 173.
R. Wilson: Introduction to Graph theory, Prentice Hall.
B. Bollobas: Modern Graph theory, Springer GTM 184.
J. A. Bondy, U.S.R. Murty: Graph theory, Springer GTM 244.
N. Biggs: Algebraic graph theory, Cambridge University Press.
C. D. Godsil, G. Royle: Algebraic Graph theory, Springer GTM 207.
J. G. Oxley: Matroid theory. Oxford graduate texts in mathematics, 3.

Linear Systems
Direst methods: Gaussian elimination. Pivoting strategies. Gaussian elimination as a factorization. Doolittle and Choleshy factorizations. Iterative methods: Jacobi, Gauss-Seidel, SOR, Richardson, and related convergence results. Comparison of direct vs iterative solvers. Stability of algorithms for the solution of linear systems.

Iterative Methods for Scalar Nonlinear Equations
The intermediate zero theorem. The algorithms of bisection, Newton, secants, chords, and related convergence results. (Reference: Chapter 1 excluding Section 1.2.3, and Appendices A.1, A.2)

Approximation of Functions
General approximation strategies. Interpolating polynomial in Lagrange and Newton form. Representation of the interpolation error. Convergence of the interpolating polynomial for analytic functions. Refinement strategies in interpolation: Chebyshev nodes, composite approximations. Error estimates. Hermite polynomial, construction and representation of the error. Least Squares approximations. (Reference: Chapter 5 excluding Section 5.2, and Appendix A.4)

Numerical Integration
General principles of numerical integration. Polya's Theorem on the convergence of interpolatory quadrature formulae. Closed and open Newton-Cotes formulae. Stability results and error estimation. Generalized Newton-Cotes formulae and their convergence. Gaussian quadratures and their convergence. (Reference: Chapter 6)

Laboratory Activity
C language coding of some of the major algorithms, and in particular: Gaussian elimination, iterative methods for linear systems and scalar equations, Lagrange/Newton interpolation with a refinement strategy.

N.B.: References are provided with respect to the course notes.

Roberto Ferretti, "Appunti del corso di Analisi Numerica", available at the address: http://www.mat.uniroma3.it/users/ferretti/corso.pdf

Slides of the lessons, available from the course page: http://www.mat.uniroma3.it/users/ferretti/bacheca.html

1. Probability

Introductory example: The branching process.
Introduction to measure theory.
Measure spaces. Events. Uniqueness and extension of measure.
Probability measures. First Borel--Cantelli lemma.
Random variables, distribution function and law. Indipendence.
Second Borel--Cantelli lemma.
0--1 law for independent random variables.

2. Integration, expected value

Introduxction to integration theory. Monotone convergence theorem.
Expectation. Taking the limit under expectation.
Jensen's inequality. L_p norms.
H\"older inequality and Cauchy-Schwarz.
Markov's inequality.
Examples of weak and strong laws of large numbers.
Product measures. Fubini theorem.
Joint laws.

3. Conditional expectation, martingales and limit theorems

Conditional expectation with respect to a sub $\sigma$--algebra. Kolmogorov existence and uniqueness theorem.
Filtrations. Martingale. Gambilng. Stopping times.
Optional stopping.
Some applications to exit times from an interval.
Convergence theorem for martingales in L_1 and in L_2. Kolmogorov's strong law of large numbers.

4. Convergence ind distribution and the central limit theorem

Characteristic functions. Inversion theorem.
Equivalence between convergence in distributiona nd pointwise convergence of characteristic functions.
Various modes of convergence for random variables. Examples.

D. Williams.
Probability with martingales
Cambridge University Press, 1991


R. Durrett
Probability: Theory and Examples
Thomson, 2000

Selected examples of partial differential equations
The weak form of boundary value problems
Galerkin method
The Finite Elements Method (FEM)
Convergence of MEF solutions
Data structure and FEM implementation
Numerical methods for the solution of algebraic systems

Mark S. Gockenbach,
Understanding and Implementing the Finite Elements Method,
SIAM 2006

LAGRANGIAN AND HAMILTONIAN MECHANICS. VARIATIONAL PRINCIPLES. CONSTRAINTS. CYCLICI VARIABLES. CONSTANTS OF MOTION AND SYMMETRIES. HAMILTONIAN FLOWS. LIOUVILLE THEOREM AND POINCARE RECURRENCE THEOREM. CANONICAL TRANSFORMATIONS. GENERATING FUNCTIONS. HAMILTON-JACOBI METHOD AND ACTION-ANGLE VARIABLES. PERTURBATION THEORY. KAM THEOREM.

[1] G. GENTILE,INTRODUZIONE AI SISTEMI DINAMICI. 1.EQUAZIONI DIFFERENZIALI ORDINARIE, ANALISI QUALITATIVA E ALCUNE APPLICAZIONI.
AVAILABLE ON-LINE: http://www.mat.uniroma3.it/users/gentile/2016-2017/FM410/testo.html
[2] G. GENTILE,INTRODUZIONE AI SISTEMI DINAMICI. 2.FORMALISMO LAGRANGIANO E HAMILTONIANO.
AVAILABLE ON-LINE:http://www.mat.uniroma3.it/users/gentile/2016-2017/FM410/testo.html
[3] G. DELL'ANTONIO, ELEMENTI DI MECCANICA. LIGUORI EDITORE, (1996).
[4] V.I. ARNOLD, METODI MATEMATICI DELLA MECCANICA CLASSICA. EDITORI RIUNITI, (1979).
[5] G. GALLAVOTTI, MECCANICA ELEMENTARE. BOLLATI-BORINGHIERI, (1980)

Graphs: basic definitions. Simple and non simple graphs, planarity, connectivity, degree, regularity, incidence and adjacency matrices. Examples of families of graphs. The '' handshaking lemma ''.
Graphs obtained from others: complement, subgraph, cancellation and contraction. Isomorphisms and automorphisms of graphs. Connectivity: paths, cycles. A graph is bipartite if and only if each cycle has equal length.
Connetivity and connected component. Connectivity for sides and vertices. Eulerian and semi-Eulerian graphs.
Euler's theorem: a connected graph is Eulerian if and only if every vertex has an even degree. Hamiltonian graphs. Sufficient conditions to guarantee that a graph is Hamiltonian: the theorems of Ore and Dirac.
the Ore theorem. Trees and forests. The cyclomatic number and the "cutset" rank of a graph.
Fundamental system of cycles and cuts associated with a generating forest. Enumeration of generating forests. The Cayley theorem.
Generating trees: the "greedy" algorithm for the "connector problem". Planar graphs. K3.3 and K5 are not planar. Statement of the theorem of Kuratovski and variations.
Euler's formula for planar graphs. The dual of a planar graph.
Correspondence between cycles and cuts for planar graphs and their dual. Dual abstract. A graph that admits a dual abstract is planar. Colorings: initial considerations and some properties.
Colorings: the 5 colors theorem. Graphs on surfaces: classification of topological surfaces.
Coloring of faces and duality between this problem and the coloring of vertices. Reduction of the proof of the 4-color theorem to the coloring of cubic graph faces. '' The marriage problem ": Hall's theorem.
Hall's theorem in the language of transversals. Criteria of existence of transversal and partial transversivities. Application to the construction of Latin squares. Direct graphs: basic notions and orientability.
The Max-Flow Min-Cut theorem and Menger's theorem.
Complexity of algorithms and applications to the theory of graphs.
Introduction to the theory of matroids: definitions using bases and independent elements. Graphical and cographic matroids, vector matroids and the problem of representability.
Definition of matroid using the cycles and the rank function. Minors of a matroid. Transverse matroids and the Rado Theorem for matroids.
Union of matroids and applications: existence of disjoint bases in a matroid. Duality for matroids and applications to graphic and cographic matroids. Planar matroids and the generalization of Kuratovski's theorem for matroids.
Elements of algebraic graph theory: the incidence matrix and the Laplacian matrix of an oriented graph. The vertex space and the space of edges of a graph. Subspaces of the cycles and subspace of the cuts of a defined oriented graph of the incidence matrix.
Basis for the space of the cycles and for the space of the cuts of a graph. The Riemann-Roch theorem for graphs.
Proof of the "generalized Matrix Tree theorem". The contraction / restriction algorithm for matroids. Examples. The number of acyclical orientations of a graph.
Graph polynomials: the chromatic polynomial, the "reliabiliaty" polynomial. Examples. The polynomial rank of a matroid. Properties and first applications.
Proof of the structure theorem for functions on matroids that satisfy contraction/restriction properties. Their incarnation through the polynomial rank.
Whitey's moves and two isomorphisms for graphs. Isomorphism between graphical matroids implies isomorphism between graphs in case the graphs are 3 connected.
Whitney's Theorem for graphic matroids: sketch of the demonstration. Characterization for minors excluded from binary and regular matroids. The theorem of Seymour.

R. Diestel: Graph theory, Spriger GTM 173.
R. Wilson: Introduction to Graph theory, Prentice Hall.
B. Bollobas: Modern Graph theory, Springer GTM 184.
J. A. Bondy, U.S.R. Murty: Graph theory, Springer GTM 244.
N. Biggs: Algebraic graph theory, Cambridge University Press.
C. D. Godsil, G. Royle: Algebraic Graph theory, Springer GTM 207.
J. G. Oxley: Matroid theory. Oxford graduate texts in mathematics, 3.

1. Abstract integration theory
Riemann integration theory. The concept of measurability. Step functions. Elementary properties of measures. Arithmetic in [0,∞]. Integration of positive functions. Integration of complex functions. Importance of sets
with null measure.
2. Positive Borel measures
Vector spaces. Topological preliminaries. Riesz representation theorem. Regularity properties of Borel measures. Lebesgue measure. Continuity properties of measurable functions.
3. L^p spaces
Inequalities and convex functions. L^p spaces. Approximation through continuous functions.
4. Basic theory of Hilbert spaces
Inner products and linear functionals. Dual space of L^2
4. Integration on product spaces
Measurability on cartesian products. Product measure. Fubini theorem.
4. Complex measures
Total variation. Absolute continuity. Radon-Nykodym theorem. Bounded linear functionals on L^p. The Riesz representation theorem.

"Analisi reale e complessa”, W. Rudin. Bollati Boringhieri.

Under the general title of this Course, we consider both, asymptotic and
numerical methods mostly to handle ordinary differential equations and systems,
and possibly some partial differential equations, and applications. Among the
asymptotic methods, the so-called ``singular perturbations'' are addressed.
These two subjects, which represent basic techniques of Applied Mathematics,
are usually taught separately. In this course we will use both, at the same
time, in the same problems.

Asymptotic Methods:

Introduction to Asymptotic Analysis. Asymptotic series. Regular and singular
perturbations of ordinary differential equations and systems.

Numerical Methods:

Numerical solution, using several algorithms, of ordinary differential
equations and systems. This will be mostly done in the computing laboratory.

Robert E. O'malley,
Singular Perturbation Methods for Ordinary Differential Equations,
Springer-Verlag, Applied Mathematical Sciences, 89, New York, 1991.

John K. Hunter,
Asymptotic Analysis and Singular Perturbation Theory,
Lecture Notes, https://www.math.ucdavis.edu/~hunter/notes/asy.pdf,
2004

Brownian motion I

Gaussian multivariate distribution. Processes with stationary and independent increments.
Definition and continuity properties of Brownian motion. Non differentiability of trajectories.
Markov and Strong Markov property.

Brownian motion II

Brownian motion in higher dimensions. Harmonic functions and the Dirichlet problem in regular domains.
The Poisson problem. Iterated logarithm law. Skorohod embedding.
Donsker's invariance principle. Applications: arcsine law and the maximum of random walks.


Stochastic integrals

The Paley-Wiener-Zygmund integral. Stochastic integral with respect to Brownian motion. It\^o formula and applications. Local time and the Tanaka formula.



Stochastic differential equations

The linear case. Existence and uniqueness theorem for stochastic differential equations.
Diffusion process in the limit of zero noise. Infinitesimal generator and partial differential equations.
Feynman-Kac formula and applications.

P. M\"orters and Y. Peres
Brownian Motion
Cambridge University Press
2010


L.C. Evans
An Introduction to Stochastic Differential Equations
AMS bookstore
2013

Introduction: Cardano equations for the solubility of third-degree equations, rings and fields, the characteristics of a field, known facts about rings of polynomials, field extensions, construction of some field extensions, the sub-ring generated by a subset, the subfield generated by a subset, algebraic and transcendent elements, algebraically closed fields.

Splitting fields: Simple extensions and maps between simple extensions, splitting fields, existence of the splitting field, uniqueness up to isomorphisms of the splitting field, multiple roots, formal derivatives, separable polynomials and perfect fields, minimal polynomials and their characterizations.

The fundamental Theorem of Galois Theory: Group of the automorphisms of a field, normal, separable and Galois extensions, characterizations of separable extensions, Fundamental Theorem of Galois Correspondence, examples, Galois group of a polynomial, Radical extensions, solvable groups and Galois's Theorem on solving equations , Theorem of the existence of primitive elements.

The computation of Galois group: Galois groups as subgroups of $ S_n $, transitive subgroups of $ S_n $, characterization of irreducibility in terms of transitivity, polynomials with Galois groups in $ A_n $, Theory of discriminants, Galois groups of polynomials with degree up to $4$, examples of polynomials with Galois group $S_p$.

Cyclotomic fields: Definitions, Galois group, maximal real subfields, quadratic subfields, Galois groups, cyclotomic polynomials and their properties, Theorem of the inverse Galois theory for abelian groups.

Finite fields: Existence and uniqueness of finite fields, Galois group of a finite field, subfields of a finite field, enumeration of irreducible polynomials on finite fields. Construction of the algebraic closure of a finite field with $ p $ elements.

Constructions with ruler and compass: Definition of constructible points of the plane, constructible real numbers, characterization of constructible points in terms of fields, constructible subfields and construction of constructible numbers, cube duplication, trisection of angles, quadrature of the circle and Gauss's theorem for the construction of regular polygons with ruler and compass.

J. S. Milne.Fields and Galois Theory. Course Notes v4.22 (March 30, 2011).
S. Gabelli. Teoria delle Equazioni e Teoria di Galois. Springer UNITEXT (La Matematica per il 3+2) 2008, XVII, 410 pagg., ISBN: 978-88-470-0618-8
E. Artin.Galois Theory. NOTRE DAME MATHEMATICAL LECTURES Number 2. 1942.
C. Procesi.Elementi di Teoria di Galois. Decibel, Zanichelli, (Seconda ristampa, 1991).

1. Abstract integration theory
Riemann integration theory. The concept of measurability. Step functions. Elementary properties of measures. Arithmetic in [0,∞]. Integration of positive functions. Integration of complex functions. Importance of sets
with null measure.
2. Positive Borel measures
Vector spaces. Topological preliminaries. Riesz representation theorem. Regularity properties of Borel measures. Lebesgue measure. Continuity properties of measurable functions.
3. L^p spaces
Inequalities and convex functions. L^p spaces. Approximation through continuous functions.
4. Basic theory of Hilbert spaces
Inner products and linear functionals. Dual space of L^2
4. Integration on product spaces
Measurability on cartesian products. Product measure. Fubini theorem.
4. Complex measures
Total variation. Absolute continuity. Radon-Nykodym theorem. Bounded linear functionals on L^p. The Riesz representation theorem.

"Analisi reale e complessa”, W. Rudin. Bollati Boringhieri.

Affine Spaces
Zariski topology. Affine closed subsets and radical ideals. Theorem of the zeros of Hilbert. Correspondence between closed subsets and radical ideals. Noetherian topological spaces. Irreducible closed subsets, irreducible components. Regular functions of affine closed subsets. Regular maps, isomorphisms. Principal open subsets. Examples. Projections are open. Finite morphisms.

Varieties
Projective spaces and their Zariski topology. Quasi-projective varieties. Rational and regular maps. Projective hypersurfaces. Birational equivalence. Principal open subsets and affine closed subsets. Affine varieties. Dimension of quasi-projective varieties. Finite and generically finite morphisms. Characterizations of birational equivalence. Characterization of generically finite morphisms. Costructible sets, Chevalley's theorem. Every variety is birational to a hypersurface.

Local geometry
Local ring of a variety in a point. Zariski cotangent space. Zariski tangent space. Singular and non singular points.

L. Caporaso
Introduction to algebraic geometry
Notes of the course

I. Shafarevich
Basic Algebraic geometry
Springer-Verlag, Berlin, 1994

Graphs: basic definitions. Simple and non simple graphs, planarity, connectivity, degree, regularity, incidence and adjacency matrices. Examples of families of graphs. The '' handshaking lemma ''.
Graphs obtained from others: complement, subgraph, cancellation and contraction. Isomorphisms and automorphisms of graphs. Connectivity: paths, cycles. A graph is bipartite if and only if each cycle has equal length.
Connetivity and connected component. Connectivity for sides and vertices. Eulerian and semi-Eulerian graphs.
Euler's theorem: a connected graph is Eulerian if and only if every vertex has an even degree. Hamiltonian graphs. Sufficient conditions to guarantee that a graph is Hamiltonian: the theorems of Ore and Dirac.
the Ore theorem. Trees and forests. The cyclomatic number and the "cutset" rank of a graph.
Fundamental system of cycles and cuts associated with a generating forest. Enumeration of generating forests. The Cayley theorem.
Generating trees: the "greedy" algorithm for the "connector problem". Planar graphs. K3.3 and K5 are not planar. Statement of the theorem of Kuratovski and variations.
Euler's formula for planar graphs. The dual of a planar graph.
Correspondence between cycles and cuts for planar graphs and their dual. Dual abstract. A graph that admits a dual abstract is planar. Colorings: initial considerations and some properties.
Colorings: the 5 colors theorem. Graphs on surfaces: classification of topological surfaces.
Coloring of faces and duality between this problem and the coloring of vertices. Reduction of the proof of the 4-color theorem to the coloring of cubic graph faces. '' The marriage problem ": Hall's theorem.
Hall's theorem in the language of transversals. Criteria of existence of transversal and partial transversivities. Application to the construction of Latin squares. Direct graphs: basic notions and orientability.
The Max-Flow Min-Cut theorem and Menger's theorem.
Complexity of algorithms and applications to the theory of graphs.
Introduction to the theory of matroids: definitions using bases and independent elements. Graphical and cographic matroids, vector matroids and the problem of representability.
Definition of matroid using the cycles and the rank function. Minors of a matroid. Transverse matroids and the Rado Theorem for matroids.
Union of matroids and applications: existence of disjoint bases in a matroid. Duality for matroids and applications to graphic and cographic matroids. Planar matroids and the generalization of Kuratovski's theorem for matroids.
Elements of algebraic graph theory: the incidence matrix and the Laplacian matrix of an oriented graph. The vertex space and the space of edges of a graph. Subspaces of the cycles and subspace of the cuts of a defined oriented graph of the incidence matrix.
Basis for the space of the cycles and for the space of the cuts of a graph. The Riemann-Roch theorem for graphs.
Proof of the "generalized Matrix Tree theorem". The contraction / restriction algorithm for matroids. Examples. The number of acyclical orientations of a graph.
Graph polynomials: the chromatic polynomial, the "reliabiliaty" polynomial. Examples. The polynomial rank of a matroid. Properties and first applications.
Proof of the structure theorem for functions on matroids that satisfy contraction/restriction properties. Their incarnation through the polynomial rank.
Whitey's moves and two isomorphisms for graphs. Isomorphism between graphical matroids implies isomorphism between graphs in case the graphs are 3 connected.
Whitney's Theorem for graphic matroids: sketch of the demonstration. Characterization for minors excluded from binary and regular matroids. The theorem of Seymour.

R. Diestel: Graph theory, Spriger GTM 173.
R. Wilson: Introduction to Graph theory, Prentice Hall.
B. Bollobas: Modern Graph theory, Springer GTM 184.
J. A. Bondy, U.S.R. Murty: Graph theory, Springer GTM 244.
N. Biggs: Algebraic graph theory, Cambridge University Press.
C. D. Godsil, G. Royle: Algebraic Graph theory, Springer GTM 207.
J. G. Oxley: Matroid theory. Oxford graduate texts in mathematics, 3.

1) Computability, complexity and representability:

- Introduction to decision problems, algorithmic and non-algorithmic procedures, deterministic computations, discrete procedures, the notion of an alphabet, the notion of a formal language. Decidability and semi-decidability of a set. Deterministic, finitary and discrete computations. Formal algorithms: formal definition of algorithm, configurations of input, output, transition function. Example of formalization of an algorithm. Decidability for finished automata. Representation of the automata by matrices. Free Monoid of words. Formal semi-rings. Non-deterministic finite automata. Regular Languages. Equivalence between deterministic and non-deterministic automata.

- Turing machines: definition, decidability for Turing machine, stopping time, stopping space. Cost of computation. Complexity: worst case and average case. Independence of decision time from a finite number of input configurations. Complexity functions, complexity classes DTIME and DSPACE (deterministic time and space). Inclusion DTIME (T (n)) ⊂ DSPACE (T (n)) ⊂ DTIME (2 ^ {cT (n)}). Pumping Lemma. Simulation of algorithms, simulation of the half tape Turing machine, simulation of a multi-tape machine. Special Turing machines. Linear Speedup theorem for Turing machines with an extended alphabet. Evaluation of acceleration coefficient in relation to alphabets. Decisions of natural number sets. Independence from representation. Considerations concerning complexity.

- Turing computability: definition of Turing computable function, characteristic functions of Turing decidable sets, the class of Turing computable functions is closed by composition, concatenation, primitive recursion and minimization. Examples of Turing computable functions. Recursive Functions: equivalence between Turing computability and recursive functions. Ackermann function ([1] chapter 1,2,3,4,5 and [4] chapter 1).

- Time-constructible functions. The notion of T-clock. Examples of some time constructible function. Closure by composition.

- Non-deterministic Turing machines: characterization through the decidability of projection sets. Definition of the class of polynomial non-deterministic functions. NP-complete problems.

2) Lambda calculus and functional programming:

- Declarative programming: historical outline on the lambda calculus, basic definitions, the terms of the lambda calculus, the simple substitution. Relations on the lambda terms. Congruences, transition to the context. α-equivalence. alpha-equivalence passes to the context. Transitive closure of a relationship, owned by Church-Rosser. Listing of lambda-terms with respect to alpha-equivalence.

- Definition of beta-reduction and beta-equivalence. Church-Rosser's theorem for beta-reduction. Normal forms for beta-reduction. Beta-reduction strategies. Normalizing strategy: left reduction (left most-outer most). Head reduction. Soluble Terms. Head Normal Forms. Solvability characterization theorem.

- Representation of the recursive functions: lambda definability theorem. The existence of the fixed point for any lambda term. Church Fixed Point and Curry fixed point.
- Representation of other data types in the lambda-calculus: pairs, lists, trees, the solution of recursive equations on lambda-terms ([2] chapters 1, 2, 5).

- P. Dehornoy, Calculabilite et Decidabilite, (1993) Springer-Verlag (in francese);
- J.-L. Krivine, Lambda Calculus: Types and Models, (1993) Ellis Horwood editore.
- M. Sipser, An introduction to the theory of computation (2005), Course Technology.

Linear Systems
Direst methods: Gaussian elimination. Pivoting strategies. Gaussian elimination as a factorization. Doolittle and Choleshy factorizations. Iterative methods: Jacobi, Gauss-Seidel, SOR, Richardson, and related convergence results. Comparison of direct vs iterative solvers. Stability of algorithms for the solution of linear systems.

Iterative Methods for Scalar Nonlinear Equations
The intermediate zero theorem. The algorithms of bisection, Newton, secants, chords, and related convergence results. (Reference: Chapter 1 excluding Section 1.2.3, and Appendices A.1, A.2)

Approximation of Functions
General approximation strategies. Interpolating polynomial in Lagrange and Newton form. Representation of the interpolation error. Convergence of the interpolating polynomial for analytic functions. Refinement strategies in interpolation: Chebyshev nodes, composite approximations. Error estimates. Hermite polynomial, construction and representation of the error. Least Squares approximations. (Reference: Chapter 5 excluding Section 5.2, and Appendix A.4)

Numerical Integration
General principles of numerical integration. Polya's Theorem on the convergence of interpolatory quadrature formulae. Closed and open Newton-Cotes formulae. Stability results and error estimation. Generalized Newton-Cotes formulae and their convergence. Gaussian quadratures and their convergence. (Reference: Chapter 6)

Laboratory Activity
C language coding of some of the major algorithms, and in particular: Gaussian elimination, iterative methods for linear systems and scalar equations, Lagrange/Newton interpolation with a refinement strategy.

N.B.: References are provided with respect to the course notes.

Roberto Ferretti, "Appunti del corso di Analisi Numerica", available at the address: http://www.mat.uniroma3.it/users/ferretti/corso.pdf

Slides of the lessons, available from the course page: http://www.mat.uniroma3.it/users/ferretti/bacheca.html

1. Probability

Introductory example: The branching process.
Introduction to measure theory.
Measure spaces. Events. Uniqueness and extension of measure.
Probability measures. First Borel--Cantelli lemma.
Random variables, distribution function and law. Indipendence.
Second Borel--Cantelli lemma.
0--1 law for independent random variables.

2. Integration, expected value

Introduxction to integration theory. Monotone convergence theorem.
Expectation. Taking the limit under expectation.
Jensen's inequality. L_p norms.
H\"older inequality and Cauchy-Schwarz.
Markov's inequality.
Examples of weak and strong laws of large numbers.
Product measures. Fubini theorem.
Joint laws.

3. Conditional expectation, martingales and limit theorems

Conditional expectation with respect to a sub $\sigma$--algebra. Kolmogorov existence and uniqueness theorem.
Filtrations. Martingale. Gambilng. Stopping times.
Optional stopping.
Some applications to exit times from an interval.
Convergence theorem for martingales in L_1 and in L_2. Kolmogorov's strong law of large numbers.

4. Convergence ind distribution and the central limit theorem

Characteristic functions. Inversion theorem.
Equivalence between convergence in distributiona nd pointwise convergence of characteristic functions.
Various modes of convergence for random variables. Examples.

D. Williams.
Probability with martingales
Cambridge University Press, 1991


R. Durrett
Probability: Theory and Examples
Thomson, 2000

LAGRANGIAN AND HAMILTONIAN MECHANICS. VARIATIONAL PRINCIPLES. CONSTRAINTS. CYCLICI VARIABLES. CONSTANTS OF MOTION AND SYMMETRIES. HAMILTONIAN FLOWS. LIOUVILLE THEOREM AND POINCARE RECURRENCE THEOREM. CANONICAL TRANSFORMATIONS. GENERATING FUNCTIONS. HAMILTON-JACOBI METHOD AND ACTION-ANGLE VARIABLES. PERTURBATION THEORY. KAM THEOREM.

[1] G. GENTILE,INTRODUZIONE AI SISTEMI DINAMICI. 1.EQUAZIONI DIFFERENZIALI ORDINARIE, ANALISI QUALITATIVA E ALCUNE APPLICAZIONI.
AVAILABLE ON-LINE: http://www.mat.uniroma3.it/users/gentile/2016-2017/FM410/testo.html
[2] G. GENTILE,INTRODUZIONE AI SISTEMI DINAMICI. 2.FORMALISMO LAGRANGIANO E HAMILTONIANO.
AVAILABLE ON-LINE: http://www.mat.uniroma3.it/users/gentile/2016-2017/FM410/testo.html
[3] G. DELL'ANTONIO, ELEMENTI DI MECCANICA. LIGUORI EDITORE, (1996)
[4] V.I. ARNOLD, METODI MATEMATICI DELLA MECCANICA CLASSICA. EDITORI RIUNITI, (1979)
[5] G. GALLAVOTTI, MECCANICA ELEMENTARE. BOLLATI-BORINGHIERI, (1980)

The purpose of this course is to give a brief introduction to the Finite Elements Method (FEM), a gold standard for the numerical solution of PDEs systems. Oddly enough, the widespread use of the FEM is not accompanied by an adequate knowledge of the mathematical framework underlying the method.

This course, starting from the weak formulation of balance equations, will give an overview of the techniques used to reduce a differential problem into an algebraic one. During the course, some selected problems in mechanics and physics will be solved, covering the three main types of equations: elliptic, parabolic and hyperbolic.
The course will cover the following topics: - Applied Linear Algebra. - Boundary Value Problems. - Initial Value Problems Moreover, the students will be introduced to the use COMSOL Multiphysics, a scientific software for numerical simulations based on the Finite Element Method.

Mark S. Gockenbach,
Understanding and Implementing the Finite Elements Method,
SIAM, 2006

Gilbert Strang
Computational Science and Engineering
Wellesley-Cambridge Press, 2007

QUANTUM MECHANICS: THE CRISIS OF CLASSICAL PHYSICS. WAVES AND PARTICLES. STATE VECTORS AND OPERATORS. MEASUREMENTS AND OBSERVABLES. THE POSITION OPERATOR. TRANSLATIONS AND MOMENTUM. TIME EVOLUTION AND THE SCHRODINGER EQUATION. PARITY. ONE-DIMENSIONAL PROBLEMS. HARMONIC OSCILLATOR. SYMMETRIES AND CONSERVATION LAWS. TIME INDEPENDENT PERTURBATION THEORY. TIME DEPENDENT PERTURBATION THEORY.

J.J. SAKURAI, J. NAPOLITANO. MECCANICA QUANTISTICA MODERNA. SECONDA EDIZIONE, ZANICHELLI, BOLOGNA, 2014

An english version of the book is also available:
J.J. SAKURAI, J. NAPOLITANO. MODERN QUANTUM MECHANICS. SECOND EDITION, PEARSON EDUCATION LIMITED 2014

Time structure of the exchange of amounts, capital and interest: exchange of amounts, time, price, price of time, convention for measuring time, deferred contracts and rights, transactions with a fixed schedule, regular investment / indebtedness transactions, laws financial, the law of simple interests, the law of compound interests, fundamental definitions based on the value function, factors rates and intensity, instantaneous intensity.
Contracts, exchange, prices: prices on the primary and secondary market, some types are bond contracts, zero coupon securities (ZCB), fixed coupon bonds (CB), accruals, and tel. Risks: time, uncertainty, risk, credit risk for mortgages and bonds,
Evaluation in conditions of certainty, the exponential law: the exponential function as a law of financial equivalence, rates and equivalent intensity in exponential and linear law, evaluation of a financial transaction based on exponential law, equity, functional properties of exponential law, breakdown of financial transactions;
annuities and amortization schedules: definitions, present value of annuities in constant installments, deferred immediate annuity of duration m, deferred perpetual annuity, immediate anticipated annuity of duration m, anticipated perpetual annuity, deferred annuities of n years, deferred (deferred) annuity, perpetual deferred payment in advance. Earnings in appearance, deferred income in constant installments, early repayments at constant installments, deferred annuities with constant capital, the repayment schedule, single repayment amortization;
internal rate of return: the case of periodic payments, the Descartes theorem, the case of a ZCB, the case of an investment transaction, the case of a financial transaction consisting of three amounts, the case of a CB quoted on par, the Newton method, the APR (Gross annual percentage rate);
theory of financial equivalence laws: the value function in a spot contract, the value function in a forward contract, the ownership of uniformity over time, discount and capitalization factors, hypothesis of consistency between spot contracts and forward contracts , the property of divisibility, interest rates and intensity, equivalent rates, the instantaneous intensity of interest, integral form of the discount factor, uniform laws, divisible laws, Cantelli's theorem, yield maturity intensity (yield to maturity) , linear and hyperbolic capitalization, linearity of present value.
Financial transactions in the market: value function and market prices: the characteristic assumptions of the market, the perfect market, the principle of non-arbitrage, the absence of arbitrage, the single price law, zero coupon coupons, decryption theorem with respect to maturity, coupon bonds nothing not unitary, amount independence theorem, ZCB portfolios with different maturity, price linearity theorem, forward contracts, implicit price theorems, implicit rates, considerations on tax effects, case of Treasury Obligation Bonds (BOT);
interest rate maturity structure: spot maturity structures, implicit maturity structures, dominance relationship between implied interest rate structure and spot rate structure, discrete time tables, discrete time schedules with a continuous pattern, parity, risky structures and credit spreads; time index and variability indices: payment flow timescales, maturity and maturity life, duration, the case of constant-rate annuities, second-order moments, duration and dispersion of portfolios, indices of variability of a payment flow , analysis of price sensitivity, semi-elasticity, elasticity, convexity, "thumb rule";
measurement of the interest rate maturity structure: methods based on the internal rate of return, methods based on linear algebra, methods based on the parity rate, swap rate as parity rate, methods based on the estimation of a model, Masera model , Nelson -Siegel-Svensson model;
arbitrage valuation of variable rate plans: random financial transactions, variable rate coupons, effects of perfect indexing of interest shares, reinvestment security, valuation of the stochastic ZCB, logic of the replicating portfolio, valuation of the individual indexed coupon, valuation of the flow of indexed coupons, valuation of the coupon at a variable rate at issue and in place, equivalence with a roll-over strategy;
interest rate sensitive contracts (outlines): the valuation of contracts dependent on nominal interest rates, recalls on the theory of the structure by maturity in certainty conditions, models of the structure by maturity in certainty conditions, examples of IRS contracts, stochastic models for contracts IRS, a class of uni models that varied over time, the basic assumptions, the dynamics of IRS contracts, the hedging argument, risk measures, the Vasicek model.

Castellani, G., De Felice, M., Moriconi, F. Manuale di finanza. Tassi d’interesse. Mutui e obbligazioni. Il Mulino, 2005
Allevi, E., Bosi, G., Riccardi, R., Zuanon, M., Matematica finanziaria e attuariale, Pearson, 2017
Luenberger, D., G., Introduzione alla matematica finanziaria, Apogeo, 2015
Cesari, R., Introduzione alla Finanza Matematica. Mercati azionari, rischi e portafogli, McGraw-Hill, 2012
Castellani, G., De Felice, M., Moriconi, F. Manuale di finanza. Tassi d’interesse. Mutui e obbligazioni. Il Mulino, 2005
Castellani, G., De Felice, M., Moriconi, F. Modelli stocastici e contratti derivati. Il Mulino, 2005
Naccarato, A., Pierini, A., “BEKK element-by-element estimation of a volatility matrix, A portfolio simulation”, in Mathematical and Statistical Methods for Actuarial Sciences and Finance, (editors Perna, C., Sibillo, M.), Springer, 2014
Lecture notes (delivered to class)

Graphs: basic definitions. Simple and non simple graphs, planarity, connectivity, degree, regularity, incidence and adjacency matrices. Examples of families of graphs. The '' handshaking lemma ''.
Graphs obtained from others: complement, subgraph, cancellation and contraction. Isomorphisms and automorphisms of graphs. Connectivity: paths, cycles. A graph is bipartite if and only if each cycle has equal length.
Connetivity and connected component. Connectivity for sides and vertices. Eulerian and semi-Eulerian graphs.
Euler's theorem: a connected graph is Eulerian if and only if every vertex has an even degree. Hamiltonian graphs. Sufficient conditions to guarantee that a graph is Hamiltonian: the theorems of Ore and Dirac.
the Ore theorem. Trees and forests. The cyclomatic number and the "cutset" rank of a graph.
Fundamental system of cycles and cuts associated with a generating forest. Enumeration of generating forests. The Cayley theorem.
Generating trees: the "greedy" algorithm for the "connector problem". Planar graphs. K3.3 and K5 are not planar. Statement of the theorem of Kuratovski and variations.
Euler's formula for planar graphs. The dual of a planar graph.
Correspondence between cycles and cuts for planar graphs and their dual. Dual abstract. A graph that admits a dual abstract is planar. Colorings: initial considerations and some properties.
Colorings: the 5 colors theorem. Graphs on surfaces: classification of topological surfaces.
Coloring of faces and duality between this problem and the coloring of vertices. Reduction of the proof of the 4-color theorem to the coloring of cubic graph faces. '' The marriage problem ": Hall's theorem.
Hall's theorem in the language of transversals. Criteria of existence of transversal and partial transversivities. Application to the construction of Latin squares. Direct graphs: basic notions and orientability.
The Max-Flow Min-Cut theorem and Menger's theorem.
Complexity of algorithms and applications to the theory of graphs.
Introduction to the theory of matroids: definitions using bases and independent elements. Graphical and cographic matroids, vector matroids and the problem of representability.
Definition of matroid using the cycles and the rank function. Minors of a matroid. Transverse matroids and the Rado Theorem for matroids.
Union of matroids and applications: existence of disjoint bases in a matroid. Duality for matroids and applications to graphic and cographic matroids. Planar matroids and the generalization of Kuratovski's theorem for matroids.
Elements of algebraic graph theory: the incidence matrix and the Laplacian matrix of an oriented graph. The vertex space and the space of edges of a graph. Subspaces of the cycles and subspace of the cuts of a defined oriented graph of the incidence matrix.
Basis for the space of the cycles and for the space of the cuts of a graph. The Riemann-Roch theorem for graphs.
Proof of the "generalized Matrix Tree theorem". The contraction / restriction algorithm for matroids. Examples. The number of acyclical orientations of a graph.
Graph polynomials: the chromatic polynomial, the "reliabiliaty" polynomial. Examples. The polynomial rank of a matroid. Properties and first applications.
Proof of the structure theorem for functions on matroids that satisfy contraction/restriction properties. Their incarnation through the polynomial rank.
Whitey's moves and two isomorphisms for graphs. Isomorphism between graphical matroids implies isomorphism between graphs in case the graphs are 3 connected.
Whitney's Theorem for graphic matroids: sketch of the demonstration. Characterization for minors excluded from binary and regular matroids. The theorem of Seymour.

R. Diestel: Graph theory, Spriger GTM 173.
R. Wilson: Introduction to Graph theory, Prentice Hall.
B. Bollobas: Modern Graph theory, Springer GTM 184.
J. A. Bondy, U.S.R. Murty: Graph theory, Springer GTM 244.
N. Biggs: Algebraic graph theory, Cambridge University Press.
C. D. Godsil, G. Royle: Algebraic Graph theory, Springer GTM 207.
J. G. Oxley: Matroid theory. Oxford graduate texts in mathematics, 3.

1) Computability, complexity and representability:

- Introduction to decision problems, algorithmic and non-algorithmic procedures, deterministic computations, discrete procedures, the notion of an alphabet, the notion of a formal language. Decidability and semi-decidability of a set. Deterministic, finitary and discrete computations. Formal algorithms: formal definition of algorithm, configurations of input, output, transition function. Example of formalization of an algorithm. Decidability for finished automata. Representation of the automata by matrices. Free Monoid of words. Formal semi-rings. Non-deterministic finite automata. Regular Languages. Equivalence between deterministic and non-deterministic automata.

- Turing machines: definition, decidability for Turing machine, stopping time, stopping space. Cost of computation. Complexity: worst case and average case. Independence of decision time from a finite number of input configurations. Complexity functions, complexity classes DTIME and DSPACE (deterministic time and space). Inclusion DTIME (T (n)) ⊂ DSPACE (T (n)) ⊂ DTIME (2 ^ {cT (n)}). Pumping Lemma. Simulation of algorithms, simulation of the half tape Turing machine, simulation of a multi-tape machine. Special Turing machines. Linear Speedup theorem for Turing machines with an extended alphabet. Evaluation of acceleration coefficient in relation to alphabets. Decisions of natural number sets. Independence from representation. Considerations concerning complexity.

- Turing computability: definition of Turing computable function, characteristic functions of Turing decidable sets, the class of Turing computable functions is closed by composition, concatenation, primitive recursion and minimization. Examples of Turing computable functions. Recursive Functions: equivalence between Turing computability and recursive functions. Ackermann function ([1] chapter 1,2,3,4,5 and [4] chapter 1).

- Time-constructible functions. The notion of T-clock. Examples of some time constructible function. Closure by composition.

- Non-deterministic Turing machines: characterization through the decidability of projection sets. Definition of the class of polynomial non-deterministic functions. NP-complete problems.

2) Lambda calculus and functional programming:

- Declarative programming: historical outline on the lambda calculus, basic definitions, the terms of the lambda calculus, the simple substitution. Relations on the lambda terms. Congruences, transition to the context. α-equivalence. alpha-equivalence passes to the context. Transitive closure of a relationship, owned by Church-Rosser. Listing of lambda-terms with respect to alpha-equivalence.

- Definition of beta-reduction and beta-equivalence. Church-Rosser's theorem for beta-reduction. Normal forms for beta-reduction. Beta-reduction strategies. Normalizing strategy: left reduction (left most-outer most). Head reduction. Soluble Terms. Head Normal Forms. Solvability characterization theorem.

- Representation of the recursive functions: lambda definability theorem. The existence of the fixed point for any lambda term. Church Fixed Point and Curry fixed point.
- Representation of other data types in the lambda-calculus: pairs, lists, trees, the solution of recursive equations on lambda-terms ([2] chapters 1, 2, 5).

- P. Dehornoy, Calculabilite et Decidabilite, (1993) Springer-Verlag (in francese);
- J.-L. Krivine, Lambda Calculus: Types and Models, (1993) Ellis Horwood editore.
- M. Sipser, An introduction to the theory of computation (2005), Course Technology.

The purpose of this course is to give a brief introduction to the Finite Elements Method (FEM), a gold standard for the numerical solution of PDEs systems. Oddly enough, the widespread use of the FEM is not accompanied by an adequate knowledge of the mathematical framework underlying the method.

This course, starting from the weak formulation of balance equations, will give an overview of the techniques used to reduce a differential problem into an algebraic one. During the course, some selected problems in mechanics and physics will be solved, covering the three main types of equations: elliptic, parabolic and hyperbolic.
The course will cover the following topics: - Applied Linear Algebra. - Boundary Value Problems. - Initial Value Problems Moreover, the students will be introduced to the use COMSOL Multiphysics, a scientific software for numerical simulations based on the Finite Element Method.

Mark S. Gockenbach,
Understanding and Implementing the Finite Elements Method,
SIAM, 2006

Gilbert Strang
Computational Science and Engineering
Wellesley-Cambridge Press, 2007

The course is organised in a combination of lectures and laboratory. The lectures start introducing basic technical topics that are fundamental in order to understand the panorama of architectural approaches. The classical components of a data acquisition system, the issues and opportunities offered by different technical choices are also discussed. A non exhaustive list of the technical topics addressed during the course are: trigger, data acquisition systems, on-line systems, real-time systems, event building, communication networks and protocols, data archiving and mass storage.

Copy of the presentations made during the lectures.

1. Introduction to graph theory: graph, directed graph, tree, rooted tree, connection, strong connection, acyclic graph; graphs isomorphisms, planarity, Kuratowski's theorem, Euler's formula; graph coloring; Eulerian paths, Hamiltonian circuits.

2. Theory of algorithms and optimization: recalls on algorithms and structured programming; computational complexity of an algorithm, complexity classes for problems, classes P, NP, NP-complete, NP-hard; decision-making, research, enumeration and optimization problems; non-linear programming problems, convex programming, linear programming and integer linear programming; combinatorial optimization problems.

Recalls on the elements of combinatorial calculation, algorithms for the generation of the set of parts of a finite set, permutations and combinations of the elements of a set, calculation of the binomial coefficient; the problem of the "Latin squares" and the game of Sudoku; a recursive algorithm for the solution of the game.

3. Optimization problems on graphs and network flows: graph visit, verification of fundamental properties of a graph: connection, strong connection, presence of cycles. Topological sorting of an directed acyclic graph.

The problem of the minimum spanning tree, Kruskal's algorithm, Prim's algorithm, formulation of the problem in terms of Integer Linear Programming.

Search for minimum cost paths on a weighed graph; minimum cost path with single source, Dijkstra's algorithm, Bellman-Ford's algorithm; minimum cost path between all pairs of vertices of the graph, dynamic programming algorithms, Floyd-Warshall's algorithm, transitive closure of a graph.

Network flows and the maximum flow on a network, Theorem of maximum flow and minimum cut, Ford-Fulkerson's algorithm, Edmonds-Karp's algorithm, preflow algorithm, "push-relabel" algorithm.

Problems of partitioning of graphs, trees and paths, problems for the optimal partitioning of trees and paths into p connected components; objective functions, algorithmic techniques for the solution of this class of problems (dynamic programming, linear programming, shifting).

Stable marriage problem, problem definition and matching stability criterion, applications, Gale and Shapley algorithm.

4. Programming laboratory for developing C language programs for the optimizations algorithms of algorithms in GNU/Linux environment and with Wolfram Mathematica software.

Cormen, Leiserson, Rivest, Stein, "Introduction to algorithms", third edition, The MIT Press, 2009

Short introduction to Julia for scientific programming. Introduction to parallel aechitectures. Designi principles of parallel algorithms. Prallel and distributed programming with Julia. Comunication and sincronization primitives: MPI paradigm. Directive-based languages: OpenMP. Performance metrics of parallel programs. Matrix operations and dense linear systems: mentions to BLAS, LAPACK, scaLAPACK. Sparse linear systems. Mentions to CombBLAS and GraphBLAS.

1. [Lecture slides and diary](https://github.com/cvdlab-courses/pdc/blob/master/schedule.md)

2. [Learning Julia](https://www.manning.com/books/julia-in-action)

3. Blaise N. Barney, [HPC Training Materials](https://computing.llnl.gov/tutorials/parallel_comp/), per gentile concessione del Lawrence Livermore National Laboratory's Computational Training Center

4. J. Dongarra, J. Kurzak, J. Demmel, M. Heroux, [Linear Algebra Libraries for High- Performance Computing: Scientific Computing with Multicore and Accelerators](http://www.netlib.org/utk/people/JackDongarra/SLIDES/sc2011-tutorial.pdf), SuperComputing 2011 (SC11)

Linear Systems
Direst methods: Gaussian elimination. Pivoting strategies. Gaussian elimination as a factorization. Doolittle and Choleshy factorizations. Iterative methods: Jacobi, Gauss-Seidel, SOR, Richardson, and related convergence results. Comparison of direct vs iterative solvers. Stability of algorithms for the solution of linear systems.

Iterative Methods for Scalar Nonlinear Equations
The intermediate zero theorem. The algorithms of bisection, Newton, secants, chords, and related convergence results. (Reference: Chapter 1 excluding Section 1.2.3, and Appendices A.1, A.2)

Approximation of Functions
General approximation strategies. Interpolating polynomial in Lagrange and Newton form. Representation of the interpolation error. Convergence of the interpolating polynomial for analytic functions. Refinement strategies in interpolation: Chebyshev nodes, composite approximations. Error estimates. Hermite polynomial, construction and representation of the error. Least Squares approximations. (Reference: Chapter 5 excluding Section 5.2, and Appendix A.4)

Numerical Integration
General principles of numerical integration. Polya's Theorem on the convergence of interpolatory quadrature formulae. Closed and open Newton-Cotes formulae. Stability results and error estimation. Generalized Newton-Cotes formulae and their convergence. Gaussian quadratures and their convergence. (Reference: Chapter 6)

Laboratory Activity
C language coding of some of the major algorithms, and in particular: Gaussian elimination, iterative methods for linear systems and scalar equations, Lagrange/Newton interpolation with a refinement strategy.

N.B.: References are provided with respect to the course notes.

Roberto Ferretti, "Appunti del corso di Analisi Numerica", available at the address: http://www.mat.uniroma3.it/users/ferretti/corso.pdf

Slides of the lessons, available from the course page: http://www.mat.uniroma3.it/users/ferretti/bacheca.html

1. Probability

Introductory example: The branching process.
Introduction to measure theory.
Measure spaces. Events. Uniqueness and extension of measure.
Probability measures. First Borel--Cantelli lemma.
Random variables, distribution function and law. Indipendence.
Second Borel--Cantelli lemma.
0--1 law for independent random variables.

2. Integration, expected value

Introduxction to integration theory. Monotone convergence theorem.
Expectation. Taking the limit under expectation.
Jensen's inequality. L_p norms.
H\"older inequality and Cauchy-Schwarz.
Markov's inequality.
Examples of weak and strong laws of large numbers.
Product measures. Fubini theorem.
Joint laws.

3. Conditional expectation, martingales and limit theorems

Conditional expectation with respect to a sub $\sigma$--algebra. Kolmogorov existence and uniqueness theorem.
Filtrations. Martingale. Gambilng. Stopping times.
Optional stopping.
Some applications to exit times from an interval.
Convergence theorem for martingales in L_1 and in L_2. Kolmogorov's strong law of large numbers.

4. Convergence ind distribution and the central limit theorem

Characteristic functions. Inversion theorem.
Equivalence between convergence in distributiona nd pointwise convergence of characteristic functions.
Various modes of convergence for random variables. Examples.

D. Williams.
Probability with martingales
Cambridge University Press, 1991


R. Durrett
Probability: Theory and Examples
Thomson, 2000

Affine Spaces
Zariski topology. Affine closed subsets and radical ideals. Theorem of the zeros of Hilbert. Correspondence between closed subsets and radical ideals. Noetherian topological spaces. Irreducible closed subsets, irreducible components. Regular functions of affine closed subsets. Regular maps, isomorphisms. Principal open subsets. Examples. Projections are open. Finite morphisms.

Varieties
Projective spaces and their Zariski topology. Quasi-projective varieties. Rational and regular maps. Projective hypersurfaces. Birational equivalence. Principal open subsets and affine closed subsets. Affine varieties. Dimension of quasi-projective varieties. Finite and generically finite morphisms. Characterizations of birational equivalence. Characterization of generically finite morphisms. Costructible sets, Chevalley's theorem. Every variety is birational to a hypersurface.

Local geometry
Local ring of a variety in a point. Zariski cotangent space. Zariski tangent space. Singular and non singular points.

L. Caporaso
Introduction to algebraic geometry
Notes of the course

I. Shafarevich
Basic Algebraic geometry
Springer-Verlag, Berlin, 1994

LAGRANGIAN AND HAMILTONIAN MECHANICS. VARIATIONAL PRINCIPLES. CONSTRAINTS. CYCLICI VARIABLES. CONSTANTS OF MOTION AND SYMMETRIES. HAMILTONIAN FLOWS. LIOUVILLE THEOREM AND POINCARE RECURRENCE THEOREM. CANONICAL TRANSFORMATIONS. GENERATING FUNCTIONS. HAMILTON-JACOBI METHOD AND ACTION-ANGLE VARIABLES. PERTURBATION THEORY. KAM THEOREM.

[1] G. GENTILE,INTRODUZIONE AI SISTEMI DINAMICI. 1.EQUAZIONI DIFFERENZIALI ORDINARIE, ANALISI QUALITATIVA E ALCUNE APPLICAZIONI.
AVAILABLE ON-LINE: http://www.mat.uniroma3.it/users/gentile/2016-2017/FM410/testo.html
[2] G. GENTILE,INTRODUZIONE AI SISTEMI DINAMICI. 2.FORMALISMO LAGRANGIANO E HAMILTONIANO.
AVAILABLE ON-LINE: http://www.mat.uniroma3.it/users/gentile/2016-2017/FM410/testo.html
[3] G. DELL'ANTONIO, ELEMENTI DI MECCANICA. LIGUORI EDITORE, (1996)
[4] V.I. ARNOLD, METODI MATEMATICI DELLA MECCANICA CLASSICA. EDITORI RIUNITI, (1979)
[5] G. GALLAVOTTI, MECCANICA ELEMENTARE. BOLLATI-BORINGHIERI, (1980)

Outline of the course; Introduction and generality; Bioinformatics and algorithms; Computational biology in the clinic and in the pharmaceutical industry; Pharmacokinetics and pharmacodynamics;

Introduction to Systems Biology: what is computational biology; The roles of mathematical modeling and bioinformatics; what is he aiming for; what are the problems; Theoretical tools used in bio-mathematics and bioinformatics.

Introduction to molecular and cellular biology (first part): basic knowledge of genetics, proteomics and cellular processes; Ecology and evolution; the basic molecule; molecular bonds; the chromosomes; DNA and its replication;

Introduction to molecular and cellular biology (second part); genomics; The central dogma of biology; The genome project; the structure of the human genome Analysis of genes; transcription of DNA; the viruses;

Laboratory: generation of random numbers; the functions srand48 and drand48; random generation of arbitrary length nucleotide strings (program1.c); random generation of amino acid strings of arbitrary length (program2.c);

Introduction to information theory; Shannon Entropy; Conditional Entropy; Mutual Information; Indices of biological diversity; Shannon Index; True diversity; Reny index;

Laboratory: the genetic code; C program of transcription DNA sequence and translation into proteins;

Introduction to stochastic processes; basic definition; examples; model of queues; Bernoulli and Poisson process; Markov processes; stochastic processes in bioinformatics and bio-mathematics; the autocorrelation; Outline of the Random Walks and the BLAST algorithm of sequence alignment as a stochastic process and principal algorithm for the consultation of biological sequence databases;

Laboratory: development of an algorithm in C for the calculation of the Shannon Entropy of a text in English (or in Italian) any (e.g., http://www.textfiles.com/etext/)

Random walks. The BLAST algorithm for aligning sequences as a random path; Laboratory: C implementation of different algorithms for the generation of a random walk in 1D and 2D on the lattice and in R or R ^ 2 signal and calculation of the mean square displacement;

Compare sequences: similarity and homology; pairwise alignment; editing distance; scoring matrices PAM and BLOSUM; Needleman-Wunsch's algorithm; local alignment; Smith-Waterman's algorithm; BLAST algorithm;

Laboratory: C implementation of an algorithm for the generation of a signal with noise and calculation of the correlogram in the presence or absence of a true signal;

Multiple Sequence Alignment; consensus sequence; star alignment algorithms; ClustalW; entropy and circular sum scoring functions;

Biological data banks; reasons; data format; taxonomy; Primary DBs; Secondary DBs; NCBI, EMBL, DDBJ; NCBI EBI-Entrez; Exact matching / string searching: general; the agony of Knuth-Morris-Pratt;

Exact matching / string searching: the Boyer-Moore agoritm;

Exercise on an implementation of the Knuth-Morris-Pratt exact matching algorithm. Exercise on biological databases; primary databases; secondary databases; NCBI, EMBL, DDBJ; NCBI EBI-Entrez; Use of the BLAST algorithm

Phylogenetic Analysis; phylogenetic trees; dimension of the research space of phylogenetic algorithms; Methods of construction of phylogenetic trees; Data used for phylogenetic analysis; The Unweighted Pair Method Method with Arithmetic mean (UPGMA) algorithm; the Neighbor Joining Method algorithm; Hidden Markov Models; decoding; the Viterbi Algorithm; Evaluation;

Laboratory: completion of the exercise on mutation, selection and evolution of nucleotide strings (genotype) translated into amino acid strings (phenotype); Selection is made based on the presence of certain substrings in the phenotype that determines the fitness value; Implemented details, display of the convergence criterion and results, discussion, etc .;

Machine Learning; generality'; supervised and unsupervised learning; model selection; undefitting; overfitting; Polynomial curve fitting; machine learning as an estimate of the parameters and the problem of overfitting; subdivision of the training set into testing and testing; concept of bias and variance trade-off; Artificial Neural Networks; definizone; the percussion of Rosenblatt; the percettrone learning algorithm; the multi-layer perceptron;

Laboratory: completion of the implementation in ANSI C of the evolutionary algorithm of nucleotide strings (genotype) translated, through the use of the genetic code, into amino acid strings (phenotype);

Hidden Markov Models; The Forward Algorithm; The Backward Algorithm; Posterior Decoding; Learning; Baum-Welch Algorithm; Use of Hidden Markov Models for the analysis of bio-sequences; gene finding;

Artificial Neural Networks; the error-back propagation algorithm for learning MLP; types of neural networks; convolution networks; reinforcement networks; unsupervised learning and self-organizing maps; Introduction to graph theory; representation, terminology, concepts; paths; cycles; connettivita '; distance; connected components; distance;

Introduction to graph theory; visit breadth-first search; depth-first search; Dijkstra's algorithm; six-degree of separation; small world networks; centrality measures; degree centrality; eigenvector centrality; betweennes centrality; closeness centrality; The network biology; generality'; concepts; types of biological data used to build networks; network biology and network medicine; problems and algorithms used; centrality measures; random networks; scale-free networks; preferential attachment; scale-free network in biology;

Laboratory: completion of the exercise on the evolutionary algorithm; Implemented details, display of the convergence criterion and results, discussion, etc .;

Bio-mathematical models; prediction using theoretical models; the itertative paradigm of mathematical modeling; data-driven models; limited and non-population growth models; analytical derivation and examples; logistics growth; ecological models limited by density; The Lotka-Volterra model; the experiment by Huffaker and Kenneth; the SIR epidemic model and some of its variants; Perelson's model for HAART; the Java Populus application for the solution of continuous models of population dynamics; hints to the numerical resolution methods of differential equation systems;

Discrete models; spin models (Ising models); Cellular automata; Boolean networks; Agent-based models; data fitting and parameter estimation; software tools available; Cellular automata; introduction and history; definition; the 1-dimensional automaton; Wolfram classification; the 2-dimensional automaton; Conway's Game of Life; Software available for CA simulation; dedicated hardware (CA-Machine); the prey-predator model as a two-dimensional cellular automaton; relationship with the system of ordinary derivation equations; stochastic models; Stochastic CAs as discrete stochastic dynamic systems and stochastic processes; example of CA: Belousov-Zabotonsky reactions;

[-] E.S. Allman, J.A. Rhodes. Mathematical Models in Biology: An Introduction (2004) Cambridge University Press.
[-] W.J. Ewens, G.R. Grant. Statistical Methods in Bioinformatics, An Introduction (2005) Springer Verlag.
[-] R. Durbin, S. Eddy, A. Krogh, G. Mitchison. Biological sequence analysis - Probabilistic models of proteins and nucleic acids (1998) Cambridge University Press.

Complex numbers: algebraic and topological properties. Geometric representation of complex numbers: polar coordinates and the complex exponential.
Complex functions with complex variables: continuity and properties, differentiability and first properties. Holomorphic function: properties and examples of holomorphic and non-holomorphic functions. Cauchy-Riemann equations. The real and imaginary parts of holomorphic function are harmonic conjugated.
Equations of Cauchy-Riemann: proof. Examples. Sequences and complex series. Properties. Power series with complex values. Abel's theorem and Hadamard's formula.
Proof of Abel's Theorem. Taylor's formula for series of complex powers. The exponential and the trigonometric functions as analytical functions. Basic properties.
Periodicity of the complex exponential function. The complex logarithm: first considerations. The ring of formal powers series with complex coefficients: basic properties. Analytical functions: definition and first properties.
Series of converging powers are analytical within the convergence region. Composition of analytical functions. Theorem of the inverse function.
Inverse by composition of a formal series and its convergence. Complex powers and properties. The binomial series and properties. Consequences of the inverse theorem: the canonical form of an analytic function.
Local properties of analytical function: open function theorem, invertibility criterion, principle of the maximum local module. The fundamental theorem of algebra. Parameterized curves. A holomorphic function with zero derivative is constant.
The place of the zeros of a non-constant analytical function is discrete. Analytical continuation of functions defined on open connected sets. Principle of the maximum global module. Integrals in paths: definition and first properties. Examples.
A continuous function in a connected open admits a primitive if and only if its integral along a closed curve is zero. Integration of uniformly converging series of functions. Examples. Local primitive of a holomorphic function.
Local primitive of a holomorphic function. The Goursat theorem. Integral of a holomorphic function along a continuous path.
The homotopical form of the Cauchy Theorem. Global primitive of a holomorphic function in a simply connected domain. Applications to the study of the logarithm.
The integral formula of Cauchy. Cauchy formula for development in series and applications: a holomorphic and analytical function; the theorem of Liouville and the fundamental theorem of algebra.
Integral formula for derivatives. The number of windings of a curve with respect to a point. Curves homologous to 0. The global formula of Cauchy.
Demonstration of the global Cauchy formula. Examples.
The first homology group of an open set with values ​​in integers. The Cauchy formula for homological invariance. Examples.
Applications of the Cauchy theorem: uniform limit on holomorphic function compacts is holomorphic. Examples. Laurent series.
Series expansion of a holomorphic function in a circular crown in the Laurent series. Isolated singularities and the field of meromorphic functions. Examples. Statement of the classification theorem of isolated singularities and residual theorem: local and global versions.
Proof of the classification theorem of isolated singularities and proof of the residues theorem. The logarithmic derivative and the principle of the argument. Calculation of residues.
Classification of the connected open of C. The Riemann map theorem and the uniformization theorem (without proof). The Riemann sphere as a compactification of the complex plane. The group of linear transformations of the projective line and the linear transformations produced by them. The group of automorphisms of the complex plane.
The lemma of Schwarz and the group of automorphisms of the unitary disc. Elements of global analytical functions and function. The logarithm as a global analytical function.
The n-th rooty as a global analytical function. The bundle of germs of analytical functions and its properties. The Riemann surface associated with a global analytical function.
Examples and properties of Riemann surface. The Riemann surface associated with an algebraic function and properties. Summary and considerations on the course program.

L. V. Ahlfors: Complex Analysis, McGraw-Hill.
S. Lang: Complex analysis, GTM 103.
E. Freitag, R. Busam: Complex Analysis, Springer.

Brownian motion I

Gaussian multivariate distribution. Processes with stationary and independent increments.
Definition and continuity properties of Brownian motion. Non differentiability of trajectories.
Markov and Strong Markov property.

Brownian motion II

Brownian motion in higher dimensions. Harmonic functions and the Dirichlet problem in regular domains.
The Poisson problem. Iterated logarithm law. Skorohod embedding.
Donsker's invariance principle. Applications: arcsine law and the maximum of random walks.


Stochastic integrals

The Paley-Wiener-Zygmund integral. Stochastic integral with respect to Brownian motion. It\^o formula and applications. Local time and the Tanaka formula.



Stochastic differential equations

The linear case. Existence and uniqueness theorem for stochastic differential equations.
Diffusion process in the limit of zero noise. Infinitesimal generator and partial differential equations.
Feynman-Kac formula and applications.

P. M\"orters and Y. Peres
Brownian Motion
Cambridge University Press
2010


L.C. Evans
An Introduction to Stochastic Differential Equations
AMS bookstore
2013

Under the general title of this Course, we consider both, asymptotic and
numerical methods mostly to handle ordinary differential equations and systems,
and possibly some partial differential equations, and applications. Among the
asymptotic methods, the so-called ``singular perturbations'' are addressed.
These two subjects, which represent basic techniques of Applied Mathematics,
are usually taught separately. In this course we will use both, at the same
time, in the same problems.

Asymptotic Methods:

Introduction to Asymptotic Analysis. Asymptotic series. Regular and singular
perturbations of ordinary differential equations and systems.

Numerical Methods:

Numerical solution, using several algorithms, of ordinary differential
equations and systems. This will be mostly done in the computing laboratory.

Robert E. O'malley,
Singular Perturbation Methods for Ordinary Differential Equations,
Springer-Verlag, Applied Mathematical Sciences, 89, New York, 1991.

John K. Hunter,
Asymptotic Analysis and Singular Perturbation Theory,
Lecture Notes, https://www.math.ucdavis.edu/~hunter/notes/asy.pdf,
2004

1. Probability

Introductory example: The branching process.
Introduction to measure theory.
Measure spaces. Events. Uniqueness and extension of measure.
Probability measures. First Borel--Cantelli lemma.
Random variables, distribution function and law. Indipendence.
Second Borel--Cantelli lemma.
0--1 law for independent random variables.

2. Integration, expected value

Introduxction to integration theory. Monotone convergence theorem.
Expectation. Taking the limit under expectation.
Jensen's inequality. L_p norms.
H\"older inequality and Cauchy-Schwarz.
Markov's inequality.
Examples of weak and strong laws of large numbers.
Product measures. Fubini theorem.
Joint laws.

3. Conditional expectation, martingales and limit theorems

Conditional expectation with respect to a sub $\sigma$--algebra. Kolmogorov existence and uniqueness theorem.
Filtrations. Martingale. Gambilng. Stopping times.
Optional stopping.
Some applications to exit times from an interval.
Convergence theorem for martingales in L_1 and in L_2. Kolmogorov's strong law of large numbers.

4. Convergence ind distribution and the central limit theorem

Characteristic functions. Inversion theorem.
Equivalence between convergence in distributiona nd pointwise convergence of characteristic functions.
Various modes of convergence for random variables. Examples.

D. Williams.
Probability with martingales
Cambridge University Press, 1991


R. Durrett
Probability: Theory and Examples
Thomson, 2000

CONGRUENCES AND POLYNOMIALS. LINEAR DIOPHANTINE EQUATIONS. SYSTEMS OF LINEAR CONGRUENCES. POLYNOMIAL CONGRUENCES. POLYNOMIAL CONGRUENCES MOD p: LAGRANGES’S THEOREM. p-ADIC APPROXIMATION. THE EXISTENCE OF PRIMITIVE ROOTS MOD p. QUADRATIC CONGRUENCES. QUADRATIC RESIDUES. THE LEGENDRE SYMBOL. GAUSS’S LEMMA AND THE LAW OF QUADRATIC RECIPROCITY.THE JACOBI SYMBOL. SUMS OF TWO, THREE, FOUR SQUARES. MULTIPLICATIVE FUNCTIONS. THE MÖBIUS INVERSION FORMULA. SOME NONLINEAR DIOPHANTINE EQUATIONS. CONTINUED FRACTIONS.

M. FONTANA, APPUNTIDEL CORSO TN1 (ARGOMENTI DELLA TEORIA CLASSICA DEI NUMERI), HTTP://WWW.MAT.UNIROMA3.IT/USERS/FONTANA/DIDATTICA/FONTANA_DIDATTICA.HTML DISPENSE.
D.M. BURTON: ELEMENTARY NUMBER THEORY, MCGRAW-HILLINTERNATIONAL EDITION, 6TH EDITION (2007), 434 PP.
G.A. JONES AND J.M. JONES, ELEMENTARY NUMBER THEORY, SPRINGER, 1ST EDITION (1998), 200 PP.
H. DAVENPORT, ARITMETICA SUPERIORE.UN?INTRODUZIONE ALLA TEORIA DEI NUMERI, ZANICHELLI, (1994), 199 PP.
G.H. HARDY AND E.M. WRIGHT, AN INTRODUCTION TO THE THEORY OH NUMBERS, THE CLARENDON PRESS, OXFORD UNIVERSITY, 5TH EDITION (1979), XVI+426 PP.
W.J. LEVEQUE, FUNDAMENTALS OF NUMBER THEORY, DOVER PUBLICATIONS, (1996)
K.H. ROSEN. ELEMENTARY NUMBER THEORY AND ITS APPLICATIONS, ADDISON-WESLEY, 6TH EDITION (2011), XV+752 PP.

Teacher's Lectures notes available on web.

Introduzione alla topologia algebrica. I simplessi e le loro proprieta'. Il complesso delle catene singolari associate ad uno spazio topologico. I gruppi di omologia singolare. Proprieta' elementari: i gruppi di omologia rispettano la decomposizione di uno spazio in componenti connesse per archi; calcolo del gruppo di omologia H_0. L'omologia di un punto. Omologia ridotta: definizione e confronto con l'omologia usuale. Categorie e funtori: definizioni ed esempi. L'omologia come functore dagli spazi topologichi ai gruppi abeliani, passando per la categoria dei complessi. Invarianza omotopica dell'omologia: funzioni continue omotope inducono mappe di complessi omotope; mappe di complessi omotope inducono la stessa mappa in omologia. Il complesso delle catene di una coppia e l'omologia di una coppia: proprieta' funtoriali e invarianza omotopica. La successione esatta lunga in coomologia associata ad una tripla (o ad una coppia). Corollario: l'omologia ridotta come omologia relativa ad un punto.
Il Teorema di escissione. L'omologia relativa ad un ricoprimento.
Coppie buone. Esempi e controesempi. L'omologia di una coppia buona e' isomorfa all'omologia ridotta del quoziente.
Successione di Mayer-Vietoris. Calcolo dell'omologia della sfera e di un disco relativo al bordo, con generatori espliciti. Applicazioni: il teorema del punto fisso di Brower, il teorema dell'invarianza della dimensione (tramite l'omologia locale di R^n).
Grado di applicazioni dalla sfera n-dimensionale in se'. Proprieta' elementari: moltiplicativita' e invarianza omotopica. Esempi: riflessioni, mappa antipodale, mappe senza punti fissi. Applicazione: la pettinabilita' della sfera n-dimensionale. Formula locale per il grado. Il caso di dimensione 1: la mappa grado definisce un isomorfismo tra il gruppo fondamentale della sfera 1-dimensionale e il gruppo degli interi.
La sospensione di spazi topologici. Il grado della sospensione di una mappa da S^n a S^n e' ugale al grado della mappa originaria. Naturalita' delle successioni esatte lunghe. Delta-complessi. Esempi in dimensione uno e due. Equivalenza tra Delta-complessi e Delta-insiemi.
Complessi simpliciali e loro equivalenza con i complessi simpliciali astratti. L'omologia simpliciale di un Delta-complesso. Esempi. L'isomorfismo tra l'omologia simpliciale e l'omologia singolare.
Complessi cellulari (o CW): definizione induttiva. Esempi. I Delta-complessi come complessi cellulari. Ogni complesso cellulare e' omotopo ad un Delta-complesso.
Proprieta' topologiche dei complessi cellulari: normalita', locale contraibilita', paracompattezza (senza dimostrazione), compattezza e locale compattezza. Le coppie cellulari sono coppie buone. Strutture cellulari e simpliciali su varieta' topologiche e differenziabili: una panoramica dei risultati. Esempi di complessi cellulari: la sfera, lo spazio proiettivo reale.
Lemma di compattezza per complessi cellulari. La caratterizzazione dei complessi cellulari tramite le mappe caratteristiche. Non esempi di complessi cellulari che violano la proprieta' della topologia debole o quella della finitezza della chiusura. Esempi di complessi cellulari: lo spazio proiettivo complesso, le superfici topologiche compatte orientate e non orientate.
Somma wedge di spazi topologici puntati e loro gruppi di omologia. Il complesso delle catene cellulari di un complesso cellulare e l'omologia cellulare. L'omologia cellulare coincide con l'omologia singolare. La formula per il differenziale cellulare. Esercizi: omologia cellulare di sfere, spazi proiettivi reali e complessi, superfici topologiche compatte orientate e non, i tori n-dimensionali.
Omologia con coefficienti. Digressione sull'algebra omologica dei gruppi abeliani: prodotto tensoriale e bifuntori Tor.
Il teorema dei coefficienti universali per l'omologia. Spazi di Moore. Lo spezzamento della successione esatta nel teorema dei coefficienti universali non e' naturale: un controesempio.
Una coppia cellulare soddisfa la proprieta' dell'estensione omotopica. Se una coppia soddisfa la proprieta' dell'estensione omotopica e il sottospazio e' contraibile, allora il quoziente per il sottospazio e' un'equivalenza omotopica. Il primo gruppo di omologia e' l'abelianizzato del gruppo fondamentale (inizio della dimostrazione).
Il primo gruppo di omologia e' l'abelianizzato del gruppo fondamentale (fine della dimostrazione). La caratteristica di Eulero. Formula della caratteristica di Eulero per un complesso cellulare finito. Esercizi.
Coomologia con coefficienti. Digressione sull'algebra omologica dei gruppi abeliani: il bifuntore Hom e il bifuntore Ext. Il teorema dei coefficienti universali per la coomologia.
Proprieta' della Coomologia: funtorialita', invarianza omotopica, successione esatta lunga di una tripla e relazione tra i morfismi di connessione in omologia e coomologia, coomologia ridotta, escissione, coomologia di una coppia buona, successione di Mayer-Vietoris.
Il prodotto cup e l'anello di coomologia.
+La coomologia simpliciale di un Delta-complesso. La coomologia cellulare di un complesso cellulare. Esempio: l'anello di coomologia delle superfici topologiche compatte orientate e non.
L'approccio assiomatico alla (co)omologia assoluta di una coppia e relativa (senza dimostrazione). Il prodotto cross in omologia e coomologia singolare e cellulare. Relazione tra il prodotto cup e il prodotto cross in coomologia. La formula di Kunneth in omologia e coomologia (senza dimostrazione). Esempio: l'anello di coomologia di un toro n-dimensionale e' l'algebra esterna di dimensione n.
Varieta' (topologiche). Orientabilita' di varieta'. Il rivestimento doppio delle orientazioni locali. R-orientabilita'. Il rivestimento delle R-classi locali. L'omologia n-esima di una n-varieta' compatta e connessa, R-orientabile e non. La R-classe fondamentale di una varieta' compatta R-orientabile.
Il prodotto cap e la sua relazione col prodotto cup. La formulazione della dualita' di Poincare' per varieta' compatte R-orientate: versione I (col prodotto cap) e versione II (col prodotto cup).
Coomologia a supporto compatto. Digressione sui limiti diretti di gruppi. L'operatore di dualita' per varieta' topologiche R-orientate non compatte. Il teorema di dualita' di Poincare' per varieta' topologiche R-orientate compatte e non: dimostrazione. Esempio: l'anello di coomologia degli spazi proeittivi reali, complessi e quaternionici.

A. Hatcher: Algebraic Topology. Cambridge University Press, 2002.

1. Probability

Introductory example: The branching process.
Introduction to measure theory.
Measure spaces. Events. Uniqueness and extension of measure.
Probability measures. First Borel--Cantelli lemma.
Random variables, distribution function and law. Indipendence.
Second Borel--Cantelli lemma.
0--1 law for independent random variables.

2. Integration, expected value

Introduxction to integration theory. Monotone convergence theorem.
Expectation. Taking the limit under expectation.
Jensen's inequality. L_p norms.
H\"older inequality and Cauchy-Schwarz.
Markov's inequality.
Examples of weak and strong laws of large numbers.
Product measures. Fubini theorem.
Joint laws.

3. Conditional expectation, martingales and limit theorems

Conditional expectation with respect to a sub $\sigma$--algebra. Kolmogorov existence and uniqueness theorem.
Filtrations. Martingale. Gambilng. Stopping times.
Optional stopping.
Some applications to exit times from an interval.
Convergence theorem for martingales in L_1 and in L_2. Kolmogorov's strong law of large numbers.

4. Convergence ind distribution and the central limit theorem

Characteristic functions. Inversion theorem.
Equivalence between convergence in distributiona nd pointwise convergence of characteristic functions.
Various modes of convergence for random variables. Examples.

D. Williams.
Probability with martingales
Cambridge University Press, 1991


R. Durrett
Probability: Theory and Examples
Thomson, 2000

Introduction to cryptography. Classic ciphers. Introduction to cryptanalysis.
Introduction to public-key cryptography.
The RSA cryptosystem.
Primality tests.
Factorization algorithms.
Some attacks on the RSA.
The discrete logarithm problem. Diffie-Hellman key exchange.
Elgamal cryptosystem.
Massey-Omura cryptosystem.
Digital signatures.
Overview of some cryptographic protocols.

Baldoni, Ciliberto, Piacentini: Aritmetica, crittografia e codici
D. Stinson: Cryptography - theory and practice

[1] Maurizio Gabbrielli, Simone Martini, Programming Languages: Principles and Paradigms, Springer (2011) ISBN 9781848829138
[2] Dean Wampler, Alex Payne, Programming Scala: Scalability = Functional Programming + Objects, O'Reilly Media; 2 edizione (2014)
[3] David Parsons, Foundational Java Key Elements and Practical Programming, Springer-Verlag, (2012)

Presentation of the problems that can be treated through integrals on large number of dimensions

Basics

Probability and Random variables

Measurement, uncertainty and its propagation

Curve-fitting, least-squares, optimization

Classical numerical integration, speed of convergence

Integration MC (Mean, variance)

Sampling Strategies

Applications

Propagation of uncertainties

Generation according to a distribution

Real World Applications

Cosmic Rays Shower

System Availabilty

Further applications

Stefan Weinzierl - Introduction to Monte Carlo methods - arXiv:hep-ph/0006269

Further Bibliography will be presented in class, according to the needs of the Participants.

Principles of Object Oriented Design
Abstraction, Polimorphism, Inheritance, Aggregation
Object Oriented Programming Models and UML
UML Use Case, Sequence, Class e Object, Deployment diagrams
Software Analysis and Developmenmt for Java Virtual Machine: I/O, Stream, Networking, Exception Handling
(Scientific, Real-time,...) Efficient Distributed Computing, Multithreading and Concurrency in Cloud e Mobile

Manuale di Java 9 De Sio Cesari Claudio Hoepli Informatica

Brownian motion I

Gaussian multivariate distribution. Processes with stationary and independent increments.
Definition and continuity properties of Brownian motion. Non differentiability of trajectories.
Markov and Strong Markov property.

Brownian motion II

Brownian motion in higher dimensions. Harmonic functions and the Dirichlet problem in regular domains.
The Poisson problem. Iterated logarithm law. Skorohod embedding.
Donsker's invariance principle. Applications: arcsine law and the maximum of random walks.


Stochastic integrals

The Paley-Wiener-Zygmund integral. Stochastic integral with respect to Brownian motion. It\^o formula and applications. Local time and the Tanaka formula.



Stochastic differential equations

The linear case. Existence and uniqueness theorem for stochastic differential equations.
Diffusion process in the limit of zero noise. Infinitesimal generator and partial differential equations.
Feynman-Kac formula and applications.

P. M\"orters and Y. Peres
Brownian Motion
Cambridge University Press
2010


L.C. Evans
An Introduction to Stochastic Differential Equations
AMS bookstore
2013

Under the general title of this Course, we consider both, asymptotic and
numerical methods mostly to handle ordinary differential equations and systems,
and possibly some partial differential equations, and applications. Among the
asymptotic methods, the so-called ``singular perturbations'' are addressed.
These two subjects, which represent basic techniques of Applied Mathematics,
are usually taught separately. In this course we will use both, at the same
time, in the same problems.

Asymptotic Methods:

Introduction to Asymptotic Analysis. Asymptotic series. Regular and singular
perturbations of ordinary differential equations and systems.

Numerical Methods:

Numerical solution, using several algorithms, of ordinary differential
equations and systems. This will be mostly done in the computing laboratory.

Robert E. O'malley,
Singular Perturbation Methods for Ordinary Differential Equations,
Springer-Verlag, Applied Mathematical Sciences, 89, New York, 1991.

John K. Hunter,
Asymptotic Analysis and Singular Perturbation Theory,
Lecture Notes, https://www.math.ucdavis.edu/~hunter/notes/asy.pdf,
2004

A) STRUCTURAL RULES EXPRESSED AS LOGICAL RULES: SEQUENT CALCULUS AND DERIVABILITY IN LINEAR LOGIC
B) POSITIVE AND NEGATIVE: FOCUSED SEQUENT CALCULUS FOR LINEAR LOGIC, LUDICS
C) IMPLICIT COMPLEXITY AND LINEAR LOGIC
D) GEOMETRY OF PROOFS: PROOF NETS IN LINEAR LOGIC
E) INVARIANTS AND DEVELOPMENT OF INTERACTION OF PROOFS: COHERENT SPACES, GEOMETRY OF INTERACTION

V. MICHELE ABRUSCI, LEZIONI DI LOGICA LINEARE, 2018 (FORTH COMING)

• Euclidean Geometry: triangles centers , circle inversion.
• Ordered Geometry and Sylvester problem.
• Projective Geometry: axioms, Desargues, collineations and
correlations
• Platonic Solids Euler formula. Politops, 4-dimensional space politops.
• Topology of surfaces and graphs 4 colors problem and 6 colors theorem, Heawood theorem.
• Delaunay triangulations.
• Newton polygons .
• introduction to tropical geometry.

1) H.S.M Coxeter Introduction to geometry, Wiley 1970; + notes.
morevover:
2) D. Hilbert, S. Cohn Vossen, Geometria intuitiva, cap. 3 , 1932, ed. varie;
3) M. Aigner, Martin, G. Ziegler, Proofs from THE BOOK, Springer, 1998;
4) J. Stillwell The Four Pillars of Geometry, Springer 2005;
5) H.D. Ebbinghaus, H. Hermes, F. Hirzebruch, et al. Numbers, GTM 123,
Springer 1990;
6) Stefano Rebay, Tecniche di Generazione di Griglia per il Calcolo
Scientico-Triangolazione di Delaunay, slides Univ. Studi di Brescia;
7) G. Fisher Plane algebraic curves, AMS Students Mathematical Library V.
15, AMS 2001.

Introduction: Cardano equations for the solubility of third-degree equations, rings and fields, the characteristics of a field, known facts about rings of polynomials, field extensions, construction of some field extensions, the sub-ring generated by a subset, the subfield generated by a subset, algebraic and transcendent elements, algebraically closed fields.

Splitting fields: Simple extensions and maps between simple extensions, splitting fields, existence of the splitting field, uniqueness up to isomorphisms of the splitting field, multiple roots, formal derivatives, separable polynomials and perfect fields, minimal polynomials and their characterizations.

The fundamental Theorem of Galois Theory: Group of the automorphisms of a field, normal, separable and Galois extensions, characterizations of separable extensions, Fundamental Theorem of Galois Correspondence, examples, Galois group of a polynomial, Radical extensions, solvable groups and Galois's Theorem on solving equations , Theorem of the existence of primitive elements.

The computation of Galois group: Galois groups as subgroups of $ S_n $, transitive subgroups of $ S_n $, characterization of irreducibility in terms of transitivity, polynomials with Galois groups in $ A_n $, Theory of discriminants, Galois groups of polynomials with degree up to $4$, examples of polynomials with Galois group $S_p$.

Cyclotomic fields: Definitions, Galois group, maximal real subfields, quadratic subfields, Galois groups, cyclotomic polynomials and their properties, Theorem of the inverse Galois theory for abelian groups.

Finite fields: Existence and uniqueness of finite fields, Galois group of a finite field, subfields of a finite field, enumeration of irreducible polynomials on finite fields. Construction of the algebraic closure of a finite field with $ p $ elements.

Constructions with ruler and compass: Definition of constructible points of the plane, constructible real numbers, characterization of constructible points in terms of fields, constructible subfields and construction of constructible numbers, cube duplication, trisection of angles, quadrature of the circle and Gauss's theorem for the construction of regular polygons with ruler and compass.

J. S. Milne.Fields and Galois Theory. Course Notes v4.22 (March 30, 2011).
S. Gabelli. Teoria delle Equazioni e Teoria di Galois. Springer UNITEXT (La Matematica per il 3+2) 2008, XVII, 410 pagg., ISBN: 978-88-470-0618-8
E. Artin.Galois Theory. NOTRE DAME MATHEMATICAL LECTURES Number 2. 1942.
C. Procesi.Elementi di Teoria di Galois. Decibel, Zanichelli, (Seconda ristampa, 1991).

1. Abstract integration theory
Riemann integration theory. The concept of measurability. Step functions. Elementary properties of measures. Arithmetic in [0,∞]. Integration of positive functions. Integration of complex functions. Importance of sets
with null measure.
2. Positive Borel measures
Vector spaces. Topological preliminaries. Riesz representation theorem. Regularity properties of Borel measures. Lebesgue measure. Continuity properties of measurable functions.
3. L^p spaces
Inequalities and convex functions. L^p spaces. Approximation through continuous functions.
4. Basic theory of Hilbert spaces
Inner products and linear functionals. Dual space of L^2
4. Integration on product spaces
Measurability on cartesian products. Product measure. Fubini theorem.
4. Complex measures
Total variation. Absolute continuity. Radon-Nykodym theorem. Bounded linear functionals on L^p. The Riesz representation theorem.

"Analisi reale e complessa”, W. Rudin. Bollati Boringhieri.

1. Preliminaries
Definition of hyper-surface. Integration on hyper-surfaces. The divergence theorem.

2. The Laplace equation
Mean value inequalities. Minimum and maximum principle. The Harnack inequality. The Green representation. The Poisson integral. Convergence theorems. Interior estimates on the derivatives. The Dirichlet problem:
the method of sub-harmonic functions.

3. The classical maximum principle
The weak maximum principle. The strong maximum principle. The Hopf lemma.

4. The Poisson equation and the Newtonian potential
Hölder-continuity. The Dirichlet problem for the Poisson equation. Hölder estimates for second derivatives. Boundary estimates. Hölder estimates for first derivatives.

5. Banach and Hilbert spaces
The contraction principle. The continuity method.

6. Classical solutions: the Schauder approach
Schauder interior estimates. Boundary and global estimates. The Dirichlet problem. Boundary and interior regularity.

“Elliptic partial differential equations of second order. Reprint of the 1998 edition”, D. Gilbarg e N.S. Trudinger. Classics in Mathematics. Springer-Verlag, Berlin, 2001.

Affine Spaces
Zariski topology. Affine closed subsets and radical ideals. Theorem of the zeros of Hilbert. Correspondence between closed subsets and radical ideals. Noetherian topological spaces. Irreducible closed subsets, irreducible components. Regular functions of affine closed subsets. Regular maps, isomorphisms. Principal open subsets. Examples. Projections are open. Finite morphisms.

Varieties
Projective spaces and their Zariski topology. Quasi-projective varieties. Rational and regular maps. Projective hypersurfaces. Birational equivalence. Principal open subsets and affine closed subsets. Affine varieties. Dimension of quasi-projective varieties. Finite and generically finite morphisms. Characterizations of birational equivalence. Characterization of generically finite morphisms. Costructible sets, Chevalley's theorem. Every variety is birational to a hypersurface.

Local geometry
Local ring of a variety in a point. Zariski cotangent space. Zariski tangent space. Singular and non singular points.

L. Caporaso
Introduction to algebraic geometry
Notes of the course

I. Shafarevich
Basic Algebraic geometry
Springer-Verlag, Berlin, 1994

Graphs: basic definitions. Simple and non simple graphs, planarity, connectivity, degree, regularity, incidence and adjacency matrices. Examples of families of graphs. The '' handshaking lemma ''.
Graphs obtained from others: complement, subgraph, cancellation and contraction. Isomorphisms and automorphisms of graphs. Connectivity: paths, cycles. A graph is bipartite if and only if each cycle has equal length.
Connetivity and connected component. Connectivity for sides and vertices. Eulerian and semi-Eulerian graphs.
Euler's theorem: a connected graph is Eulerian if and only if every vertex has an even degree. Hamiltonian graphs. Sufficient conditions to guarantee that a graph is Hamiltonian: the theorems of Ore and Dirac.
the Ore theorem. Trees and forests. The cyclomatic number and the "cutset" rank of a graph.
Fundamental system of cycles and cuts associated with a generating forest. Enumeration of generating forests. The Cayley theorem.
Generating trees: the "greedy" algorithm for the "connector problem". Planar graphs. K3.3 and K5 are not planar. Statement of the theorem of Kuratovski and variations.
Euler's formula for planar graphs. The dual of a planar graph.
Correspondence between cycles and cuts for planar graphs and their dual. Dual abstract. A graph that admits a dual abstract is planar. Colorings: initial considerations and some properties.
Colorings: the 5 colors theorem. Graphs on surfaces: classification of topological surfaces.
Coloring of faces and duality between this problem and the coloring of vertices. Reduction of the proof of the 4-color theorem to the coloring of cubic graph faces. '' The marriage problem ": Hall's theorem.
Hall's theorem in the language of transversals. Criteria of existence of transversal and partial transversivities. Application to the construction of Latin squares. Direct graphs: basic notions and orientability.
The Max-Flow Min-Cut theorem and Menger's theorem.
Complexity of algorithms and applications to the theory of graphs.
Introduction to the theory of matroids: definitions using bases and independent elements. Graphical and cographic matroids, vector matroids and the problem of representability.
Definition of matroid using the cycles and the rank function. Minors of a matroid. Transverse matroids and the Rado Theorem for matroids.
Union of matroids and applications: existence of disjoint bases in a matroid. Duality for matroids and applications to graphic and cographic matroids. Planar matroids and the generalization of Kuratovski's theorem for matroids.
Elements of algebraic graph theory: the incidence matrix and the Laplacian matrix of an oriented graph. The vertex space and the space of edges of a graph. Subspaces of the cycles and subspace of the cuts of a defined oriented graph of the incidence matrix.
Basis for the space of the cycles and for the space of the cuts of a graph. The Riemann-Roch theorem for graphs.
Proof of the "generalized Matrix Tree theorem". The contraction / restriction algorithm for matroids. Examples. The number of acyclical orientations of a graph.
Graph polynomials: the chromatic polynomial, the "reliabiliaty" polynomial. Examples. The polynomial rank of a matroid. Properties and first applications.
Proof of the structure theorem for functions on matroids that satisfy contraction/restriction properties. Their incarnation through the polynomial rank.
Whitey's moves and two isomorphisms for graphs. Isomorphism between graphical matroids implies isomorphism between graphs in case the graphs are 3 connected.
Whitney's Theorem for graphic matroids: sketch of the demonstration. Characterization for minors excluded from binary and regular matroids. The theorem of Seymour.

R. Diestel: Graph theory, Spriger GTM 173.
R. Wilson: Introduction to Graph theory, Prentice Hall.
B. Bollobas: Modern Graph theory, Springer GTM 184.
J. A. Bondy, U.S.R. Murty: Graph theory, Springer GTM 244.
N. Biggs: Algebraic graph theory, Cambridge University Press.
C. D. Godsil, G. Royle: Algebraic Graph theory, Springer GTM 207.
J. G. Oxley: Matroid theory. Oxford graduate texts in mathematics, 3.

1) Computability, complexity and representability:

- Introduction to decision problems, algorithmic and non-algorithmic procedures, deterministic computations, discrete procedures, the notion of an alphabet, the notion of a formal language. Decidability and semi-decidability of a set. Deterministic, finitary and discrete computations. Formal algorithms: formal definition of algorithm, configurations of input, output, transition function. Example of formalization of an algorithm. Decidability for finished automata. Representation of the automata by matrices. Free Monoid of words. Formal semi-rings. Non-deterministic finite automata. Regular Languages. Equivalence between deterministic and non-deterministic automata.

- Turing machines: definition, decidability for Turing machine, stopping time, stopping space. Cost of computation. Complexity: worst case and average case. Independence of decision time from a finite number of input configurations. Complexity functions, complexity classes DTIME and DSPACE (deterministic time and space). Inclusion DTIME (T (n)) ⊂ DSPACE (T (n)) ⊂ DTIME (2 ^ {cT (n)}). Pumping Lemma. Simulation of algorithms, simulation of the half tape Turing machine, simulation of a multi-tape machine. Special Turing machines. Linear Speedup theorem for Turing machines with an extended alphabet. Evaluation of acceleration coefficient in relation to alphabets. Decisions of natural number sets. Independence from representation. Considerations concerning complexity.

- Turing computability: definition of Turing computable function, characteristic functions of Turing decidable sets, the class of Turing computable functions is closed by composition, concatenation, primitive recursion and minimization. Examples of Turing computable functions. Recursive Functions: equivalence between Turing computability and recursive functions. Ackermann function ([1] chapter 1,2,3,4,5 and [4] chapter 1).

- Time-constructible functions. The notion of T-clock. Examples of some time constructible function. Closure by composition.

- Non-deterministic Turing machines: characterization through the decidability of projection sets. Definition of the class of polynomial non-deterministic functions. NP-complete problems.

2) Lambda calculus and functional programming:

- Declarative programming: historical outline on the lambda calculus, basic definitions, the terms of the lambda calculus, the simple substitution. Relations on the lambda terms. Congruences, transition to the context. α-equivalence. alpha-equivalence passes to the context. Transitive closure of a relationship, owned by Church-Rosser. Listing of lambda-terms with respect to alpha-equivalence.

- Definition of beta-reduction and beta-equivalence. Church-Rosser's theorem for beta-reduction. Normal forms for beta-reduction. Beta-reduction strategies. Normalizing strategy: left reduction (left most-outer most). Head reduction. Soluble Terms. Head Normal Forms. Solvability characterization theorem.

- Representation of the recursive functions: lambda definability theorem. The existence of the fixed point for any lambda term. Church Fixed Point and Curry fixed point.
- Representation of other data types in the lambda-calculus: pairs, lists, trees, the solution of recursive equations on lambda-terms ([2] chapters 1, 2, 5).

- P. Dehornoy, Calculabilite et Decidabilite, (1993) Springer-Verlag (in francese);
- J.-L. Krivine, Lambda Calculus: Types and Models, (1993) Ellis Horwood editore.
- M. Sipser, An introduction to the theory of computation (2005), Course Technology.

Linear Systems
Direst methods: Gaussian elimination. Pivoting strategies. Gaussian elimination as a factorization. Doolittle and Choleshy factorizations. Iterative methods: Jacobi, Gauss-Seidel, SOR, Richardson, and related convergence results. Comparison of direct vs iterative solvers. Stability of algorithms for the solution of linear systems.

Iterative Methods for Scalar Nonlinear Equations
The intermediate zero theorem. The algorithms of bisection, Newton, secants, chords, and related convergence results. (Reference: Chapter 1 excluding Section 1.2.3, and Appendices A.1, A.2)

Approximation of Functions
General approximation strategies. Interpolating polynomial in Lagrange and Newton form. Representation of the interpolation error. Convergence of the interpolating polynomial for analytic functions. Refinement strategies in interpolation: Chebyshev nodes, composite approximations. Error estimates. Hermite polynomial, construction and representation of the error. Least Squares approximations. (Reference: Chapter 5 excluding Section 5.2, and Appendix A.4)

Numerical Integration
General principles of numerical integration. Polya's Theorem on the convergence of interpolatory quadrature formulae. Closed and open Newton-Cotes formulae. Stability results and error estimation. Generalized Newton-Cotes formulae and their convergence. Gaussian quadratures and their convergence. (Reference: Chapter 6)

Laboratory Activity
C language coding of some of the major algorithms, and in particular: Gaussian elimination, iterative methods for linear systems and scalar equations, Lagrange/Newton interpolation with a refinement strategy.

N.B.: References are provided with respect to the course notes.

Roberto Ferretti, "Appunti del corso di Analisi Numerica", available at the address: http://www.mat.uniroma3.it/users/ferretti/corso.pdf

Slides of the lessons, available from the course page: http://www.mat.uniroma3.it/users/ferretti/bacheca.html

1. Probability

Introductory example: The branching process.
Introduction to measure theory.
Measure spaces. Events. Uniqueness and extension of measure.
Probability measures. First Borel--Cantelli lemma.
Random variables, distribution function and law. Indipendence.
Second Borel--Cantelli lemma.
0--1 law for independent random variables.

2. Integration, expected value

Introduxction to integration theory. Monotone convergence theorem.
Expectation. Taking the limit under expectation.
Jensen's inequality. L_p norms.
H\"older inequality and Cauchy-Schwarz.
Markov's inequality.
Examples of weak and strong laws of large numbers.
Product measures. Fubini theorem.
Joint laws.

3. Conditional expectation, martingales and limit theorems

Conditional expectation with respect to a sub $\sigma$--algebra. Kolmogorov existence and uniqueness theorem.
Filtrations. Martingale. Gambilng. Stopping times.
Optional stopping.
Some applications to exit times from an interval.
Convergence theorem for martingales in L_1 and in L_2. Kolmogorov's strong law of large numbers.

4. Convergence ind distribution and the central limit theorem

Characteristic functions. Inversion theorem.
Equivalence between convergence in distributiona nd pointwise convergence of characteristic functions.
Various modes of convergence for random variables. Examples.

D. Williams.
Probability with martingales
Cambridge University Press, 1991


R. Durrett
Probability: Theory and Examples
Thomson, 2000

LAGRANGIAN AND HAMILTONIAN MECHANICS. VARIATIONAL PRINCIPLES. CONSTRAINTS. CYCLICI VARIABLES. CONSTANTS OF MOTION AND SYMMETRIES. HAMILTONIAN FLOWS. LIOUVILLE THEOREM AND POINCARE RECURRENCE THEOREM. CANONICAL TRANSFORMATIONS. GENERATING FUNCTIONS. HAMILTON-JACOBI METHOD AND ACTION-ANGLE VARIABLES. PERTURBATION THEORY. KAM THEOREM.

[1] G. GENTILE,INTRODUZIONE AI SISTEMI DINAMICI. 1.EQUAZIONI DIFFERENZIALI ORDINARIE, ANALISI QUALITATIVA E ALCUNE APPLICAZIONI.
AVAILABLE ON-LINE: http://www.mat.uniroma3.it/users/gentile/2016-2017/FM410/testo.html
[2] G. GENTILE,INTRODUZIONE AI SISTEMI DINAMICI. 2.FORMALISMO LAGRANGIANO E HAMILTONIANO.
AVAILABLE ON-LINE: http://www.mat.uniroma3.it/users/gentile/2016-2017/FM410/testo.html
[3] G. DELL'ANTONIO, ELEMENTI DI MECCANICA. LIGUORI EDITORE, (1996)
[4] V.I. ARNOLD, METODI MATEMATICI DELLA MECCANICA CLASSICA. EDITORI RIUNITI, (1979)
[5] G. GALLAVOTTI, MECCANICA ELEMENTARE. BOLLATI-BORINGHIERI, (1980)

The purpose of this course is to give a brief introduction to the Finite Elements Method (FEM), a gold standard for the numerical solution of PDEs systems. Oddly enough, the widespread use of the FEM is not accompanied by an adequate knowledge of the mathematical framework underlying the method.

This course, starting from the weak formulation of balance equations, will give an overview of the techniques used to reduce a differential problem into an algebraic one. During the course, some selected problems in mechanics and physics will be solved, covering the three main types of equations: elliptic, parabolic and hyperbolic.
The course will cover the following topics: - Applied Linear Algebra. - Boundary Value Problems. - Initial Value Problems Moreover, the students will be introduced to the use COMSOL Multiphysics, a scientific software for numerical simulations based on the Finite Element Method.

Mark S. Gockenbach,
Understanding and Implementing the Finite Elements Method,
SIAM, 2006

Gilbert Strang
Computational Science and Engineering
Wellesley-Cambridge Press, 2007

QUANTUM MECHANICS: THE CRISIS OF CLASSICAL PHYSICS. WAVES AND PARTICLES. STATE VECTORS AND OPERATORS. MEASUREMENTS AND OBSERVABLES. THE POSITION OPERATOR. TRANSLATIONS AND MOMENTUM. TIME EVOLUTION AND THE SCHRODINGER EQUATION. PARITY. ONE-DIMENSIONAL PROBLEMS. HARMONIC OSCILLATOR. SYMMETRIES AND CONSERVATION LAWS. TIME INDEPENDENT PERTURBATION THEORY. TIME DEPENDENT PERTURBATION THEORY.

J.J. SAKURAI, J. NAPOLITANO. MECCANICA QUANTISTICA MODERNA. SECONDA EDIZIONE, ZANICHELLI, BOLOGNA, 2014

An english version of the book is also available:
J.J. SAKURAI, J. NAPOLITANO. MODERN QUANTUM MECHANICS. SECOND EDITION, PEARSON EDUCATION LIMITED 2014

Time structure of the exchange of amounts, capital and interest: exchange of amounts, time, price, price of time, convention for measuring time, deferred contracts and rights, transactions with a fixed schedule, regular investment / indebtedness transactions, laws financial, the law of simple interests, the law of compound interests, fundamental definitions based on the value function, factors rates and intensity, instantaneous intensity.
Contracts, exchange, prices: prices on the primary and secondary market, some types are bond contracts, zero coupon securities (ZCB), fixed coupon bonds (CB), accruals, and tel. Risks: time, uncertainty, risk, credit risk for mortgages and bonds,
Evaluation in conditions of certainty, the exponential law: the exponential function as a law of financial equivalence, rates and equivalent intensity in exponential and linear law, evaluation of a financial transaction based on exponential law, equity, functional properties of exponential law, breakdown of financial transactions;
annuities and amortization schedules: definitions, present value of annuities in constant installments, deferred immediate annuity of duration m, deferred perpetual annuity, immediate anticipated annuity of duration m, anticipated perpetual annuity, deferred annuities of n years, deferred (deferred) annuity, perpetual deferred payment in advance. Earnings in appearance, deferred income in constant installments, early repayments at constant installments, deferred annuities with constant capital, the repayment schedule, single repayment amortization;
internal rate of return: the case of periodic payments, the Descartes theorem, the case of a ZCB, the case of an investment transaction, the case of a financial transaction consisting of three amounts, the case of a CB quoted on par, the Newton method, the APR (Gross annual percentage rate);
theory of financial equivalence laws: the value function in a spot contract, the value function in a forward contract, the ownership of uniformity over time, discount and capitalization factors, hypothesis of consistency between spot contracts and forward contracts , the property of divisibility, interest rates and intensity, equivalent rates, the instantaneous intensity of interest, integral form of the discount factor, uniform laws, divisible laws, Cantelli's theorem, yield maturity intensity (yield to maturity) , linear and hyperbolic capitalization, linearity of present value.
Financial transactions in the market: value function and market prices: the characteristic assumptions of the market, the perfect market, the principle of non-arbitrage, the absence of arbitrage, the single price law, zero coupon coupons, decryption theorem with respect to maturity, coupon bonds nothing not unitary, amount independence theorem, ZCB portfolios with different maturity, price linearity theorem, forward contracts, implicit price theorems, implicit rates, considerations on tax effects, case of Treasury Obligation Bonds (BOT);
interest rate maturity structure: spot maturity structures, implicit maturity structures, dominance relationship between implied interest rate structure and spot rate structure, discrete time tables, discrete time schedules with a continuous pattern, parity, risky structures and credit spreads; time index and variability indices: payment flow timescales, maturity and maturity life, duration, the case of constant-rate annuities, second-order moments, duration and dispersion of portfolios, indices of variability of a payment flow , analysis of price sensitivity, semi-elasticity, elasticity, convexity, "thumb rule";
measurement of the interest rate maturity structure: methods based on the internal rate of return, methods based on linear algebra, methods based on the parity rate, swap rate as parity rate, methods based on the estimation of a model, Masera model , Nelson -Siegel-Svensson model;
arbitrage valuation of variable rate plans: random financial transactions, variable rate coupons, effects of perfect indexing of interest shares, reinvestment security, valuation of the stochastic ZCB, logic of the replicating portfolio, valuation of the individual indexed coupon, valuation of the flow of indexed coupons, valuation of the coupon at a variable rate at issue and in place, equivalence with a roll-over strategy;
interest rate sensitive contracts (outlines): the valuation of contracts dependent on nominal interest rates, recalls on the theory of the structure by maturity in certainty conditions, models of the structure by maturity in certainty conditions, examples of IRS contracts, stochastic models for contracts IRS, a class of uni models that varied over time, the basic assumptions, the dynamics of IRS contracts, the hedging argument, risk measures, the Vasicek model.

Castellani, G., De Felice, M., Moriconi, F. Manuale di finanza. Tassi d’interesse. Mutui e obbligazioni. Il Mulino, 2005
Allevi, E., Bosi, G., Riccardi, R., Zuanon, M., Matematica finanziaria e attuariale, Pearson, 2017
Luenberger, D., G., Introduzione alla matematica finanziaria, Apogeo, 2015
Cesari, R., Introduzione alla Finanza Matematica. Mercati azionari, rischi e portafogli, McGraw-Hill, 2012
Castellani, G., De Felice, M., Moriconi, F. Manuale di finanza. Tassi d’interesse. Mutui e obbligazioni. Il Mulino, 2005
Castellani, G., De Felice, M., Moriconi, F. Modelli stocastici e contratti derivati. Il Mulino, 2005
Naccarato, A., Pierini, A., “BEKK element-by-element estimation of a volatility matrix, A portfolio simulation”, in Mathematical and Statistical Methods for Actuarial Sciences and Finance, (editors Perna, C., Sibillo, M.), Springer, 2014
Lecture notes (delivered to class)

Graphs: basic definitions. Simple and non simple graphs, planarity, connectivity, degree, regularity, incidence and adjacency matrices. Examples of families of graphs. The '' handshaking lemma ''.
Graphs obtained from others: complement, subgraph, cancellation and contraction. Isomorphisms and automorphisms of graphs. Connectivity: paths, cycles. A graph is bipartite if and only if each cycle has equal length.
Connetivity and connected component. Connectivity for sides and vertices. Eulerian and semi-Eulerian graphs.
Euler's theorem: a connected graph is Eulerian if and only if every vertex has an even degree. Hamiltonian graphs. Sufficient conditions to guarantee that a graph is Hamiltonian: the theorems of Ore and Dirac.
the Ore theorem. Trees and forests. The cyclomatic number and the "cutset" rank of a graph.
Fundamental system of cycles and cuts associated with a generating forest. Enumeration of generating forests. The Cayley theorem.
Generating trees: the "greedy" algorithm for the "connector problem". Planar graphs. K3.3 and K5 are not planar. Statement of the theorem of Kuratovski and variations.
Euler's formula for planar graphs. The dual of a planar graph.
Correspondence between cycles and cuts for planar graphs and their dual. Dual abstract. A graph that admits a dual abstract is planar. Colorings: initial considerations and some properties.
Colorings: the 5 colors theorem. Graphs on surfaces: classification of topological surfaces.
Coloring of faces and duality between this problem and the coloring of vertices. Reduction of the proof of the 4-color theorem to the coloring of cubic graph faces. '' The marriage problem ": Hall's theorem.
Hall's theorem in the language of transversals. Criteria of existence of transversal and partial transversivities. Application to the construction of Latin squares. Direct graphs: basic notions and orientability.
The Max-Flow Min-Cut theorem and Menger's theorem.
Complexity of algorithms and applications to the theory of graphs.
Introduction to the theory of matroids: definitions using bases and independent elements. Graphical and cographic matroids, vector matroids and the problem of representability.
Definition of matroid using the cycles and the rank function. Minors of a matroid. Transverse matroids and the Rado Theorem for matroids.
Union of matroids and applications: existence of disjoint bases in a matroid. Duality for matroids and applications to graphic and cographic matroids. Planar matroids and the generalization of Kuratovski's theorem for matroids.
Elements of algebraic graph theory: the incidence matrix and the Laplacian matrix of an oriented graph. The vertex space and the space of edges of a graph. Subspaces of the cycles and subspace of the cuts of a defined oriented graph of the incidence matrix.
Basis for the space of the cycles and for the space of the cuts of a graph. The Riemann-Roch theorem for graphs.
Proof of the "generalized Matrix Tree theorem". The contraction / restriction algorithm for matroids. Examples. The number of acyclical orientations of a graph.
Graph polynomials: the chromatic polynomial, the "reliabiliaty" polynomial. Examples. The polynomial rank of a matroid. Properties and first applications.
Proof of the structure theorem for functions on matroids that satisfy contraction/restriction properties. Their incarnation through the polynomial rank.
Whitey's moves and two isomorphisms for graphs. Isomorphism between graphical matroids implies isomorphism between graphs in case the graphs are 3 connected.
Whitney's Theorem for graphic matroids: sketch of the demonstration. Characterization for minors excluded from binary and regular matroids. The theorem of Seymour.

R. Diestel: Graph theory, Spriger GTM 173.
R. Wilson: Introduction to Graph theory, Prentice Hall.
B. Bollobas: Modern Graph theory, Springer GTM 184.
J. A. Bondy, U.S.R. Murty: Graph theory, Springer GTM 244.
N. Biggs: Algebraic graph theory, Cambridge University Press.
C. D. Godsil, G. Royle: Algebraic Graph theory, Springer GTM 207.
J. G. Oxley: Matroid theory. Oxford graduate texts in mathematics, 3.

1) Computability, complexity and representability:

- Introduction to decision problems, algorithmic and non-algorithmic procedures, deterministic computations, discrete procedures, the notion of an alphabet, the notion of a formal language. Decidability and semi-decidability of a set. Deterministic, finitary and discrete computations. Formal algorithms: formal definition of algorithm, configurations of input, output, transition function. Example of formalization of an algorithm. Decidability for finished automata. Representation of the automata by matrices. Free Monoid of words. Formal semi-rings. Non-deterministic finite automata. Regular Languages. Equivalence between deterministic and non-deterministic automata.

- Turing machines: definition, decidability for Turing machine, stopping time, stopping space. Cost of computation. Complexity: worst case and average case. Independence of decision time from a finite number of input configurations. Complexity functions, complexity classes DTIME and DSPACE (deterministic time and space). Inclusion DTIME (T (n)) ⊂ DSPACE (T (n)) ⊂ DTIME (2 ^ {cT (n)}). Pumping Lemma. Simulation of algorithms, simulation of the half tape Turing machine, simulation of a multi-tape machine. Special Turing machines. Linear Speedup theorem for Turing machines with an extended alphabet. Evaluation of acceleration coefficient in relation to alphabets. Decisions of natural number sets. Independence from representation. Considerations concerning complexity.

- Turing computability: definition of Turing computable function, characteristic functions of Turing decidable sets, the class of Turing computable functions is closed by composition, concatenation, primitive recursion and minimization. Examples of Turing computable functions. Recursive Functions: equivalence between Turing computability and recursive functions. Ackermann function ([1] chapter 1,2,3,4,5 and [4] chapter 1).

- Time-constructible functions. The notion of T-clock. Examples of some time constructible function. Closure by composition.

- Non-deterministic Turing machines: characterization through the decidability of projection sets. Definition of the class of polynomial non-deterministic functions. NP-complete problems.

2) Lambda calculus and functional programming:

- Declarative programming: historical outline on the lambda calculus, basic definitions, the terms of the lambda calculus, the simple substitution. Relations on the lambda terms. Congruences, transition to the context. α-equivalence. alpha-equivalence passes to the context. Transitive closure of a relationship, owned by Church-Rosser. Listing of lambda-terms with respect to alpha-equivalence.

- Definition of beta-reduction and beta-equivalence. Church-Rosser's theorem for beta-reduction. Normal forms for beta-reduction. Beta-reduction strategies. Normalizing strategy: left reduction (left most-outer most). Head reduction. Soluble Terms. Head Normal Forms. Solvability characterization theorem.

- Representation of the recursive functions: lambda definability theorem. The existence of the fixed point for any lambda term. Church Fixed Point and Curry fixed point.
- Representation of other data types in the lambda-calculus: pairs, lists, trees, the solution of recursive equations on lambda-terms ([2] chapters 1, 2, 5).

- P. Dehornoy, Calculabilite et Decidabilite, (1993) Springer-Verlag (in francese);
- J.-L. Krivine, Lambda Calculus: Types and Models, (1993) Ellis Horwood editore.
- M. Sipser, An introduction to the theory of computation (2005), Course Technology.

The purpose of this course is to give a brief introduction to the Finite Elements Method (FEM), a gold standard for the numerical solution of PDEs systems. Oddly enough, the widespread use of the FEM is not accompanied by an adequate knowledge of the mathematical framework underlying the method.

This course, starting from the weak formulation of balance equations, will give an overview of the techniques used to reduce a differential problem into an algebraic one. During the course, some selected problems in mechanics and physics will be solved, covering the three main types of equations: elliptic, parabolic and hyperbolic.
The course will cover the following topics: - Applied Linear Algebra. - Boundary Value Problems. - Initial Value Problems Moreover, the students will be introduced to the use COMSOL Multiphysics, a scientific software for numerical simulations based on the Finite Element Method.

Mark S. Gockenbach,
Understanding and Implementing the Finite Elements Method,
SIAM, 2006

Gilbert Strang
Computational Science and Engineering
Wellesley-Cambridge Press, 2007

The course is organised in a combination of lectures and laboratory. The lectures start introducing basic technical topics that are fundamental in order to understand the panorama of architectural approaches. The classical components of a data acquisition system, the issues and opportunities offered by different technical choices are also discussed. A non exhaustive list of the technical topics addressed during the course are: trigger, data acquisition systems, on-line systems, real-time systems, event building, communication networks and protocols, data archiving and mass storage.

Copy of the presentations made during the lectures.

1. Introduction to graph theory: graph, directed graph, tree, rooted tree, connection, strong connection, acyclic graph; graphs isomorphisms, planarity, Kuratowski's theorem, Euler's formula; graph coloring; Eulerian paths, Hamiltonian circuits.

2. Theory of algorithms and optimization: recalls on algorithms and structured programming; computational complexity of an algorithm, complexity classes for problems, classes P, NP, NP-complete, NP-hard; decision-making, research, enumeration and optimization problems; non-linear programming problems, convex programming, linear programming and integer linear programming; combinatorial optimization problems.

Recalls on the elements of combinatorial calculation, algorithms for the generation of the set of parts of a finite set, permutations and combinations of the elements of a set, calculation of the binomial coefficient; the problem of the "Latin squares" and the game of Sudoku; a recursive algorithm for the solution of the game.

3. Optimization problems on graphs and network flows: graph visit, verification of fundamental properties of a graph: connection, strong connection, presence of cycles. Topological sorting of an directed acyclic graph.

The problem of the minimum spanning tree, Kruskal's algorithm, Prim's algorithm, formulation of the problem in terms of Integer Linear Programming.

Search for minimum cost paths on a weighed graph; minimum cost path with single source, Dijkstra's algorithm, Bellman-Ford's algorithm; minimum cost path between all pairs of vertices of the graph, dynamic programming algorithms, Floyd-Warshall's algorithm, transitive closure of a graph.

Network flows and the maximum flow on a network, Theorem of maximum flow and minimum cut, Ford-Fulkerson's algorithm, Edmonds-Karp's algorithm, preflow algorithm, "push-relabel" algorithm.

Problems of partitioning of graphs, trees and paths, problems for the optimal partitioning of trees and paths into p connected components; objective functions, algorithmic techniques for the solution of this class of problems (dynamic programming, linear programming, shifting).

Stable marriage problem, problem definition and matching stability criterion, applications, Gale and Shapley algorithm.

4. Programming laboratory for developing C language programs for the optimizations algorithms of algorithms in GNU/Linux environment and with Wolfram Mathematica software.

Cormen, Leiserson, Rivest, Stein, "Introduction to algorithms", third edition, The MIT Press, 2009

Short introduction to Julia for scientific programming. Introduction to parallel aechitectures. Designi principles of parallel algorithms. Prallel and distributed programming with Julia. Comunication and sincronization primitives: MPI paradigm. Directive-based languages: OpenMP. Performance metrics of parallel programs. Matrix operations and dense linear systems: mentions to BLAS, LAPACK, scaLAPACK. Sparse linear systems. Mentions to CombBLAS and GraphBLAS.

1. [Lecture slides and diary](https://github.com/cvdlab-courses/pdc/blob/master/schedule.md)

2. [Learning Julia](https://www.manning.com/books/julia-in-action)

3. Blaise N. Barney, [HPC Training Materials](https://computing.llnl.gov/tutorials/parallel_comp/), per gentile concessione del Lawrence Livermore National Laboratory's Computational Training Center

4. J. Dongarra, J. Kurzak, J. Demmel, M. Heroux, [Linear Algebra Libraries for High- Performance Computing: Scientific Computing with Multicore and Accelerators](http://www.netlib.org/utk/people/JackDongarra/SLIDES/sc2011-tutorial.pdf), SuperComputing 2011 (SC11)

1. Preliminaries
Definition of hyper-surface. Integration on hyper-surfaces. The divergence theorem.

2. The Laplace equation
Mean value inequalities. Minimum and maximum principle. The Harnack inequality. The Green representation. The Poisson integral. Convergence theorems. Interior estimates on the derivatives. The Dirichlet problem:
the method of sub-harmonic functions.

3. The classical maximum principle
The weak maximum principle. The strong maximum principle. The Hopf lemma.

4. The Poisson equation and the Newtonian potential
Hölder-continuity. The Dirichlet problem for the Poisson equation. Hölder estimates for second derivatives. Boundary estimates. Hölder estimates for first derivatives.

5. Banach and Hilbert spaces
The contraction principle. The continuity method.

6. Classical solutions: the Schauder approach
Schauder interior estimates. Boundary and global estimates. The Dirichlet problem. Boundary and interior regularity.

“Elliptic partial differential equations of second order. Reprint of the 1998 edition”, D. Gilbarg e N.S. Trudinger. Classics in Mathematics. Springer-Verlag, Berlin, 2001.

Linear Systems
Direst methods: Gaussian elimination. Pivoting strategies. Gaussian elimination as a factorization. Doolittle and Choleshy factorizations. Iterative methods: Jacobi, Gauss-Seidel, SOR, Richardson, and related convergence results. Comparison of direct vs iterative solvers. Stability of algorithms for the solution of linear systems.

Iterative Methods for Scalar Nonlinear Equations
The intermediate zero theorem. The algorithms of bisection, Newton, secants, chords, and related convergence results. (Reference: Chapter 1 excluding Section 1.2.3, and Appendices A.1, A.2)

Approximation of Functions
General approximation strategies. Interpolating polynomial in Lagrange and Newton form. Representation of the interpolation error. Convergence of the interpolating polynomial for analytic functions. Refinement strategies in interpolation: Chebyshev nodes, composite approximations. Error estimates. Hermite polynomial, construction and representation of the error. Least Squares approximations. (Reference: Chapter 5 excluding Section 5.2, and Appendix A.4)

Numerical Integration
General principles of numerical integration. Polya's Theorem on the convergence of interpolatory quadrature formulae. Closed and open Newton-Cotes formulae. Stability results and error estimation. Generalized Newton-Cotes formulae and their convergence. Gaussian quadratures and their convergence. (Reference: Chapter 6)

Laboratory Activity
C language coding of some of the major algorithms, and in particular: Gaussian elimination, iterative methods for linear systems and scalar equations, Lagrange/Newton interpolation with a refinement strategy.

N.B.: References are provided with respect to the course notes.

Roberto Ferretti, "Appunti del corso di Analisi Numerica", available at the address: http://www.mat.uniroma3.it/users/ferretti/corso.pdf

Slides of the lessons, available from the course page: http://www.mat.uniroma3.it/users/ferretti/bacheca.html

1. Probability

Introductory example: The branching process.
Introduction to measure theory.
Measure spaces. Events. Uniqueness and extension of measure.
Probability measures. First Borel--Cantelli lemma.
Random variables, distribution function and law. Indipendence.
Second Borel--Cantelli lemma.
0--1 law for independent random variables.

2. Integration, expected value

Introduxction to integration theory. Monotone convergence theorem.
Expectation. Taking the limit under expectation.
Jensen's inequality. L_p norms.
H\"older inequality and Cauchy-Schwarz.
Markov's inequality.
Examples of weak and strong laws of large numbers.
Product measures. Fubini theorem.
Joint laws.

3. Conditional expectation, martingales and limit theorems

Conditional expectation with respect to a sub $\sigma$--algebra. Kolmogorov existence and uniqueness theorem.
Filtrations. Martingale. Gambilng. Stopping times.
Optional stopping.
Some applications to exit times from an interval.
Convergence theorem for martingales in L_1 and in L_2. Kolmogorov's strong law of large numbers.

4. Convergence ind distribution and the central limit theorem

Characteristic functions. Inversion theorem.
Equivalence between convergence in distributiona nd pointwise convergence of characteristic functions.
Various modes of convergence for random variables. Examples.

D. Williams.
Probability with martingales
Cambridge University Press, 1991


R. Durrett
Probability: Theory and Examples
Thomson, 2000

Affine Spaces
Zariski topology. Affine closed subsets and radical ideals. Theorem of the zeros of Hilbert. Correspondence between closed subsets and radical ideals. Noetherian topological spaces. Irreducible closed subsets, irreducible components. Regular functions of affine closed subsets. Regular maps, isomorphisms. Principal open subsets. Examples. Projections are open. Finite morphisms.

Varieties
Projective spaces and their Zariski topology. Quasi-projective varieties. Rational and regular maps. Projective hypersurfaces. Birational equivalence. Principal open subsets and affine closed subsets. Affine varieties. Dimension of quasi-projective varieties. Finite and generically finite morphisms. Characterizations of birational equivalence. Characterization of generically finite morphisms. Costructible sets, Chevalley's theorem. Every variety is birational to a hypersurface.

Local geometry
Local ring of a variety in a point. Zariski cotangent space. Zariski tangent space. Singular and non singular points.

L. Caporaso
Introduction to algebraic geometry
Notes of the course

I. Shafarevich
Basic Algebraic geometry
Springer-Verlag, Berlin, 1994

LAGRANGIAN AND HAMILTONIAN MECHANICS. VARIATIONAL PRINCIPLES. CONSTRAINTS. CYCLICI VARIABLES. CONSTANTS OF MOTION AND SYMMETRIES. HAMILTONIAN FLOWS. LIOUVILLE THEOREM AND POINCARE RECURRENCE THEOREM. CANONICAL TRANSFORMATIONS. GENERATING FUNCTIONS. HAMILTON-JACOBI METHOD AND ACTION-ANGLE VARIABLES. PERTURBATION THEORY. KAM THEOREM.

[1] G. GENTILE,INTRODUZIONE AI SISTEMI DINAMICI. 1.EQUAZIONI DIFFERENZIALI ORDINARIE, ANALISI QUALITATIVA E ALCUNE APPLICAZIONI.
AVAILABLE ON-LINE: http://www.mat.uniroma3.it/users/gentile/2016-2017/FM410/testo.html
[2] G. GENTILE,INTRODUZIONE AI SISTEMI DINAMICI. 2.FORMALISMO LAGRANGIANO E HAMILTONIANO.
AVAILABLE ON-LINE: http://www.mat.uniroma3.it/users/gentile/2016-2017/FM410/testo.html
[3] G. DELL'ANTONIO, ELEMENTI DI MECCANICA. LIGUORI EDITORE, (1996)
[4] V.I. ARNOLD, METODI MATEMATICI DELLA MECCANICA CLASSICA. EDITORI RIUNITI, (1979)
[5] G. GALLAVOTTI, MECCANICA ELEMENTARE. BOLLATI-BORINGHIERI, (1980)

Outline of the course; Introduction and generality; Bioinformatics and algorithms; Computational biology in the clinic and in the pharmaceutical industry; Pharmacokinetics and pharmacodynamics;

Introduction to Systems Biology: what is computational biology; The roles of mathematical modeling and bioinformatics; what is he aiming for; what are the problems; Theoretical tools used in bio-mathematics and bioinformatics.

Introduction to molecular and cellular biology (first part): basic knowledge of genetics, proteomics and cellular processes; Ecology and evolution; the basic molecule; molecular bonds; the chromosomes; DNA and its replication;

Introduction to molecular and cellular biology (second part); genomics; The central dogma of biology; The genome project; the structure of the human genome Analysis of genes; transcription of DNA; the viruses;

Laboratory: generation of random numbers; the functions srand48 and drand48; random generation of arbitrary length nucleotide strings (program1.c); random generation of amino acid strings of arbitrary length (program2.c);

Introduction to information theory; Shannon Entropy; Conditional Entropy; Mutual Information; Indices of biological diversity; Shannon Index; True diversity; Reny index;

Laboratory: the genetic code; C program of transcription DNA sequence and translation into proteins;

Introduction to stochastic processes; basic definition; examples; model of queues; Bernoulli and Poisson process; Markov processes; stochastic processes in bioinformatics and bio-mathematics; the autocorrelation; Outline of the Random Walks and the BLAST algorithm of sequence alignment as a stochastic process and principal algorithm for the consultation of biological sequence databases;

Laboratory: development of an algorithm in C for the calculation of the Shannon Entropy of a text in English (or in Italian) any (e.g., http://www.textfiles.com/etext/)

Random walks. The BLAST algorithm for aligning sequences as a random path; Laboratory: C implementation of different algorithms for the generation of a random walk in 1D and 2D on the lattice and in R or R ^ 2 signal and calculation of the mean square displacement;

Compare sequences: similarity and homology; pairwise alignment; editing distance; scoring matrices PAM and BLOSUM; Needleman-Wunsch's algorithm; local alignment; Smith-Waterman's algorithm; BLAST algorithm;

Laboratory: C implementation of an algorithm for the generation of a signal with noise and calculation of the correlogram in the presence or absence of a true signal;

Multiple Sequence Alignment; consensus sequence; star alignment algorithms; ClustalW; entropy and circular sum scoring functions;

Biological data banks; reasons; data format; taxonomy; Primary DBs; Secondary DBs; NCBI, EMBL, DDBJ; NCBI EBI-Entrez; Exact matching / string searching: general; the agony of Knuth-Morris-Pratt;

Exact matching / string searching: the Boyer-Moore agoritm;

Exercise on an implementation of the Knuth-Morris-Pratt exact matching algorithm. Exercise on biological databases; primary databases; secondary databases; NCBI, EMBL, DDBJ; NCBI EBI-Entrez; Use of the BLAST algorithm

Phylogenetic Analysis; phylogenetic trees; dimension of the research space of phylogenetic algorithms; Methods of construction of phylogenetic trees; Data used for phylogenetic analysis; The Unweighted Pair Method Method with Arithmetic mean (UPGMA) algorithm; the Neighbor Joining Method algorithm; Hidden Markov Models; decoding; the Viterbi Algorithm; Evaluation;

Laboratory: completion of the exercise on mutation, selection and evolution of nucleotide strings (genotype) translated into amino acid strings (phenotype); Selection is made based on the presence of certain substrings in the phenotype that determines the fitness value; Implemented details, display of the convergence criterion and results, discussion, etc .;

Machine Learning; generality'; supervised and unsupervised learning; model selection; undefitting; overfitting; Polynomial curve fitting; machine learning as an estimate of the parameters and the problem of overfitting; subdivision of the training set into testing and testing; concept of bias and variance trade-off; Artificial Neural Networks; definizone; the percussion of Rosenblatt; the percettrone learning algorithm; the multi-layer perceptron;

Laboratory: completion of the implementation in ANSI C of the evolutionary algorithm of nucleotide strings (genotype) translated, through the use of the genetic code, into amino acid strings (phenotype);

Hidden Markov Models; The Forward Algorithm; The Backward Algorithm; Posterior Decoding; Learning; Baum-Welch Algorithm; Use of Hidden Markov Models for the analysis of bio-sequences; gene finding;

Artificial Neural Networks; the error-back propagation algorithm for learning MLP; types of neural networks; convolution networks; reinforcement networks; unsupervised learning and self-organizing maps; Introduction to graph theory; representation, terminology, concepts; paths; cycles; connettivita '; distance; connected components; distance;

Introduction to graph theory; visit breadth-first search; depth-first search; Dijkstra's algorithm; six-degree of separation; small world networks; centrality measures; degree centrality; eigenvector centrality; betweennes centrality; closeness centrality; The network biology; generality'; concepts; types of biological data used to build networks; network biology and network medicine; problems and algorithms used; centrality measures; random networks; scale-free networks; preferential attachment; scale-free network in biology;

Laboratory: completion of the exercise on the evolutionary algorithm; Implemented details, display of the convergence criterion and results, discussion, etc .;

Bio-mathematical models; prediction using theoretical models; the itertative paradigm of mathematical modeling; data-driven models; limited and non-population growth models; analytical derivation and examples; logistics growth; ecological models limited by density; The Lotka-Volterra model; the experiment by Huffaker and Kenneth; the SIR epidemic model and some of its variants; Perelson's model for HAART; the Java Populus application for the solution of continuous models of population dynamics; hints to the numerical resolution methods of differential equation systems;

Discrete models; spin models (Ising models); Cellular automata; Boolean networks; Agent-based models; data fitting and parameter estimation; software tools available; Cellular automata; introduction and history; definition; the 1-dimensional automaton; Wolfram classification; the 2-dimensional automaton; Conway's Game of Life; Software available for CA simulation; dedicated hardware (CA-Machine); the prey-predator model as a two-dimensional cellular automaton; relationship with the system of ordinary derivation equations; stochastic models; Stochastic CAs as discrete stochastic dynamic systems and stochastic processes; example of CA: Belousov-Zabotonsky reactions;

[-] E.S. Allman, J.A. Rhodes. Mathematical Models in Biology: An Introduction (2004) Cambridge University Press.
[-] W.J. Ewens, G.R. Grant. Statistical Methods in Bioinformatics, An Introduction (2005) Springer Verlag.
[-] R. Durbin, S. Eddy, A. Krogh, G. Mitchison. Biological sequence analysis - Probabilistic models of proteins and nucleic acids (1998) Cambridge University Press.

Complex numbers: algebraic and topological properties. Geometric representation of complex numbers: polar coordinates and the complex exponential.
Complex functions with complex variables: continuity and properties, differentiability and first properties. Holomorphic function: properties and examples of holomorphic and non-holomorphic functions. Cauchy-Riemann equations. The real and imaginary parts of holomorphic function are harmonic conjugated.
Equations of Cauchy-Riemann: proof. Examples. Sequences and complex series. Properties. Power series with complex values. Abel's theorem and Hadamard's formula.
Proof of Abel's Theorem. Taylor's formula for series of complex powers. The exponential and the trigonometric functions as analytical functions. Basic properties.
Periodicity of the complex exponential function. The complex logarithm: first considerations. The ring of formal powers series with complex coefficients: basic properties. Analytical functions: definition and first properties.
Series of converging powers are analytical within the convergence region. Composition of analytical functions. Theorem of the inverse function.
Inverse by composition of a formal series and its convergence. Complex powers and properties. The binomial series and properties. Consequences of the inverse theorem: the canonical form of an analytic function.
Local properties of analytical function: open function theorem, invertibility criterion, principle of the maximum local module. The fundamental theorem of algebra. Parameterized curves. A holomorphic function with zero derivative is constant.
The place of the zeros of a non-constant analytical function is discrete. Analytical continuation of functions defined on open connected sets. Principle of the maximum global module. Integrals in paths: definition and first properties. Examples.
A continuous function in a connected open admits a primitive if and only if its integral along a closed curve is zero. Integration of uniformly converging series of functions. Examples. Local primitive of a holomorphic function.
Local primitive of a holomorphic function. The Goursat theorem. Integral of a holomorphic function along a continuous path.
The homotopical form of the Cauchy Theorem. Global primitive of a holomorphic function in a simply connected domain. Applications to the study of the logarithm.
The integral formula of Cauchy. Cauchy formula for development in series and applications: a holomorphic and analytical function; the theorem of Liouville and the fundamental theorem of algebra.
Integral formula for derivatives. The number of windings of a curve with respect to a point. Curves homologous to 0. The global formula of Cauchy.
Demonstration of the global Cauchy formula. Examples.
The first homology group of an open set with values ​​in integers. The Cauchy formula for homological invariance. Examples.
Applications of the Cauchy theorem: uniform limit on holomorphic function compacts is holomorphic. Examples. Laurent series.
Series expansion of a holomorphic function in a circular crown in the Laurent series. Isolated singularities and the field of meromorphic functions. Examples. Statement of the classification theorem of isolated singularities and residual theorem: local and global versions.
Proof of the classification theorem of isolated singularities and proof of the residues theorem. The logarithmic derivative and the principle of the argument. Calculation of residues.
Classification of the connected open of C. The Riemann map theorem and the uniformization theorem (without proof). The Riemann sphere as a compactification of the complex plane. The group of linear transformations of the projective line and the linear transformations produced by them. The group of automorphisms of the complex plane.
The lemma of Schwarz and the group of automorphisms of the unitary disc. Elements of global analytical functions and function. The logarithm as a global analytical function.
The n-th rooty as a global analytical function. The bundle of germs of analytical functions and its properties. The Riemann surface associated with a global analytical function.
Examples and properties of Riemann surface. The Riemann surface associated with an algebraic function and properties. Summary and considerations on the course program.

L. V. Ahlfors: Complex Analysis, McGraw-Hill.
S. Lang: Complex analysis, GTM 103.
E. Freitag, R. Busam: Complex Analysis, Springer.

Introduzione alla topologia algebrica. I simplessi e le loro proprieta'. Il complesso delle catene singolari associate ad uno spazio topologico. I gruppi di omologia singolare. Proprieta' elementari: i gruppi di omologia rispettano la decomposizione di uno spazio in componenti connesse per archi; calcolo del gruppo di omologia H_0. L'omologia di un punto. Omologia ridotta: definizione e confronto con l'omologia usuale. Categorie e funtori: definizioni ed esempi. L'omologia come functore dagli spazi topologichi ai gruppi abeliani, passando per la categoria dei complessi. Invarianza omotopica dell'omologia: funzioni continue omotope inducono mappe di complessi omotope; mappe di complessi omotope inducono la stessa mappa in omologia. Il complesso delle catene di una coppia e l'omologia di una coppia: proprieta' funtoriali e invarianza omotopica. La successione esatta lunga in coomologia associata ad una tripla (o ad una coppia). Corollario: l'omologia ridotta come omologia relativa ad un punto.
Il Teorema di escissione. L'omologia relativa ad un ricoprimento.
Coppie buone. Esempi e controesempi. L'omologia di una coppia buona e' isomorfa all'omologia ridotta del quoziente.
Successione di Mayer-Vietoris. Calcolo dell'omologia della sfera e di un disco relativo al bordo, con generatori espliciti. Applicazioni: il teorema del punto fisso di Brower, il teorema dell'invarianza della dimensione (tramite l'omologia locale di R^n).
Grado di applicazioni dalla sfera n-dimensionale in se'. Proprieta' elementari: moltiplicativita' e invarianza omotopica. Esempi: riflessioni, mappa antipodale, mappe senza punti fissi. Applicazione: la pettinabilita' della sfera n-dimensionale. Formula locale per il grado. Il caso di dimensione 1: la mappa grado definisce un isomorfismo tra il gruppo fondamentale della sfera 1-dimensionale e il gruppo degli interi.
La sospensione di spazi topologici. Il grado della sospensione di una mappa da S^n a S^n e' ugale al grado della mappa originaria. Naturalita' delle successioni esatte lunghe. Delta-complessi. Esempi in dimensione uno e due. Equivalenza tra Delta-complessi e Delta-insiemi.
Complessi simpliciali e loro equivalenza con i complessi simpliciali astratti. L'omologia simpliciale di un Delta-complesso. Esempi. L'isomorfismo tra l'omologia simpliciale e l'omologia singolare.
Complessi cellulari (o CW): definizione induttiva. Esempi. I Delta-complessi come complessi cellulari. Ogni complesso cellulare e' omotopo ad un Delta-complesso.
Proprieta' topologiche dei complessi cellulari: normalita', locale contraibilita', paracompattezza (senza dimostrazione), compattezza e locale compattezza. Le coppie cellulari sono coppie buone. Strutture cellulari e simpliciali su varieta' topologiche e differenziabili: una panoramica dei risultati. Esempi di complessi cellulari: la sfera, lo spazio proiettivo reale.
Lemma di compattezza per complessi cellulari. La caratterizzazione dei complessi cellulari tramite le mappe caratteristiche. Non esempi di complessi cellulari che violano la proprieta' della topologia debole o quella della finitezza della chiusura. Esempi di complessi cellulari: lo spazio proiettivo complesso, le superfici topologiche compatte orientate e non orientate.
Somma wedge di spazi topologici puntati e loro gruppi di omologia. Il complesso delle catene cellulari di un complesso cellulare e l'omologia cellulare. L'omologia cellulare coincide con l'omologia singolare. La formula per il differenziale cellulare. Esercizi: omologia cellulare di sfere, spazi proiettivi reali e complessi, superfici topologiche compatte orientate e non, i tori n-dimensionali.
Omologia con coefficienti. Digressione sull'algebra omologica dei gruppi abeliani: prodotto tensoriale e bifuntori Tor.
Il teorema dei coefficienti universali per l'omologia. Spazi di Moore. Lo spezzamento della successione esatta nel teorema dei coefficienti universali non e' naturale: un controesempio.
Una coppia cellulare soddisfa la proprieta' dell'estensione omotopica. Se una coppia soddisfa la proprieta' dell'estensione omotopica e il sottospazio e' contraibile, allora il quoziente per il sottospazio e' un'equivalenza omotopica. Il primo gruppo di omologia e' l'abelianizzato del gruppo fondamentale (inizio della dimostrazione).
Il primo gruppo di omologia e' l'abelianizzato del gruppo fondamentale (fine della dimostrazione). La caratteristica di Eulero. Formula della caratteristica di Eulero per un complesso cellulare finito. Esercizi.
Coomologia con coefficienti. Digressione sull'algebra omologica dei gruppi abeliani: il bifuntore Hom e il bifuntore Ext. Il teorema dei coefficienti universali per la coomologia.
Proprieta' della Coomologia: funtorialita', invarianza omotopica, successione esatta lunga di una tripla e relazione tra i morfismi di connessione in omologia e coomologia, coomologia ridotta, escissione, coomologia di una coppia buona, successione di Mayer-Vietoris.
Il prodotto cup e l'anello di coomologia.
+La coomologia simpliciale di un Delta-complesso. La coomologia cellulare di un complesso cellulare. Esempio: l'anello di coomologia delle superfici topologiche compatte orientate e non.
L'approccio assiomatico alla (co)omologia assoluta di una coppia e relativa (senza dimostrazione). Il prodotto cross in omologia e coomologia singolare e cellulare. Relazione tra il prodotto cup e il prodotto cross in coomologia. La formula di Kunneth in omologia e coomologia (senza dimostrazione). Esempio: l'anello di coomologia di un toro n-dimensionale e' l'algebra esterna di dimensione n.
Varieta' (topologiche). Orientabilita' di varieta'. Il rivestimento doppio delle orientazioni locali. R-orientabilita'. Il rivestimento delle R-classi locali. L'omologia n-esima di una n-varieta' compatta e connessa, R-orientabile e non. La R-classe fondamentale di una varieta' compatta R-orientabile.
Il prodotto cap e la sua relazione col prodotto cup. La formulazione della dualita' di Poincare' per varieta' compatte R-orientate: versione I (col prodotto cap) e versione II (col prodotto cup).
Coomologia a supporto compatto. Digressione sui limiti diretti di gruppi. L'operatore di dualita' per varieta' topologiche R-orientate non compatte. Il teorema di dualita' di Poincare' per varieta' topologiche R-orientate compatte e non: dimostrazione. Esempio: l'anello di coomologia degli spazi proeittivi reali, complessi e quaternionici.

A. Hatcher: Algebraic Topology. Cambridge University Press, 2002.

Brownian motion I

Gaussian multivariate distribution. Processes with stationary and independent increments.
Definition and continuity properties of Brownian motion. Non differentiability of trajectories.
Markov and Strong Markov property.

Brownian motion II

Brownian motion in higher dimensions. Harmonic functions and the Dirichlet problem in regular domains.
The Poisson problem. Iterated logarithm law. Skorohod embedding.
Donsker's invariance principle. Applications: arcsine law and the maximum of random walks.


Stochastic integrals

The Paley-Wiener-Zygmund integral. Stochastic integral with respect to Brownian motion. It\^o formula and applications. Local time and the Tanaka formula.



Stochastic differential equations

The linear case. Existence and uniqueness theorem for stochastic differential equations.
Diffusion process in the limit of zero noise. Infinitesimal generator and partial differential equations.
Feynman-Kac formula and applications.

P. M\"orters and Y. Peres
Brownian Motion
Cambridge University Press
2010


L.C. Evans
An Introduction to Stochastic Differential Equations
AMS bookstore
2013

Under the general title of this Course, we consider both, asymptotic and
numerical methods mostly to handle ordinary differential equations and systems,
and possibly some partial differential equations, and applications. Among the
asymptotic methods, the so-called ``singular perturbations'' are addressed.
These two subjects, which represent basic techniques of Applied Mathematics,
are usually taught separately. In this course we will use both, at the same
time, in the same problems.

Asymptotic Methods:

Introduction to Asymptotic Analysis. Asymptotic series. Regular and singular
perturbations of ordinary differential equations and systems.

Numerical Methods:

Numerical solution, using several algorithms, of ordinary differential
equations and systems. This will be mostly done in the computing laboratory.

Robert E. O'malley,
Singular Perturbation Methods for Ordinary Differential Equations,
Springer-Verlag, Applied Mathematical Sciences, 89, New York, 1991.

John K. Hunter,
Asymptotic Analysis and Singular Perturbation Theory,
Lecture Notes, https://www.math.ucdavis.edu/~hunter/notes/asy.pdf,
2004

1. Probability

Introductory example: The branching process.
Introduction to measure theory.
Measure spaces. Events. Uniqueness and extension of measure.
Probability measures. First Borel--Cantelli lemma.
Random variables, distribution function and law. Indipendence.
Second Borel--Cantelli lemma.
0--1 law for independent random variables.

2. Integration, expected value

Introduxction to integration theory. Monotone convergence theorem.
Expectation. Taking the limit under expectation.
Jensen's inequality. L_p norms.
H\"older inequality and Cauchy-Schwarz.
Markov's inequality.
Examples of weak and strong laws of large numbers.
Product measures. Fubini theorem.
Joint laws.

3. Conditional expectation, martingales and limit theorems

Conditional expectation with respect to a sub $\sigma$--algebra. Kolmogorov existence and uniqueness theorem.
Filtrations. Martingale. Gambilng. Stopping times.
Optional stopping.
Some applications to exit times from an interval.
Convergence theorem for martingales in L_1 and in L_2. Kolmogorov's strong law of large numbers.

4. Convergence ind distribution and the central limit theorem

Characteristic functions. Inversion theorem.
Equivalence between convergence in distributiona nd pointwise convergence of characteristic functions.
Various modes of convergence for random variables. Examples.

D. Williams.
Probability with martingales
Cambridge University Press, 1991


R. Durrett
Probability: Theory and Examples
Thomson, 2000

CONGRUENCES AND POLYNOMIALS. LINEAR DIOPHANTINE EQUATIONS. SYSTEMS OF LINEAR CONGRUENCES. POLYNOMIAL CONGRUENCES. POLYNOMIAL CONGRUENCES MOD p: LAGRANGES’S THEOREM. p-ADIC APPROXIMATION. THE EXISTENCE OF PRIMITIVE ROOTS MOD p. QUADRATIC CONGRUENCES. QUADRATIC RESIDUES. THE LEGENDRE SYMBOL. GAUSS’S LEMMA AND THE LAW OF QUADRATIC RECIPROCITY.THE JACOBI SYMBOL. SUMS OF TWO, THREE, FOUR SQUARES. MULTIPLICATIVE FUNCTIONS. THE MÖBIUS INVERSION FORMULA. SOME NONLINEAR DIOPHANTINE EQUATIONS. CONTINUED FRACTIONS.

M. FONTANA, APPUNTIDEL CORSO TN1 (ARGOMENTI DELLA TEORIA CLASSICA DEI NUMERI), HTTP://WWW.MAT.UNIROMA3.IT/USERS/FONTANA/DIDATTICA/FONTANA_DIDATTICA.HTML DISPENSE.
D.M. BURTON: ELEMENTARY NUMBER THEORY, MCGRAW-HILLINTERNATIONAL EDITION, 6TH EDITION (2007), 434 PP.
G.A. JONES AND J.M. JONES, ELEMENTARY NUMBER THEORY, SPRINGER, 1ST EDITION (1998), 200 PP.
H. DAVENPORT, ARITMETICA SUPERIORE.UN?INTRODUZIONE ALLA TEORIA DEI NUMERI, ZANICHELLI, (1994), 199 PP.
G.H. HARDY AND E.M. WRIGHT, AN INTRODUCTION TO THE THEORY OH NUMBERS, THE CLARENDON PRESS, OXFORD UNIVERSITY, 5TH EDITION (1979), XVI+426 PP.
W.J. LEVEQUE, FUNDAMENTALS OF NUMBER THEORY, DOVER PUBLICATIONS, (1996)
K.H. ROSEN. ELEMENTARY NUMBER THEORY AND ITS APPLICATIONS, ADDISON-WESLEY, 6TH EDITION (2011), XV+752 PP.

Teacher's Lectures notes available on web.

Introduzione alla topologia algebrica. I simplessi e le loro proprieta'. Il complesso delle catene singolari associate ad uno spazio topologico. I gruppi di omologia singolare. Proprieta' elementari: i gruppi di omologia rispettano la decomposizione di uno spazio in componenti connesse per archi; calcolo del gruppo di omologia H_0. L'omologia di un punto. Omologia ridotta: definizione e confronto con l'omologia usuale. Categorie e funtori: definizioni ed esempi. L'omologia come functore dagli spazi topologichi ai gruppi abeliani, passando per la categoria dei complessi. Invarianza omotopica dell'omologia: funzioni continue omotope inducono mappe di complessi omotope; mappe di complessi omotope inducono la stessa mappa in omologia. Il complesso delle catene di una coppia e l'omologia di una coppia: proprieta' funtoriali e invarianza omotopica. La successione esatta lunga in coomologia associata ad una tripla (o ad una coppia). Corollario: l'omologia ridotta come omologia relativa ad un punto.
Il Teorema di escissione. L'omologia relativa ad un ricoprimento.
Coppie buone. Esempi e controesempi. L'omologia di una coppia buona e' isomorfa all'omologia ridotta del quoziente.
Successione di Mayer-Vietoris. Calcolo dell'omologia della sfera e di un disco relativo al bordo, con generatori espliciti. Applicazioni: il teorema del punto fisso di Brower, il teorema dell'invarianza della dimensione (tramite l'omologia locale di R^n).
Grado di applicazioni dalla sfera n-dimensionale in se'. Proprieta' elementari: moltiplicativita' e invarianza omotopica. Esempi: riflessioni, mappa antipodale, mappe senza punti fissi. Applicazione: la pettinabilita' della sfera n-dimensionale. Formula locale per il grado. Il caso di dimensione 1: la mappa grado definisce un isomorfismo tra il gruppo fondamentale della sfera 1-dimensionale e il gruppo degli interi.
La sospensione di spazi topologici. Il grado della sospensione di una mappa da S^n a S^n e' ugale al grado della mappa originaria. Naturalita' delle successioni esatte lunghe. Delta-complessi. Esempi in dimensione uno e due. Equivalenza tra Delta-complessi e Delta-insiemi.
Complessi simpliciali e loro equivalenza con i complessi simpliciali astratti. L'omologia simpliciale di un Delta-complesso. Esempi. L'isomorfismo tra l'omologia simpliciale e l'omologia singolare.
Complessi cellulari (o CW): definizione induttiva. Esempi. I Delta-complessi come complessi cellulari. Ogni complesso cellulare e' omotopo ad un Delta-complesso.
Proprieta' topologiche dei complessi cellulari: normalita', locale contraibilita', paracompattezza (senza dimostrazione), compattezza e locale compattezza. Le coppie cellulari sono coppie buone. Strutture cellulari e simpliciali su varieta' topologiche e differenziabili: una panoramica dei risultati. Esempi di complessi cellulari: la sfera, lo spazio proiettivo reale.
Lemma di compattezza per complessi cellulari. La caratterizzazione dei complessi cellulari tramite le mappe caratteristiche. Non esempi di complessi cellulari che violano la proprieta' della topologia debole o quella della finitezza della chiusura. Esempi di complessi cellulari: lo spazio proiettivo complesso, le superfici topologiche compatte orientate e non orientate.
Somma wedge di spazi topologici puntati e loro gruppi di omologia. Il complesso delle catene cellulari di un complesso cellulare e l'omologia cellulare. L'omologia cellulare coincide con l'omologia singolare. La formula per il differenziale cellulare. Esercizi: omologia cellulare di sfere, spazi proiettivi reali e complessi, superfici topologiche compatte orientate e non, i tori n-dimensionali.
Omologia con coefficienti. Digressione sull'algebra omologica dei gruppi abeliani: prodotto tensoriale e bifuntori Tor.
Il teorema dei coefficienti universali per l'omologia. Spazi di Moore. Lo spezzamento della successione esatta nel teorema dei coefficienti universali non e' naturale: un controesempio.
Una coppia cellulare soddisfa la proprieta' dell'estensione omotopica. Se una coppia soddisfa la proprieta' dell'estensione omotopica e il sottospazio e' contraibile, allora il quoziente per il sottospazio e' un'equivalenza omotopica. Il primo gruppo di omologia e' l'abelianizzato del gruppo fondamentale (inizio della dimostrazione).
Il primo gruppo di omologia e' l'abelianizzato del gruppo fondamentale (fine della dimostrazione). La caratteristica di Eulero. Formula della caratteristica di Eulero per un complesso cellulare finito. Esercizi.
Coomologia con coefficienti. Digressione sull'algebra omologica dei gruppi abeliani: il bifuntore Hom e il bifuntore Ext. Il teorema dei coefficienti universali per la coomologia.
Proprieta' della Coomologia: funtorialita', invarianza omotopica, successione esatta lunga di una tripla e relazione tra i morfismi di connessione in omologia e coomologia, coomologia ridotta, escissione, coomologia di una coppia buona, successione di Mayer-Vietoris.
Il prodotto cup e l'anello di coomologia.
+La coomologia simpliciale di un Delta-complesso. La coomologia cellulare di un complesso cellulare. Esempio: l'anello di coomologia delle superfici topologiche compatte orientate e non.
L'approccio assiomatico alla (co)omologia assoluta di una coppia e relativa (senza dimostrazione). Il prodotto cross in omologia e coomologia singolare e cellulare. Relazione tra il prodotto cup e il prodotto cross in coomologia. La formula di Kunneth in omologia e coomologia (senza dimostrazione). Esempio: l'anello di coomologia di un toro n-dimensionale e' l'algebra esterna di dimensione n.
Varieta' (topologiche). Orientabilita' di varieta'. Il rivestimento doppio delle orientazioni locali. R-orientabilita'. Il rivestimento delle R-classi locali. L'omologia n-esima di una n-varieta' compatta e connessa, R-orientabile e non. La R-classe fondamentale di una varieta' compatta R-orientabile.
Il prodotto cap e la sua relazione col prodotto cup. La formulazione della dualita' di Poincare' per varieta' compatte R-orientate: versione I (col prodotto cap) e versione II (col prodotto cup).
Coomologia a supporto compatto. Digressione sui limiti diretti di gruppi. L'operatore di dualita' per varieta' topologiche R-orientate non compatte. Il teorema di dualita' di Poincare' per varieta' topologiche R-orientate compatte e non: dimostrazione. Esempio: l'anello di coomologia degli spazi proeittivi reali, complessi e quaternionici.

A. Hatcher: Algebraic Topology. Cambridge University Press, 2002.

1. Probability

Introductory example: The branching process.
Introduction to measure theory.
Measure spaces. Events. Uniqueness and extension of measure.
Probability measures. First Borel--Cantelli lemma.
Random variables, distribution function and law. Indipendence.
Second Borel--Cantelli lemma.
0--1 law for independent random variables.

2. Integration, expected value

Introduxction to integration theory. Monotone convergence theorem.
Expectation. Taking the limit under expectation.
Jensen's inequality. L_p norms.
H\"older inequality and Cauchy-Schwarz.
Markov's inequality.
Examples of weak and strong laws of large numbers.
Product measures. Fubini theorem.
Joint laws.

3. Conditional expectation, martingales and limit theorems

Conditional expectation with respect to a sub $\sigma$--algebra. Kolmogorov existence and uniqueness theorem.
Filtrations. Martingale. Gambilng. Stopping times.
Optional stopping.
Some applications to exit times from an interval.
Convergence theorem for martingales in L_1 and in L_2. Kolmogorov's strong law of large numbers.

4. Convergence ind distribution and the central limit theorem

Characteristic functions. Inversion theorem.
Equivalence between convergence in distributiona nd pointwise convergence of characteristic functions.
Various modes of convergence for random variables. Examples.

D. Williams.
Probability with martingales
Cambridge University Press, 1991


R. Durrett
Probability: Theory and Examples
Thomson, 2000

Introduction to cryptography. Classic ciphers. Introduction to cryptanalysis.
Introduction to public-key cryptography.
The RSA cryptosystem.
Primality tests.
Factorization algorithms.
Some attacks on the RSA.
The discrete logarithm problem. Diffie-Hellman key exchange.
Elgamal cryptosystem.
Massey-Omura cryptosystem.
Digital signatures.
Overview of some cryptographic protocols.

Baldoni, Ciliberto, Piacentini: Aritmetica, crittografia e codici
D. Stinson: Cryptography - theory and practice

[1] Maurizio Gabbrielli, Simone Martini, Programming Languages: Principles and Paradigms, Springer (2011) ISBN 9781848829138
[2] Dean Wampler, Alex Payne, Programming Scala: Scalability = Functional Programming + Objects, O'Reilly Media; 2 edizione (2014)
[3] David Parsons, Foundational Java Key Elements and Practical Programming, Springer-Verlag, (2012)

Presentation of the problems that can be treated through integrals on large number of dimensions

Basics

Probability and Random variables

Measurement, uncertainty and its propagation

Curve-fitting, least-squares, optimization

Classical numerical integration, speed of convergence

Integration MC (Mean, variance)

Sampling Strategies

Applications

Propagation of uncertainties

Generation according to a distribution

Real World Applications

Cosmic Rays Shower

System Availabilty

Further applications

Stefan Weinzierl - Introduction to Monte Carlo methods - arXiv:hep-ph/0006269

Further Bibliography will be presented in class, according to the needs of the Participants.

Principles of Object Oriented Design
Abstraction, Polimorphism, Inheritance, Aggregation
Object Oriented Programming Models and UML
UML Use Case, Sequence, Class e Object, Deployment diagrams
Software Analysis and Developmenmt for Java Virtual Machine: I/O, Stream, Networking, Exception Handling
(Scientific, Real-time,...) Efficient Distributed Computing, Multithreading and Concurrency in Cloud e Mobile

Manuale di Java 9 De Sio Cesari Claudio Hoepli Informatica

Brownian motion I

Gaussian multivariate distribution. Processes with stationary and independent increments.
Definition and continuity properties of Brownian motion. Non differentiability of trajectories.
Markov and Strong Markov property.

Brownian motion II

Brownian motion in higher dimensions. Harmonic functions and the Dirichlet problem in regular domains.
The Poisson problem. Iterated logarithm law. Skorohod embedding.
Donsker's invariance principle. Applications: arcsine law and the maximum of random walks.


Stochastic integrals

The Paley-Wiener-Zygmund integral. Stochastic integral with respect to Brownian motion. It\^o formula and applications. Local time and the Tanaka formula.



Stochastic differential equations

The linear case. Existence and uniqueness theorem for stochastic differential equations.
Diffusion process in the limit of zero noise. Infinitesimal generator and partial differential equations.
Feynman-Kac formula and applications.

P. M\"orters and Y. Peres
Brownian Motion
Cambridge University Press
2010


L.C. Evans
An Introduction to Stochastic Differential Equations
AMS bookstore
2013

Under the general title of this Course, we consider both, asymptotic and
numerical methods mostly to handle ordinary differential equations and systems,
and possibly some partial differential equations, and applications. Among the
asymptotic methods, the so-called ``singular perturbations'' are addressed.
These two subjects, which represent basic techniques of Applied Mathematics,
are usually taught separately. In this course we will use both, at the same
time, in the same problems.

Asymptotic Methods:

Introduction to Asymptotic Analysis. Asymptotic series. Regular and singular
perturbations of ordinary differential equations and systems.

Numerical Methods:

Numerical solution, using several algorithms, of ordinary differential
equations and systems. This will be mostly done in the computing laboratory.

Robert E. O'malley,
Singular Perturbation Methods for Ordinary Differential Equations,
Springer-Verlag, Applied Mathematical Sciences, 89, New York, 1991.

John K. Hunter,
Asymptotic Analysis and Singular Perturbation Theory,
Lecture Notes, https://www.math.ucdavis.edu/~hunter/notes/asy.pdf,
2004

A) STRUCTURAL RULES EXPRESSED AS LOGICAL RULES: SEQUENT CALCULUS AND DERIVABILITY IN LINEAR LOGIC
B) POSITIVE AND NEGATIVE: FOCUSED SEQUENT CALCULUS FOR LINEAR LOGIC, LUDICS
C) IMPLICIT COMPLEXITY AND LINEAR LOGIC
D) GEOMETRY OF PROOFS: PROOF NETS IN LINEAR LOGIC
E) INVARIANTS AND DEVELOPMENT OF INTERACTION OF PROOFS: COHERENT SPACES, GEOMETRY OF INTERACTION

V. MICHELE ABRUSCI, LEZIONI DI LOGICA LINEARE, 2018 (FORTH COMING)