## Contents of the courses

**1st Semester**

(J. Asch, C.-A. Pillet) Local and global curve theory, classical surface theory, the inner geometry of surfaces, theorema egregium, geometry and topology, Gauss-Bonnet theorem.

(P. Briet) Hilbert spaces, orthogonal projections onto closed convex sets and consequences, orthonormal basis, weak convergence in Hilbert space and its properties (weak Bolzano-Weierstrass), Riesz representation theorem; Linear continuous (bounded) operators on Hilbert space, connection with weak convergence, self-adjoint operators, compact operators, norm limit of finite rank operators; Elements of spectral theory, spectral localization and Lax-Milgram theorem, diagonalization of compact self-adjoint operators; Banach spaces, Hahn-Banach theorem, linear continuous functionals on Banach space, weak convergence, Banach-Steinhaus theorem and its consequences, open mapping theorem, closed graph theorem.

(S. Vaienti) Recalls on probability spaces and random variables: general definitions, monotone continuity properties of probability measures, Borel-Cantelli theorem, monotone classes and Dynkin's classes (statements only) and applications, discrete probability (characterization); Random variables (r.v.): general definitions, discrete, continuous and absolutely continuous r.v., distribution functions, Chebychev nequality, Skorokhod representation theorem on the existence of r.v.'s from a given law, calculations of probability distributions; Generating and characteristic functions of a r.v.; Gaussian vectors; Conditional probabilities and conditional expections; Sequences and sums of random variables: different kinds of convergence: almost sure, in probability, in distribution and relationship with the convergence of characteristics functions, Paul Levy's theorem, Skorokhod representation theorem; Limit theorems: laws of large numbers and central theorem limit, large deviations for Bernoulli r.v., some notions on random walks and Markov chains, construction of Brownian motion.

**English - 2 ECTS -- 18h**

(F. Armao) This class will focus on oral and written comprehension/production of English with a strong emphasis on oral interaction. Students will be asked to do an oral presentation connected with their field of study; this presentation will lead to a debate in English between students. Further details will be given during the first class (which should not be missed).

Additionally, we will work on other scientific themes, mainly through the prism of video and written documents. Students are required to attend class and to actively participate.

**TICE - 1 ECTS -- 10h**

(G. Faccanoni - 2014/15) Learning LaTeX: a software package particularly well-suited to prepare documents containing mathematical formulas.

(Y. Aubry) Generalities on groups: finitely generated abelian groups, group actions on sets, Sylow theory, cyclic groups, symmetric group, diedral groups, orthogonal and unitary groups, topological groups; Linear representations of finite groups: permutation representation, regular representation, irreducible representation, character of a representation, orthogonality of characters, theorems of Maschke and Frobenius, Burnside formula, applications.

(P. Seppecher) Career of researchers in mathematics, structuration of research in France, publishing techniques. Management of a research project. As an example: complete study of a transport problem (bibliography, optimization methods, convexity, measure theory, academic examples, numerical methods, description of self-similar solutions, publication).

(W. Aschbacher - 2014/15) Matrix Lie groups: definitions, classical groups, compactness, connectedness, homomorphisms; Lie algebras and the exponential mapping: matrix exponential, Lie algebras, abstract Lie algebras, complexification; Lie algebras vs. Lie groups:

Baker-Campbell-Hausdorff formula, Lie group and Lie algebra homomorphisms; Basic representation theory: defintions, examples, Schur's lemma, direct sum of representations;

Irreducible representations of SU(2): construction of some representations of SU(2), irreducible representations of su(2), representations of Lie groups vs. representations of Lie

algebras.

**2nd Semester**

(W. Aschbacher) Spaces of test functions: locally convex and separated topological vector spaces, convergence and continuity, most important test function spaces; Distributions:

defintions, convergence of of distributions; Basic operations on distributions: derivatives, multiplication by a function, support and singular support; Convolution: convolution of functions, regularisation, convolution of distributions; Fundamental solutions: definition, fundamental solutions of important differential operators; Tempered distributions: Fourier transform, tempered distributions, Fourier transform of tempered distributions, applications.

(G. Bouchitté - 2013/14) Distributions on R^N: jump formula, PDEs; Fourier transform in L^2:

Plancherel theorem, periodic distributions, Fourier coefficients; Introduction to Sobolev spaces: variational formulation of boundary value problems, Lax-Milgram theorem.

(C. Galusinski) Approximation of elliptic PDEs: finite difference, finite element, and finite volume method; Evolution problems and stability: parabolic and hyperbolic problems;

Applications to image restoration.

**Optimization - 3 ECTS -- 30h**

(J.-J. Alibert) Minimization of convex functionals (in particular of positive quadratic functionals) over closed convex subsets of a Hilbert space, characterization of the solutions by means of variational inequalities; Lebesgue and Sobolev spaces (in 1 dimension) endowed with a Hilbert space structure; Exhaustive study of a large number of optimization problems.

**Mathematical Physics - 3 ECTS -- 30h**

(P. Briet) Recalls on Fourier transformation; Basic axioms of non relativistic quantum mechanics: state space; Unbounded self-adjoint linear operators and observables:

position and momentum observables; Quantum Hamiltonian and examples;

Basic notions of perturbation theory; One-dimensional quantum systems: construction of bound states and discret spectrum, construction of wave packages and continuous spectrum, time evolution.

(M. Rouleux) Spin models from the viewpoint of rigorous Statistical Mechanics. The mean field model for scalar spins on Z^d: magnetisation, thermodynamical limit; The microcanonical ensemble: review on probability theory, Gibbs postulate, statistical entropy,

sub-additivity, concavity, the maximum entropy criterion, partition function, basics of statistical thermodynamics; Other examples in the discrete case: 2-level systems, energy exchange; Introduction to Ising model; Spins with continuous symmetry on a 2-D lattice:

Villain model, behavior at high temperature, decay of correlations; Outline of the quantum case: Von Neumann entropy.

(T. Champion) Numerical optimization*. *Theoretical and numerical aspects; Optimization with and without constraints, optimality conditions, Kuhn-Tucker theorem; Descent algorithms: gradient, conjugate gradient, Newton, quasi-Newton.

(M. Ersoy - 2014/16) Numerical optimization*. *Theoretical and numerical aspects; Optimization with and without constraints, optimality conditions, Kuhn-Tucker theorem; Descent algorithms: gradient, conjugate gradient, Newton, quasi-Newton.

**TER (Master thesis) - 5 ECTS**

The Master thesis has to be written within six weeks' time. You can choose your advisor from a research lab (CPT, IMATH, etc.), from an engineering school (SeaTech, etc.), or from an exterior company.

**English - 2 ECTS -- 18h**

See 1st Semester.

**UE10** : The Master thesis (TER – Travail Encadré de Recherche — advised research work) has to be written within six weeks’ time. You can choose your advisor from a research lab (CPT, IMATH, etc.), from an engineering school (SeaTech, etc.), or from an exterior company.