## Contents of the courses

**3rd Semester**

(W. Aschbacher) C* algebras: definitions, spectral analysis, representations and states; W* algebras: operator topologies, commutant; Tomita-Takesaki theory: modular operators, modular group.

(A. Novotny) Hilbert spaces, orthogonal projections, dual spaces and representation theorem, strong and weak convergence, weak topology, closed convex sets, Lax-Milgram theorem, Stampacchia theorem, compact operators, Fredholm theory, Sobolev spaces (density, continuous and compact injections, traces), second-order elliptic partial differential equations, variational formualtion, weak solutions, energy inequality, Fredholm alternative,

regularity of weak solutions, maximum principle, eigenvalues and eigenvectors, Galerkine method.

(S. Vaienti) Ordinary differential equations: qualitative study, complete study of linear systems in any dimension, classifications of equilibrium points, local stability of sinks, a simple versions of the Hartman-Grobman theorem, proof of the Poincaré-Bendixons theorem, flows on the torus, introduction to local bifurcation theory; Discrete dynamical systems: topological dynamics, orbits transitivity and non-wandering sets, invariant measures, Krylov-Bogolyubov theorem on the existence of invariant measures, recurrence and Poincaré theorem; Ergodicity and mixing: definitions and main properties;

Simple systems: irrational rotations, symbolic dynamics and sub shift of finite type,

Bernoulli and Markov maps with the corresponding measures, proof of invariance, ergodicity and (eventually) mixing for the preceding transformations;

Proof of Birkhoff's theorem: ergodicity of other simple systems like algebraic automorphisms of the torus and skew-systems; Introduction to the spectral theory of the Perron- Frobenius

operator; Entropy, Lyapunov exponents and dimensions: sketch the theory of entropy and Lyapunov exponents in dimension 2, introduction to fractal analysis and its links with dynamics; Statistical properties of dynamical systems: limit theorems, central limit theorem, large deviations extreme values theory, introduction to random dynamical systems.

(Y. Aubry) Finite fields: field theory, construction of finite fields, Wedderburn theorem, Frobenius endomorphism, factorisation of polynomials, cyclotomic cosets, equations over finite fields, Legendre and Jacobi symbols, quadratic reciprocity law; Algebraic geometry: projective spaces, smooth absolutely irreducible projective algebraic curves defined over finite fields, genus of a plane curve, rational points, zeta function, Riemann hypothesis, Serre-Weil bound.

(P. Briet) Densely defined operators, closed operators (closure), adjoint of a closed operator, selfadjoint extensions of symmetric operators, elements from spectral theory for closed (selfadjoint) operators; Discrete spectrum, Weyl's characterization of the essential spectrum;

Spectral measures: definition, spectral theorem, absolutely continuous spectrum; Relatively bounded operators, relatively compact operators, Weyl's stability theorem.

(M. Rouleux) Perturbation theory: spectral stability for self-adjoint operators; finite dimensional case: Puiseux series; introduction and examples: regular perturbation theory, perturbation of the Laplace operator by a confining square inegrable potential, singular perturbation theory, anharmonic oscillator; perturbation series (regular case): analytic familes of type A, Kato-Rellich theorem, perturbation of the ground state, Rayleigh-Schrodinger series, application to the spectrum of Helium atom; singular perturbation theory: asymptotic series, Rayleigh-Ritz techniques, return to the an harmonic oscillator.

(C. Galusinski) We are interested in numerical analysis of convex optimization problem under constraint, for infinite dimensional problem. Existence results are proposed as well as algorithms to solve such problems. Fluid mechanic problems under incompressibility constraint are treated by introducing Lagrangians. Algorithms based on optimization are compared to projection methods.

(C. Galusinski - 2014/15) This lecture (Partial differential equations of physics), shared with the school of engineering SeaTech, addresses various partial differential equations such as the heat equation, the equations from fluid mechanics, the equation for surface waves... For the different classes of partial differential equations treated (parabolic equations, hyperbolic equations, dispersive equations), numerical methods are presented and implemented on the computer.

**4th Semester**

**Option 1 - 5 ECTS -- 20h**

**Duality methods in mass transport theory. **

(G.Bouchitte) We will illustrate how classical tools of functional analysis, such as duality and convex analysis, can be efficiently used in order to treat important problems appearing in optmal transport theory. After revisiting some basic tools in measure theory and functional analysis, we will introduce a class of optimal transports problems in R^d associated with a general cost c(x,y): R^d x R^d -> [0,+infty] for which a duality principle is established. In the quadratric case c(x,y)= |x-y|^2, we we will derive the existence of a unique optimal transport map and recover the celebrated Brenier's polar factorization Theorem. We will also present an equivalent time dependent formulation.

The second part of the course will be devoted to the Monge cost c(x,y)=|x-y| and the construction of optimal transport plans.

Eventually some applications will be presented as isoperimetric inequalities, optimal location problems, optimal design.

**Option 2 - 5 ECTS -- 20h**

**Geometrical and topological methods in quantum physics.**

(J. Asch) The goal of the lecture consists in providing the student with selected mathematical tools of modern quantum physics.

**Hyperbolic problems, finite volumes.**

(M. Ersoy) Conservation laws, theoretical and numerical analysis of hyperbolic equations and of systems of hyperbolic equations (transport, Burgers, traffic, Saint-Venant, gas dynamics,...), existence and uniqueness, weak solutions, weak entropy solutions, Kruzkhov theorem, Rankine-Hugoniot jump conditions, Lax criterion, Riemann problem, conservative numerical schemes, first order finite-volume method, numerical stability, consistency, monotony, total variation diminishing, entropy, convergence, exact and approximate Riemann solvers.

**Introduction to Monte-Carlo methods.**

(S. Maire) Basic concepts; review of probability theory: law of large numbers, central limit theorem; Introduction to the method using some examples: computing volumes, integrals, escape times of graphs, European options; Random generators of different laws: inversion method, rejection sampling, variance reduction, Quasi-Monte Carlo method, uncertainty quantification; Monte Carlo methods for diffusion; Markov chaines and finite differences for the Laplacian; Poisson equation in bounded domains: Euler method, random walk on spheres; More general diffusion, evolution problems (heat equation); Rare events: splitting method for the computation of the principal eigensystem.

The Master thesis has to be written within 3 months' time. You can choose your advisor from one of the two research labs CPT (located on the campus of La Garde and on the campus of Luminy inMarseille) and IMATH (located on the campus of La Garde), or from an exterior company.

In the context of the Master thesis (TER), not only the student is led to read scientific documents in English, but he also has to write the introduction of the thesis as well as the beginning of his defense in that language.

## Remark UE7

This compulsory UE, typically divided into two parts, consists of 40 hours. According to the two thematic orientations *Mathematical physics* and *Optimization and applied analysis*, the corresponding lectures may be shared with the Master Theoretical and mathematical physics, particle physics, and astrophysics (P3TMA) of Aix Marseille University, with the Master Mathematics of AixMarseille University and the University of Nice, and with the school of engineering SeaTech, respectively.

## Remark Master Thesis (UE8)

The Master thesis has to be written within 3 months’ time. You can choose your advisor from one of the two research labs CPT (located on the campus of La Garde and on the campus of Luminy inMarseille) and IMATH (located on the campus of La Garde), or from an exterior company.