3rd Semester<\/strong><\/em><\/p>\n (C-A. Pillet<\/a>) C* algebras: definitions, spectral analysis, representations and states; W* algebras: operator topologies, commutant; Tomita-Takesaki theory: modular operators, modular group.<\/p>\n <\/div>\r\n \r\n (A. Novotny<\/a>) 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, (S. Vaienti<\/a>) 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 Poincar\u00e9-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 Poincar\u00e9 theorem; Ergodicity and mixing: definitions and main properties; (Y. Aubry<\/a>) 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>\n <\/div>\r\n \r\n (A. Panati<\/a>) Compact operators, closed operators (closure), adjoint of a closed operator, selfadjoint extensions, Dunford calculus, Stone theorem, elements from spectral theory for closed (selfadjoint) operators, nature of the spectrum.<\/p>\n (P. Briet<\/a>) Perturbation theory: caracterization of selfadjoint operators, relatively bounded and relatively compact operators. Kato-Rellich theorem. Spectral localization of selfadjoint operators : essential spectrum (Weyl's caracterization), discrete spectrum. Weyl's stability theorem. Theory of regular perturbations.<\/p>\n <\/div>\r\n \r\n (T. Champion<\/a>) We are interested in numerical analysis of convex optimization problem (with or without constraint) in infinite dimension. Existence results are proposed as well as algorithms to solve such problems.<\/p>\n <\/div>\r\n \r\n Student seminar - 2 ECTS In this seminar, each student has to present to the class a little chapter, research article or modelization problem proposed by one of the teachers of the classes.<\/p>\n Scientific english - 3 ECTS -- 18h<\/strong><\/p>\n 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. 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.<\/p>\n Technics for job search - 1 ECTS - 10h<\/strong><\/p>\n Course under the responsability of Service d\u2019Accompagnement en Orientation et Insertion (SAOI).<\/p>\n <\/div>\r\n <\/div>\r\n\r\n\r\n\r\n <\/div>\n 4th Semester<\/strong><\/em><\/p>\n This module is made of two parts, each devoted to a recent research topic of the research laboratories CPT and IMATH.<\/p>\n Applied analysis - 5 ECTS -- 20h<\/strong><\/p>\n Geometric control (F. Chittaro<\/a>)<\/em> Controlability, optimal control.<\/p>\n
\nregularity of weak solutions, maximum principle, eigenvalues and eigenvectors, Galerkin method.<\/p>\n <\/div>\r\n \r\n
\nSimple systems: irrational rotations, symbolic dynamics and sub shift of finite type,
\nBernoulli and Markov maps with the corresponding measures, proof of invariance, ergodicity and (eventually) mixing for the preceding transformations;
\nProof 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
\noperator; 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.<\/p>\n <\/div>\r\n \r\n
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