1st Semester<\/strong><\/em><\/p>\n (J. Asch<\/a>, C.-A. Pillet<\/a>) Local and global curve theory, classical surface theory, the inner geometry of surfaces, theorema egregium, geometry and topology, Gauss-Bonnet theorem.<\/p>\n <\/div>\r\n \r\n (P. Briet<\/a>) 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.<\/p>\n <\/div>\r\n \r\n (S. Vaienti<\/a>) 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.<\/p>\n <\/div>\r\n \r\n English - 2 ECTS -- 18h<\/strong><\/p>\n (F. Armao<\/a>) 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). TICE - 1 ECTS -- 10h<\/strong><\/p>\n (G. Faccanoni<\/a> - 2014\/15) Learning LaTeX: a software package particularly well-suited to prepare documents containing mathematical formulas.<\/p>\n <\/div>\r\n \r\n (Y. Aubry<\/a>) 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>\n (P. Seppecher<\/a>) 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).<\/p>\n (W. Aschbacher<\/a> - 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: 2nd Semester<\/strong><\/em><\/p>\n (W. Aschbacher<\/a>) Spaces of test functions: locally convex and separated topological vector spaces, convergence and continuity, most important test function spaces; Distributions: definitions, 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.<\/p>\n (G. Bouchitt\u00e9<\/a> - 2013\/14) Distributions on R^N: jump formula, PDEs; Fourier transform in L^2: (C. Galusinski<\/a>) Approximation of elliptic PDEs: finite difference, finite element, and finite volume method; Evolution problems and stability: parabolic and hyperbolic problems; Optimization - 3 ECTS -- 30h<\/strong><\/p>\n (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.<\/p>\n
\nAdditionally, 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
\n
\n
\nBaker-Campbell-Hausdorff formula, Lie group and Lie algebra homomorphisms; Basic representation theory: defintions, examples, Schur's lemma, direct sum of representations;
\nIrreducible representations of SU(2): construction of some representations of SU(2), irreducible representations of su(2), representations of Lie groups vs. representations of Lie
\nalgebras.<\/p>\n <\/div>\r\n <\/div>\r\n\r\n\r\n\r\n <\/div><\/p>\n
\n
\nPlancherel theorem, periodic distributions, Fourier coefficients; Introduction to Sobolev spaces: variational formulation of boundary value problems, Lax-Milgram theorem.<\/p>\n <\/div>\r\n \r\n
\nApplications to image restoration.<\/p>\n <\/div>\r\n \r\n