[{"command":"insert","method":null,"selector":".course-modal-ajax-placeholder-957865","data":"\u003Cdiv class=\u0022container content-rte\u0022\u003E\n \u003Cdiv class=\u0022row\u0022\u003E\n \u003Cdiv class=\u0022box col-md-6 bg-primary text-white\u0022\u003E\n \u003Ch2 class=\u0022h1 text-left\u0022\u003E Analyse globale et diffusion 2025-2026\n\u003C\/h2\u003E\n \u003Cdiv class=\u0022field-spacing\u0022\u003E\n \u003Cdiv class=\u0022field__label h3 dash\u0022\u003EECTS :\n 7.5\n \u003C\/div\u003E\n \u003C\/div\u003E\n \u003Cdiv class=\u0022field-spacing\u0022\u003E\n \u003Cdiv class=\u0022field__label h3 dash\u0022\u003ESemestre calendaire :\n Semester 1\n \u003C\/div\u003E\n \u003C\/div\u003E\n \u003Cdiv class=\u0022field-spacing\u0022\u003E\n \u003Cdiv class=\u0022field__label h3 dash\u0022\u003ED\u00e9tail du volume horaire\u003C\/div\u003E\n \u003Cdiv class=\u0022mt-5 text-justify\u0022\u003E\n \u003Cspan class=\u0022field__label h4 mt-5 \u0022\u003ECours magistraux :\u003C\/span\u003E\n \u003Cspan\u003E25\u003C\/span\u003E\n \u003C\/div\u003E\n \u003Cdiv class=\u0022mt-5 text-justify\u0022\u003E\n \u003Cspan class=\u0022field__label h4 mt-5 \u0022\u003ETravaux dirig\u00e9s :\u003C\/span\u003E\n \u003Cspan\u003E12.5\u003C\/span\u003E\n \u003C\/div\u003E\n \u003C\/div\u003E\n \u003Cdiv class=\u0022field-spacing\u0022\u003E\n \u003Cdiv class=\u0022field__label h3 dash\u0022\u003ELangue d\u0026#039;enseignement\u003C\/div\u003E\n \u003Cdiv class=\u0022mt-5 text-justify\u0022\u003E\n \u003Cspan\u003EAnglais\u003C\/span\u003E\n \u003C\/div\u003E\n \u003C\/div\u003E\n \u003Cdiv class=\u0022field-spacing\u0022\u003E\n \u003Cdiv class=\u0022field__label h3 dash\u0022\u003EEnseignement \u00e0 distance\u003C\/div\u003E\n \u003Cdiv class=\u0022mt-5 text-justify\u0022\u003E\n \u003Cspan\u003Enon\u003C\/span\u003E\n \u003C\/div\u003E\n \u003C\/div\u003E\n \u003C\/div\u003E\n \u003Cdiv class=\u0022box col-md-6\u0022\u003E\n \u003Cdiv class=\u0022field-spacing\u0022\u003E\n \u003Cdiv class=\u0022field__label h3 dash\u0022\u003EPr\u00e9requis\u003C\/div\u003E\n \u003Cdiv class=\u0022mt-5 text-justify\u0022\u003E\n \u003Cspan\u003E\u003Cp\u003ELes pr\u00e9requis sont ceux d\u2019une premi\u00e8re ann\u00e9e de master de math\u00e9matiques fondamentales avec notamment la th\u00e9orie de la mesure, calcul diff\u00e9rentiel et topologie, analyse de Fourier, distributions et \u00e9l\u00e9ments de th\u00e9orie des op\u00e9rateurs.\u0026nbsp;\u003C\/p\u003E\n\u003Cp\u003EThe prerequisites correspond to those of a first year Master\u0027s degree in pure mathematics, including measure theory, differential calculus and topology, Fourier analysis, distributions, and elements of operator theory.\u003C\/p\u003E\u003C\/span\u003E\n \u003C\/div\u003E\n \u003C\/div\u003E\n \u003Cdiv class=\u0022field-spacing\u0022\u003E\n \u003Cdiv class=\u0022field__label h3 dash\u0022\u003EProgramme \/ plan \/ contenus\u003C\/div\u003E\n \u003Cdiv class=\u0022mt-5 text-justify\u0022\u003E\n \u003Cspan\u003E\u003Cp\u003ECe cours fondamental aborde des sujets qui prolongent la formation en analyse (analyse des EDP lin\u00e9aires et non-lin\u00e9aires, analyse et g\u00e9om\u00e9trie\u2026) et qui peut d\u00e9boucher sur beaucoup de sujets pour une poursuite en doctorat. On s\u0027int\u00e9resse aux interactions entre l\u0027\u00e9quation de la chaleur sur l\u0027espace euclidien et l\u0027existence d\u0027in\u00e9galit\u00e9s fonctionnelles d\u0027origine g\u00e9om\u00e9trique. Pour ce faire, on abordera la notion de noyau de la chaleur et l\u0027on \u00e9tudiera le comportement en temps longs d\u0027une telle \u00e9quation. L\u0027unicit\u00e9 des solutions sous certaines contraintes \u00e0 l\u0027infini sera \u00e9galement trait\u00e9e et sera le pr\u00e9texte pour \u00e9tablir des principes du maximum scalaires puis sur les tenseurs sym\u00e9triques : on verra ses cons\u00e9quences \u00e0 travers les in\u00e9galit\u00e9s de Li-Yau et d\u0027Hamilton. La notion d\u0027in\u00e9galit\u00e9s de Harnack paraboliques sera centrale ici. Enfin, nous \u00e9tablirons les liens entre bornes ponctuelles sur le noyau de la chaleur et l\u0027existence d\u0027in\u00e9galit\u00e9s fonctionnelles (in\u00e9galit\u00e9 de Sobolev, de log-Sobolev et de Nash entre autres). Ce cours est une initiation \u00e0 des techniques d\u0027analyse globale qui ne sont pas sensibles \u00e0 la structure euclidienne et qui permettent de s\u0027appliquer \u00e0 des espaces non lin\u00e9aires telles que les vari\u00e9t\u00e9s riemanniennes.\u003C\/p\u003E\n\u003Cp\u003EThis fundamental course covers topics that extend analysis courses (analysis of linear and nonlinear PDEs, analysis and geometry, etc.) and can lead to a wide range of subjects suitable for pursuing a PhD. The course focuses on the interactions between the heat equation on Euclidean space and the existence of functional inequalities with a geometric flavor. To this end, we will introduce the concept of the heat kernel and study the long-time behavior of such an equation. The uniqueness of solutions under certain conditions at infinity will also be addressed, serving as a starting point for establishing maximum principles\u2014first in the scalar case, then for symmetric tensors. The consequences of these principles will be illustrated through the Li\u2013Yau and Hamilton inequalities. The concept of parabolic Harnack inequalities will play a central role. Finally, we will explore the connections between pointwise bounds on the heat kernel and the existence of functional inequalities (including Sobolev, log-Sobolev, and Nash inequalities, among others). This course serves as an introduction to techniques in global analysis that are independent of Euclidean structure and can be applied to nonlinear spaces such as Riemannian manifolds.\u003C\/p\u003E\u003C\/span\u003E\n \u003C\/div\u003E\n \u003C\/div\u003E\n \u003Cdiv class=\u0022field-spacing\u0022\u003E\n \u003Cdiv class=\u0022field__label h3 dash\u0022\u003ENature de l\u0026#039;\u00e9valuation\u003C\/div\u003E\n \u003Cdiv class=\u0022mt-5 text-justify\u0022\u003E\n \u003Cspan\u003EEvaluation Continue non Int\u00e9grale\u003C\/span\u003E\n \u003C\/div\u003E\n \u003C\/div\u003E\n \u003C\/div\u003E\n \u003C\/div\u003E\n\u003C\/div\u003E\n","settings":null}]