Ce séminaire a été créé en pensant aux mathématicien·ne·s, et notamment aux plus jeunes. Des collègues y présentent le contexte mathématique de certains exposés du Séminaire Bourbaki, pour les rendre plus accessibles ; ils pourront aussi en introduire quelques outils ou des motivations plus lointaines.
L'entrée est libre.
Le Séminaire Bourbaki du vendredi a lieu à l'Institut Henri Poincaré (IHP, amphithéatre Charles Hermite), 11 rue Pierre et Marie Curie, Paris 5e. — [iCal] [Affiche] [Résumés]
In this seminar, I will provide an overview of the so-called geometric methods in Arnold diffusion. Arnold diffusion occurs in perturbations of integrable systems. In such systems, when written in action-angle variables, the action does not evolve under the flow (it is a first integral of motion). Roughly speaking, a system undergoes Arnold diffusion when, for arbitrarily small "typical" perturbations, the change in actions becomes of order one (independent of the parameter values). Several methods have been used to detect this phenomenon. In this talk, we will review the geometric methods, introducing concepts such as Normally Hyperbolic Invariant Manifolds (NHIM) and their stable and unstable manifolds. When these manifolds intersect along a "homoclinic channel", one can define the "Scattering map", which encodes the heteroclinic connections between points on the NHIM. By combining iterations of the Scattering map with the internal dynamics of the NHIM, we will find "pseudo-orbits" along which the action increases. These orbits will be followed by real orbits where the action also increases. Finally, we will apply this methodology to the famous example provided by Arnold in 1964.
The Langlands correspondence was originally formulated as a conjectural link between representations of Galois groups of number fields and automorphic data. Translating these conjectures into statements for function fields of curves many of the objects admit natural geometric interpretations and the geometry then helps to prove results. For example some expected invariance properties of functions turned into descent properties of sheaves that sometimes follow from topological considerations. In this talk we will try to give some background on the first examples (\(GL_1\) and \(GL_2\)) where this had been observed first by Deligne and Drinfeld and introduce some of the structures that occur in the more general results.
En 1917, Kakeya demandait quelle est l’aire minimale pour retourner une aiguille. Au même moment, pour résoudre un problème d’analyse réelle, Besicovitch démontrait qu’il existait des ensembles d’aire nulle qui contiennent une droite dans chaque direction. Le problème initial de Kakeya était résolu, mais le problème est devenu : quelle est la dimension d’un ensemble contenant une droite dans chaque direction dans l’espace euclidien de dimension n ? Nous expliquerons la solution pour \(n=2\), et le lien entre ce problème et diverses questions en analyse harmonique, équations aux dérivées partielles, analyse complexe, théorie géométrique de la mesure, ... En particulier, nous verrons le lien avec des problèmes "simples" de géométrie du plan, non résolus.
Le séminaire Bourbaki a été fondé en 1948. Au rythme de quatre séances par an, il tente d'offrir un panorama des développements mathématiques actuels.
L'entrée est libre.
Le Séminaire N. Bourbaki a lieu à l'Institut Henri Poincaré (IHP, amphithéatre Charles Hermite), 11 rue Pierre et Marie Curie, Paris 5e. — [iCal] [Affiche] [Résumés]
Le problème planétaire à \(1+n\) corps est un système d’équations différentielles modélisant l’évolution du système solaire, à savoir le mouvement de \(n\) corps (planètes) en interaction gravitationnelle autour d’un corps plus massif (Soleil). Les mathématiciens et astronomes ont cru pendant longtemps à sa stabilité: les trajectoires elliptiques (képlériennes), présentes lorsque les planètes n’interagissent pas entre elles, ne sont que légèrement déformées lorsque l’on restaure cette interaction (théorème de stabilité de Laplace–Lagrange). Mais depuis les travaux de Poincaré et d’Arnold, on s’attend au contraire à de l’instabilité, bien que cela ait résisté aux efforts des mathématiciens. Le but de l’exposé est d’expliquer un résultat de Clarke, Fejoz et Guardia qui montre que pour le problème à \(1+n=4\) corps (ou plus), il existe des mouvements dont le demi-grand axe d’un des corps a une variation aussi grande que l’on veut. Ceci montre que la conclusion du théorème de stabilité de Laplace–Lagrange n’est pas valide pour le problème planétaire, et résout une conjecture d’Arnold.
The Kakeya conjecture is a question about how thin tubes can overlap with each other in Euclidean space. It connects to problems in Fourier analysis and PDE, such as understanding \(L^p\) type estimates for solutions of the wave equation. This connection has prompted a lot of work on the problem in the harmonic analysis community.
The \(2\)-dimensional case of the Kakeya conjecture has a \(1\)-page proof and has been known since the 1970s. Recently, Wang and Zahl proved the \(3\)-dimensional case of the Kakeya conjecture. Their work builds on contributions of many people, including Bourgain, Wolff, Katz, Laba, Tao, Orponen, and Shmerkin.
There are interesting examples related to algebraic geometry which show that some cousins of the Kakeya conjecture are false. These counterexamples are also important clues for understanding the problem. In the proof of Kakeya, we imagine a hypothetical counterexample, we prove that it would have to have a great deal of algebraic structure, and we finally show that it cannot exist at all. A hypothetical counterexample to the Kakeya conjecture is just a set of thin tubes in \(3\)-dimensional space which overlap each other a lot. We will try to describe how to prove that such a set has algebraic structure.
In 1983, Henri Cohen and Hendrik Lenstra proposed a conjecture about the distribution of the \(N\)-torsion of the class group of a random quadratic field, supported by what was at the time a large amount of computational evidence. The Cohen–Lenstra heuristics, which are still almost entirely unproven, have become one of the central foundational problems in arithmetic statistics. Recent years have seen a rapidly accelerated pace of development in Cohen-Lenstra problems. I will give a tour of these developments, including the work of Wood and her collaborators developing a fully fleshed out roster of generalized Cohen–Lenstra conjectures, with support from topology; Smith’s theorems proving the Cohen–Lenstra conjectures for the 2-primary part of the class group, as part of more general theorems about Selmer groups in quadratic twists, leading to a resolution of the minimalist conjecture for elliptic curves; and recent work by Koymans and Pagano in the \(2\)-primary case, expanding on Smith’s work and proving Stevenhagen’s conjecture on the negative Pell equation.
At its core, the Langlands program seeks to give a description of the vector space of automorphic forms. This is a space of functions on a locally symmetric space \(X/\Gamma\), where \(\Gamma=G(\mathbb Z)\subset G(\mathbb R)\) is an arithmetic group, equipped with its Hecke symmetries. One seeks a “dual” description, in terms of \(L\)-parameters, which are roughly representations of the absolute Galois group with values in the Langlands dual group.
Similar conjectures exist in the function-field case; in this setting, the quotient \(X/\Gamma\) can be interpreted as the \(\mathbb F_p\)-points of the stack \(\mathrm{Bun}_G\) of \(G\)-bundles on a curve \(C\) over \(\mathbb F_p\). Gaitsgory–Raskin, partly in large collaborations, proved a fine version of Langlands’ conjecture in this setting, in the everywhere unramified case. Notably, they define a vector space on the Galois side, as global sections of a sheaf on the moduli stack of \(L\)-parameters, and relate this to the space of automorphic forms, compatibly with the Hecke symmetries.
Their proof is a culmination of many decades of work in the geometric Langlands program, and consists of two key steps: 1) The space of automorphic forms is the trace of Frobenius on a category of \(\ell\)-adic sheaves on \(\mathrm{Bun}_G\). 2) This category of \(\ell\)-adic sheaves on \(\mathrm{Bun}_G\) is equivalent to a category of coherent sheaves on the stack of \(L\)-parameters. The starting point for 2) is the observation that this conjecture makes sense for curves over any field, and any sheaf theory, and is known as the geometric Langlands equivalence. The proof starts over the complex numbers, and uses \(D\)-modules.
Des brochures contenant les exposés du Séminaire Bourbaki seront distribuées au début de chaque séance.
Pour recevoir à l'avance le programme et les résumés de chaque séminaire, veuillez vous abonner en
envoyant un mail à
Pour recevoir les annonces des prochains séminaires : écrire un message à
cette adresse.
Vous pouvez aussi ajouter à vos calendriers électroniques les agendas hébergés sur le portail Indico : Séminaire Bourbaki du vendredi et Séminaire Bourbaki (format iCalendar)
Un soutien du CNRS couvre une partie des frais d'organisation de ce Séminaire.