Séminaires

La construction des nombres surréels de Conway donne également une nouvelle approche aux nombres réels, que je qualifierais de « quantique », opposée à  l’approche « usuelle », que je qualifierais de « classique ». J’essaierai d’expliquer cette opposition « quantique-classique » en la mettant dans le contexte de la théorie des ensembles (je résumerai en partie mon cours « Ensembles et nombres » que je donne actuellement dans le cadre de notre école doctorale). Cette séance se veut plus un forum de discussion qu’une présentation formelle de résultats : le sujet du « quantique » est pour ainsi dire la musique de fond du travail pour beaucoup d’entre nous, et il me semble intéressant de poser et de discuter des questions sur ce fond. 

Dans un travail récent, McNew, Pollack et Singha Roy obtiennent plusieurs résultats relatifs à la distribution du facteur premier médian des entiers lorsque ce dernier est défini en tenant compte de la multiplicité. En particulier, le comportement asymptotique des lois locales est étudié et fait apparaître une transition de phase qui n’est pas décrite. Dans cet exposé, nous présenterons une partie des améliorations et des résultats obtenus pour les lois locales et certaines applications.

In this talk, we will discuss diophantine approximations of irrational numbers by rational numbers, where the denominators are taken from certain interesting subsets of the positive integers. First, we will consider Diophantine approximations in which the denominators are drawn from the set of positive integers represented by a given positive definite integral binary quadratic form. Next, we will discuss Diophantine approximations where the denominators are restricted to the set of y-smooth (or friable) numbers for some given y > 0. Finally, we will outline some of the proofs.

Assume that $\omega_1, \ldots, \omega_n$ are i.i.d. uniform random points in $[0,1]$, which serve as the centers of shrinking intervals of given lengths $\ell_1 \ge \cdots \ge \ell_n$. The Dvoretzky covering problem asks for necessary and sufficient conditions on the sequence $(\ell_n)$ under which these random intervals cover $[0,1]$ infinitely often, almost surely. The problem was solved in 1972 by Shepp, and his work has since been generalized in several directions.

In this talk, I will discuss some deterministic analogues of Shepp’s result and their applications to the Littlewood Conjecture.

[Exposé en ligne diffusé dans la salle de séminaire]

A common question for geometric structures is that of rigidity. Infinitesimally rigidity is often controlled by the vanishing of a cohomology group. A common question then becomes when infinitesimal rigidity actually implies rigitidy. In this talk we discuss the case of regular foliations.

In the first part we define the terms involved and give an overview of the current state of the art. Moreover, we highlight a relation with the rigidity of group actions.

One encounters two issues in the literature. Many results require the foliation to have compact leaves and there is a general lack of examples.

In the second part of the talk, we take a step towards addressing those issues, by outlining a construction of infinitesimally rigid foliations with dense leaves. This is based on joint work with Stephane Geudens.

In this talk I will introduce Quantum Field Theory, its Path Integral formulation, and Segal’s Axioms. Then I will indicate some recent progress on rigorous construction of some concrete models using probability. After that I slightly extend Segal’s framework and show how it can be related to a useful physical quantity called “entanglement entropy”, where I suggest a geometric way of rigorously deriving relevant formulae based on recent works of the speaker and B. Estienne (2501.19014).

Nous développons une nouvelle technique pour minorer des sommes d’exponentielle de la forme $\sum f(n) e^{2i\pi\alpha n}$ pour tout $\alpha.$
Nous montrerons en particulier que la somme $\sum_{p\le x} e^{2i\pi\alpha p}$ est non bornée pour tout $\alpha,$ et plus précisément diverge au moins comme $x^{1/6-\varepsilon}$ pour une suite de $x$ tendant vers l’infini, uniformément en $\alpha.$

The geometry of the flag manifold G/P associated to a parabolic subgroup P of a semisimple Lie group G gives rise to a G-equivariant complex of elliptic operators (the BGG complex) which satisfy the property of maximal hypoellipticity, as shown by the work of Dave and Haller. In the case of the group SU(n,1), the BGG complex is the Rumin complex associated to the contact structure on the sphere S^{2n-1}. We shall describe the quaternionic analogue (i.e. for G=Sp(n,1)), and compute the bundles and operators involved in the BGG complex thanks to the Kostant theorem of 1961 (generalized by Cap and Slovak) on the cohomology of the maximal nilpotent subalgebra of a parabolic subalgebra of a semisimple Lie algebra.
We shall also explain that, in the context of Non Commutative Geometry, the BGG complexes are a crucial ingredient for the construction of the so called Kasparov gamma-element, which is the obstruction to the subjectivity of the Baum-Connes map.

The phenomenon known as ‘the Mackey analogy’ is a correspondance between the tempered dual of a Lie group $G$ and the unitary dual of its associated motion group $G_0$.  The precise statement, formulated and proven by Higson in the case of complex groups, and by Afgoustidis in the general case of real groups, has long been known to be intimately connected to the Connes-Kasparov isomorphism relating the K-theory of the reduced C$^\ast$-algebras of $G$ and $G_0$ via a deformation argument.
In this talk, I will report on joint work with Higson and Román, aimed at understanding the Mackey analogy directly at the level of the group C$^\ast$-algebras.  The main result is an embedding of the C$^\ast$-algebra of $G_0$ into the reduced C$^\ast$-algebra of $G$, which is proven to characterize the bijection of Afgoustidis and to induce the Connes-Kasparov isomorphism.

Anton Kapustin, Nikita Sopenko and others have studied the group of “locally generated automorphisms” of a UHF C*-algebra whose matrix algebra factors are labelled by the points of a coarse space.  This talk will be about a related Lie algebra of “locally generated derivations.”  Specifically it will be about the problem of computing the homology of this Lie algebra.  I shall describe a set of techniques that may be employed to solve the problem, at least for lattices in Euclidean space and similar spaces. Some are from the coarse Baum-Connes conjecture, and some, from homological algebra, are much older. This is joint work with Tsuyoshi Kato.