[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).

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.

We construct a Poisson algebra bundle whose distributional sections are suitable to represent multilocal observables in classical field theory. To do this, we work with vector bundles over the unordered configuration space of a manifold M and consider the structure of a 2-monoidal category given by the usual (Hadamard) tensor product of bundles and a new (Cauchy) tensor product which provides a symmetrized version of the usual external tensor product of vector bundles on M.

In « $L_\infty$-algebras and higher analogues of Dirac structures and Courant algebroids » (arXiv:1003.1004), Marco Zambon constructed an explicit $L_\infty$-morphism between the Courant $r=1$-Lie algebra of a smooth manifold $M$, twisted by a closed 2-form $\sigma$, and the untwisted Courant $r=2$-Lie algebra of $M$. He left open the question of whether similar canonical $L_\infty$-morphisms exist in higher degrees — that is, between the Courant $r$-Lie algebra twisted by a closed $(r+1)$-form $\sigma$ and the untwisted Courant $(r+1)$-Lie algebra. In this talk, we present a general framework that naturally accounts for the existence of such morphisms for arbitrary $r$. This is joint work with Domenico Fiorenza.

Les surfaces minimales sont les surfaces qui sont points critiques de la fonctionnelle d’aire à bord fixé. Elles sont caractérisées par le fait que leur courbure moyenne est nulle. Un problème posé par Ricci est de déterminer quelles surfaces riemanniennes peuvent être immergées (localement) isométriquement comme surfaces minimales de l’espace euclidien de dimension 3. Ricci a donné une caractérisation dans le cas où la surface est à courbure strictement négative. A. et S. Moroianu ont donné une caractérisation complète sans cette hypothèse et ont introduit la notion de surface de Ricci. Nous verrons des généralisations de cette notion, nous intéresserons aux surfaces de Ricci généralisées compactes et verrons le lien avec les surfaces à courbure constante et singularités coniques. Il s’agit d’un travail en commun avec Yiming Zang.

Vincent Lafforgue a montré que tout coefficient matriciel SO(2)-fini d’une représentation unitaire de SO(3) est 1/2-Hölderien – en dehors de certains points singuliers. Ce seul résultat joue un rôle important dans la preuve de la propriété (T) renforcée pour SL(3,R) et d’autres avancées récentes en algèbre d’opérateurs. Dans cet exposé, j’expliquerai comment ce résultat de régularité peut s’interpréter en terme de paires de Gelfand et de fonctions sphériques. Grâce à cela, je montrerai qu’on peut le généraliser à tous les groupes de Lie semi-simples en étudiant le comportement asymptotique de ces fonctions. Dans le cas non-compact, la structure des groupes de Lie en donne une représentation intégrale aisément manipulable. Le cas compact est étonnement plus difficile et nécessite de passer par l’analyse complexe.

Let G be a semisimple real Lie group with Lie algebra g. We will show how the universal enveloping algebra U(g) naturally fits into a one-parameter family of algebras U_q(g) with interesting structure. Any of these algebras U_q(g) moreover allows for an associated C*-algebra, whose representation category closely resembles that of G. We mainly explain these ideas and results in the concrete case of SL(2,R). This is based on joint work with Joel Right Dzokou Talla.

I will talk about a refinement of the Connes-Kasparov isomorphism, which is proved by understanding the structure of the Casselman algebra of rapidly decreasing functions on a real reductive group. I show that this Casselman algebra, which encodes nonunitary representation theory, and the reduced group C^*-algebra, which encodes tempered unitary representation theory, are built in very similar ways from similar elementary components. The structure of the Casselman algebra is understood using techniques from Delorme’s proof of the Paley-Wiener theorem for real reductive groups, which describes the Fourier transform of compactly supported smooth functions. Thanks to the similar structures of the two algebras, it becomes straightforward to prove that the two algebras, once cut down to certain finite sets of K-types, have isomorphic K-theory, which is the refinement of Connes-Kasparov. This work is essentially my thesis at Penn State.

Let $M$ be a manifold with a geometric structure and sufficiently nice $G$ a symmetry group, often the geometric structure can be transferred to $M/G$. In (multi-)symplectic geometry, reduction procedures permit to transfer the differential form to an even smaller space. However, all approaches working directly on the space have very strong regularity requirements.
We present an approach to reducing the algebra of (multi-)symplectic observables for general (covariant) moment maps, without any regularity assumptions of the level sets (and the symmetries).
Based on joint work with Casey Blacker and Antonio Miti.

We consider the problem of quantizing the positive nilradical of a complex semisimple Lie algebra of finite rank, together with a certain fixed direct sum decomposition. The decompositions we consider are in one-to-one correspondence with total orders on the simple roots, and exhibit the nilradical as a direct sum of graded modules for appropriate Levi factors. We show that this situation can be quantized equivariantly as a finite-dimensional subspace within the positive part of the corresponding quantized enveloping algebra. Furthermore, we show that such subspaces give rise to left coideals, with the possible exception of components corresponding to some exceptional Lie algebras, and this property singles them out uniquely. Finally, we discuss how to use these quantizations to construct covariant first-order differential calculi on quantum flag manifolds, which coincide with those introduced by Heckenberger-Kolb in the irreducible case.