Schedule – Arbre de Noël GNC 2023
Talks will take place at the Saulcy Campus of the Université de Lorraine at Metz, in the UFR Arts, Lettres & Langues (map, Google maps)
- Thursday: Room A-227 (2nd floor)
- Friday/Saturday: Room A-32 (ground floor)
- Coffee breaks: Room A-41 (ground floor)
| Thursday | Friday | Saturday | |||
| 9h-10h | Voigt 2 | 9h-10h | Azzali 3 | ||
| 10h | Café | 10h10-10h40 | Schanz | 10h10-10h40 | Bhattacharjee |
| 10h30-11h30 | Voigt 1 | Café | Café | ||
| 11h40-12h10 | Göbel | 11h10-12h10 | Azzali 2 | 11h10-11h50 | Contini |
| Lunch | Lunch | 12h-12h30 | Zhou | ||
| 14h-14h40 | Cren | 14h-14h40 | Cipriana | ||
| 14h50-15h20 | Naraghi | 14h50-15h20 | Le Clainche | ||
| 15h30-16h | Schäfer | 15h30-16h | Hataishi | ||
| Café | Café | ||||
| 16h30-17h30 | Azzali 1 | 16h30-17h30 | Voigt 3 | ||
| 17h40-18h10 | Delhaye | 17h40-18h10 | Troupel | ||
Titles & Abstracts
Mini-courses
Sara Azzali (University of Bari Aldo Moro)
Index theory and applications in geometry
An index theorem is a relation between an analytic quantity (in its fundamental form, the space of solutions of an elliptic equation) and topological data associated with the underlying space. By their nature, index theorems can be used to study manifolds. Combined with techniques of operator algebras, index theory has successfully addressed profound questions, including various cases of the Novikov conjecture about the homotopy invariance of higher signatures. In these lectures, we’ll introduce the topic, with a focus on the signature operator, and describe some classical and newer developments.
Christian Voigt (University of Glasgow)
Quantum symmetries
There is a variety of problems in modern mathematics in which “quantum symmetries” play a role, ranging from quantum invariants in knot theory, to subfactors, representation theory and topological field theory, up to quantum information. The aim of these lectures is to give an introduction to this topic starting from the concept of a C*-tensor category. We will discuss some concrete examples of such categories, and their relation to the theory of discrete quantum groups. The main focus will be on the study of quantum symmetries related to infinite graphs.
Research talks
Cipriana Anghel (Universität Göttingen)
Title: On the Spectrum of the Dirac operator on degenerating Riemannian surfaces
Abstract: We study the behavior of the spectrum of the Dirac operator on degenerating families of Riemannian surfaces, when the length of a simple closed geodesic shrinks to zero. We work under the hypothesis that the spin structure along the pinched geodesic is non-trivial. It is well-known that the spectrum of an elliptic differential operator on a compact manifold varies continuously under smooth perturbations of the metric. The difficulty of our problem arises from the non-compactness of the limit surface, which is of finite area with two cusps. The main tool for this investigation is to construct an adapted pseudodifferential calculus (in the spirit of the celebrated b-algebra of Melrose) which includes both the family of Dirac operators on the family of compact surfaces and the Dirac operator on the limit (non-compact) surface, together with their resolvents.
Arnab Bhattacharjee (Charles University, Prague)
Title: Kostant differential for quantum Grassmannian
Abstract: In the seminal article of Kostant in 1961, the author described a Levi equivari- ant differential complex of twisted $U_q(\mathfrak{l})$-modules, where l is a levi subalgebra of a semi-simple lie algebra g. In this talk, we shall discuss, an analogous ver- sion of this equivariant differential complex over a q-deformed Drinfeld-Jimbo quantum groups $U_q(\mathfrak{g})$ for the case of Quantum Grassmannian. We shall show a $U_q(\mathfrak{l})$-equivariant Berenstein-Gelfand-Gelfand (BGG) sequence coming out of this q-deformed differential complex. Then We shall also discuss the q-deformed twisted de-Rham complex for quantum Grassmannian and using this frame- work we shall lift this $U_q(\mathfrak{l})$-equivariant BGG sequence to a BGG sequence of $U_q(\mathfrak{g})$-modules in terms of Verma modules. This is an ongoing work with Petr Somberg.
Alessandro Contini (Leibniz Universität Hannover)
Title: Isomorphisms of algebras of global pseudo-differential operators.
Abstract: In this talk I will be concerned with the so-called SG-pseudo-differential operators. This a class obtained by refining the bounds on the symbols by allowing decay in the space variables, which leads to a full parametrix construction on a large class of non-compact manifolds (including, luckily, R^n). In trying to characterise, in this setting, the filtration-preserving isomorphisms (following ideas of Eidelheit and Duistermaat and Singer) of this algebra, we are confronted by the delicate interplay between symplectic geometry and manifolds with corners. I will try to sketch how we can hope to prove that every such isomorphism is given by conjugation with a so-called Fourier Integral Operator, and hopefully illustrate where the difficulties lie and how we managed to overcome them.
Clément Cren (Universität Göttingen)
Title: Transverse BGG sequences
Abstract: BGG sequences are originally an algebraic tool to study irreducible representations of semi-simple Lie groups. They have been extended to a more geometric setting of manifolds with parabolic geometry, these are manifolds that look like the homogeneous space G/P for G a semi-simple Lie group and P a parabolic subgroup. The BGG sequence then becomes a sequence of differential operators whose analytical properties are tied to the representations of G.
Here we define such operators for a foliated manifolds with transverse parabolic geometry. This means that we ask now the « space of leaves » of the foliation to look like G/P. If time permits I will explain how these operators form an elliptic complex for a modified notion of ellipticity.
Jean Delhaye (Paris Saclay)
Title: Construction and asymptotic study of the Brownian motion on free unitary quantum groups.
Abstract: We construct an analogue of the Brownian motion on free unitary groups and find its cutoff time. We further compute its limit profile which involves free Poisson distributions and the semi-circle distribution. This is a work in progress.
Lucas Hataishi (Universitet i Oslo)
Title: $C^*$-algebraic factorization homology and crossed producs by unitary tensor categories.
Abstract: This talk is about producing extensions of operator algebras making use of geometrical and topological data. This task is accomplished by employing a TQFT machinery known as factorization homology, together with generalized crossed products or realizations of internal operator algebras in $C^*$-tensor categories.
Julien Le Clainche (Université Clermont Auvergne)
Title: Homological duality for Hopf Galois extensions
Abstract: Homological duality is a general phenomenon producing a duality between the cohomology and homology spaces of an algebraic system, analogous to Poincaré duality for closed varieties in algebraic topology. The first occurrence of homological duality in Hochschild cohomology for algebras is due to Van den Bergh.
In this talk, I work in a very general framework of homological duality due to Kowalzig-Krähmer in order to show a result of homological duality for Hopf-Galois extensions using in particular the Stefan spectral sequence.
Michelle Göbel (Universität Göttingen)
Title: Induced representations and C*-hulls
Abstract: The problem with representations of *-algebras is that they are possibly unbounded.Given a C*-hull we can describe these representations using representations of a C*-algebra, which are always bounded. A C*-hull is a C*-algebra together with a family of bijectionsbetween its representations and a subclass of the representations of the *-algebra. The Induction Theorem uses induction of representations to construct a C*-hull for a graded *-algebra, given a C*-hull of its unit fibre. We will generalize this result for a *-algebra with an action of a compact quantum group.
Mahsa Naraghi (Paris Cité)
Title: The $C^*$-algebras of 0-Order Pseudodifferential Operators on a Lie Groupoid
Abstract: Pseudodifferential operators with compact support on a Lie groupoid $G$ act as multipliers on the space $C_c^\infty (G)$. When these operators have an order $\leq 0$, they can be extended to act as multipliers on the $C^*$-algebra of $G$ with respect to both the maximal and the reduced norm.
In this talk, we see that the completion of $0$-order pseudodifferential operators in the multiplier algebra of $C^*(G)$ and $C^*_r(G)$ are, in fact, isomorphic modulo $C^*(G)$ and $C^*_r(G)$ respectively.
Björn Schäfer (Université des Saarlandes)
Title: Nuclearity of Hypergraph C*-Algebras
Abstract: Hypergraph $C^∗$-algebras are a recent generalization of graph $C^∗$-algebras. Unlike for graph $C^∗$-algebras, there are non-nuclear hypergraph $C^∗$-algebras. In this talk, we present results towards a characterization of nuclear hypergraph $C^∗$-algebras. In the spirit of graph minor theory, we propose a tailor-made hypergraph minor relation together with a set of forbidden minors whose prese- cence ensures non-nuclearity of a hypergraph $C^∗$-algebra.
Julien Schanz (Universität des Saarlandes)
Title: Quantum Symmetries of Graphs
Abstract: I will introduce quantum automorphism groups of graphs as defined by Banica and give a small survey about the existing results on them, including some results for quantum automorphism groups of vertex-transitive graphs and examples of graphs with interesting quantum automorphism groups.
Arthur Troupel (Paris Cité)
Title: Graphs of C*-algebras and free wreath products of quantum groups.
Abstract: The free wreath product of a compact quantum group by the quantum permutation group $S_N^+$ has been introduced by Bichon in order to give a quantum counterpart of the classical wreath product. The representation theory of such quantum groups is well-known, as well as some approximation properties, but some results about their operator algebras were still open, for example stability of Haagerup property, of K-amenability or factoriality of the von Neumann algebra. I will present a joint work with Pierre Fima in which we identify these algebras with the fundamental $C^*$-algebras of certain graphs of $C^*$-algebras, and we deduce these properties from these constructions.
Shouxing Zhou (École Normale Supérieure)
Title: Entropy theory of noncommutative Poisson boundaries
Abstract: In this talk, I will give a brief introduction to the classical Poisson boundary theory of groups and Das-Peterson’s noncommutative Poisson boundary theory of tracial von Neumann algebras. Then I will talk about my recent work on Kaimanovich-Vershik’s fundamental theorems regarding noncommutative entropy.