Séminaires

Après avoir introduit le groupe des transformations birationnelles du plan projectif complexe, j’en donnerai quelques propriétés en faisant un parallèle avec les groupes linéaires.

On a surface M with strict negative curvature given two closed curves $c_1,c_2$, the Poincaré series is a complex function counting orthogeodesic arcs joining the two curves, in the same way the Riemann zeta function counts the primes. I will first discuss the meromorphic continuation of the Poincaré series and when the curves are homologically trivial, I will explain why the value at 0 is a well–defined rational number which can be interpreted as linking of Legendrian knots. A corollary of our result is that for any pair of points (x,y) in M x M, the lenghts of the geodesics joining the two points determine the genus of M.

Zoom Meeting: Meeting ID: 895 2739 9138, Passcode: 7ni0ti

13:45 – 14:40: Alessandra IOZZI (ETH Zürich)

 The real spectrum compactification of character varieties: characterizations and applications

We describe properties of a compactification of general character varieties with good topological properties and give various interpretations of its ideal points. We relate this to the Thurston-Parreau compactification and apply our
results to the theory of maximal representations.

This is a joint work with Marc Burger, Anne Parreau and Maria Beatrice Pozzetti.

15:00 – 15:55: Raphaël BEUZART-PLESSIS (Aix-Marseille Université and CNRS)

Multipliers and isolation of the cuspidal spectrum by convolution operators

Let $G$ be a real reductive algebraic group and $\Gamma$; be an arithmetic lattice of $G$.
In this talk, I will explain how to generalize a construction of Lindenstrauss-Venkatesh giving rise to certain operators on $L^2(\Gamma\backslash G)$ with image in the cuspidal subspace. These operators can be written, in the adelic setting, as combinations of convolution operators at Archimedean places and $p$-adic places (Hecke operators). A crucial ingredient of the proof is the existence of sufficiently many multipliers of $G$ acting on the space of smooth functions with rapid decay (but not necessarily $K$-finite).
Time permitting, I will also describe one application of this construction to the global Gan-Gross-Prasad conjecture for unitary groups.
This talk is based on joint work with Yifeng Liu, Wei Zhang and Xinwen Zhu.

16:15 – 17:10: Erik VAN DEN BAN (University of Utrecht)

The Harish-Chandra transform for Whittaker functions

 I will discuss the role of the descent transform in Harish-Chandra’s approach to the Plancherel formula for Whittaker functions, presented in the posthumous volume 5 of his collected works (Springer 2018). At an earlier occasion I explained how the proof of the Plancherel theorem can be completed by using a Paley-Wiener shift technique. In the present talk I will explain how the proof can be completed in a more straightforward way, by using a suitable result on wave packets of Whittaker functions.

Webpage

Contacts

To register as a participant or for further information, please contact one of the organizers: Salah Mehdi or Angela Pasquale.

We consider a Dirac system confined to a bounded domain in the plane. This amounts to a family of boundary conditions. There are two extreme cases, zig-zig and Infinite-mass boundary conditions. Consider a magnetic field perpendicular to the plane. I will present results on accurate asymptotics of the energy spectrum of the underlying Hamiltonian in the strong magnetic field limit. We will compare the results for different boundary conditions.

(This is based on joint collaboration with Jean-Marie Barbaroux, Loic Le Treust and Nicolas Raymond)

Zoom Meeting: Meeting ID: 895 2739 9138, Passcode: 7ni0ti

We first describe an efficient algorithm to compute
$L’/L(1,\chi)$, where $\chi$ is a non-principal Dirichlet character
mod q, and q is an odd prime. We then discuss
some results on the distribution of
$m_q := \min_{\chi\ne \chi_0} \vert L’/L(1,\chi) \vert $
and about the Euler-Kronecker constants for cyclotomic fields.

Résumé

A classic paper of Salem and Zygmund investigates the distribution of trigonometric polynomials whose coefficients are chosen randomly (say +1 or -1 with equal probability) and independently. Salem and Zygmund characterized the typical distribution of such polynomials (gaussian) and the typical magnitude of their sup-norms (a degree N polynomial typically has sup-norm of size $\sqrt{N \log N}$ for large N). In this talk we will explore what happens when a weak dependence is introduced between coefficients of the polynomials; namely we consider polynomials with coefficients given by random multiplicative functions. We consider analogues of Salem and Zygmund’s results, exploring similarities and some differences.

Special attention will be given to a beautiful point-counting argument introduced by Vaughan and Wooley which ends up being useful.

This is joint work with Jacques Benatar and Alon Nishry.

Les équations de Painlevé, tout comme beaucoup d’autres équations intégrables, admettent des généralisations au cadre non-commutatif, où la variable dépendante est remplacée, par exemple, par une matrice ou un opérateur. Cette extension au cadre non-commutatif a joué un rôle centrale dans ma collaboration avec Bertola et Roubtsov sur l’étude des systèmes de Calogero-Painlevé et, plus récemment, dans ma collaboration avec Bothner et Tarricone sur les équations de Painlevé de type intégrodifférentiel et leur applications aux probabilités intégrables. Dans mon séminaire, j’essaierai d’illustrer les résultats que nous avons obtenus dans les deux cas, en soulignant leur points communs.

Les équations de Painlevé, tout comme beaucoup d’autres équations intégrables, admettent des généralisations au cadre non-commutatif, o๠la variable dépendante est remplacée, par exemple, par une matrice ou un opérateur. Cette extension au cadre non-commutatif a joué un rôle centrale dans ma collaboration avec Bertola et Roubtsov sur l’étude des systèmes de Calogero-Painlevé et, plus récemment, dans ma collaboration avec Bothner et Tarricone sur les équations de Painlevé de type intégrodifférentiel et leur applications aux probabilités intégrables. Dans mon séminaire, j’essaierai d’illustrer les résultats que nous avons obtenus dans les deux cas, en soulignant leur points communs.