CNRS Paris Diderot Sorbonne Université

Equipe Analyse Algébrique

Responsable d’équipe : François Loeser
Responsables adjoints : Penka GEORGIEVA, Maxime ZAVIDOVIQUE
Gestionnaire : Julienne PASSAVE

Adresse postale :

Université Pierre et Marie Curie
Boite courrier 247
Couloir 15-25 5e étage
4 place Jussieu, 75252 Paris Cedex 05

L’après-midi de l’équipe, le 20/10/2022

Programme de l’après-midi de l’équipe AA, le 20/10/2022

En salle 15.16.413

15h00 – 17h30 : Exposés des post-doctorants de l’équipe (titres et résumés plus bas)

15h00 – 15h40 : Konstantinos Kartas

15h50 – 16h30 : Owen Rouille       

16h40 – 17h20 : Vukasin Stojisavljevic

17h30 – 18h : Réunion de l’équipe

À partir de 18h00 : pot dans la salle 15.25.502

Résumés des exposés :

Konstantinos Kartas

Title : Some model theory of the tilting correspondence

Abstract : The idea that p-adic fields are in many ways similar to Laurent series over finite fields is a powerful philosophy. This philosophy has had two formal justifications. On one hand, the classical model-theoretic work by Ax-Kochen/Ershov in the ’60s achieves a transfer principle when p goes to infinity.  On the other hand, perfectoid geometry suggests replacing local fields with certain highly ramified extensions; this has the effect of making the similarities between mixed and positive characteristic even stronger. I will first survey those two approaches and then mention some recent work with F. Jahnke in which we give a model-theoretic generalization of the Fontaine-Wintenberger theorem. As a new arithmetic application, we provide some examples in mixed characteristic verifying the Lang-Manin conjecture on the existence of rational points on nearly rational varieties over C1 fields. 

Owen Rouille

Title : Experimenting in mathematics: examples of data generation and visualisation.

Abstract : Mathematics include many complex objects which are difficult to understand. In particular, intuition is crucial in teaching and research, allowing to state and prove conjectures. Building this intuition requires looking at and analysing many examples, which constitutes an experimental approach to mathematics. Computers are very important tools at our disposal: they allow to generate and analyse large quantities of data, and visualise patterns that would be difficult to draw by hand. A notable recent example of the use of computers in mathematics consisted in using AI to look for correlations between quantities in knot theory, this resulted in a conjecture that was then proved by specialists of the domain [Davies 21]. However, using computers is not straightforward, and implementation raises many new questions, among which the encoding and the performances for instances.

The talk is dedicated to the use of computers to help with mathematics. In the first part, I will present part of the work I did during my PhD concerning the computation of two topological invariants for 3-manifolds (Turaev–Viro invariants and hyperbolic volume). These projects illustrate the use of computers in mathematics in a domain where the computations can be difficult and the dataset very large. In the second part, I will present the main project of my postdoc: the computation and the representation of limit sets in S^3, with a focus on triangular groups.

Vukasin Stojisavljevic

Title : Coarse nodal geometry and topological persistence

Abstract : Given a Laplace-Beltrami eigenfunction on a closed manifold,its nodal domains are connected components of the complement of its zero set. Courant’s nodal domain theorem is a classical result in spectral geometry which, together with Weyl’s law, provides an upper bound on the total number of nodal domains in terms of the corresponding eigenvalue. A well-known question asks whether this result generalizes to linear combinations of eigenfunctions. While direct generalizations fail to be true, we will show that by counting nodal domains in a coarse way, i.e. by ignoring small oscillations, one may prove a version of Courant’s theorem for linear combinations as well. In order to prove this version of Courant’s theorem, we use the theory of persistence modules and barcodes, combined with multiscale polynomial approximation in Sobolev spaces. The same method allows us to prove a coarse version of Bézout’s theorem for linear combinations of Laplace-Beltrami eigenfunctions. The talk is based on a joint work with L. Buhovsky, J. Payette, I. Polterovich, L. Polterovich and E. Shelukhin.

L’après-midi de l’équipe, le 10/03/2022

Programme de l’après-midi de l’équipe AA, le 10/03/2022

En salle 15.16.413

16h-17h30 : Exposés des post-doctorants de l’équipe (titres et résumés plus bas)
16h00 – 16h40 Erman Cineli16h50 – 17h30 Marvin Hahn
17h40 – 18h : Réunion de l’équipe

À partir de 18h15 : pot dans la salle de convivialité

Résumés des exposés :
Erman Cineli 
Title: Topological entropy of Hamiltonian diffeomorphisms: a persistence homology and Floer theory perspective

Abstract: In this talk I will introduce barcode entropy and discuss its connections to topological entropy. The barcode entropy is a Floer-theoretic invariant of a compactly supported Hamiltonian diffeomorphism, measuring, roughly speaking, the exponential growth under iterations of the number of not-too-short bars in the barcode of the Floer complex. The topological entropy bounds from above the barcode entropy and, conversely, the barcode entropy is bounded from below by the topological entropy of any hyperbolic locally maximal invariant set. As a consequence, the two quantities are equal for Hamiltonian diffeomorphisms of closed surfaces. The talk is based on a joint work with Viktor Ginzburg and Basak Gurel.

Marvin Hahn 
Title: Quasimodularity of weighted Hurwitz numbers
Abstract: Hurwitz numbers enumerate branched mophisms between Riemann surfaces with fixed numerical data. When the target surface is an elliptic curve, these enumerative invariants are intimately related to mirror symmetry, which e.g. predicts a quasimodular structure of the generating series of elliptic Hurwitz numbers. This prediction was confirmed in seminal work of Dijkgraaf in 1995. In the past years, several variants of Hurwitz numbers were introduced that arise in various different contexts, e.g. monotone Hurwitz numbers in random matrix theory or strictly monotone Hurwitz numbers in the theory of Grothendieck dessins d’enfants. Recently, in work of Guay-Paquet and Harnard a unified framework for these different variants was introduced under the name of weighted Hurwitz numbers. Here the idea is to consider a Hurwitz numbers-like enumeration that depends on a weight function. For different choices of this weight function, weighted Hurwitz numbers specialise to essentially all known variants of Hurwitz numbers. In this talk, we present results regarding the structure of elliptic Hurwitz numbers. In particular, we generalise Dijkgraaf’s work to this case and derive a quasimodular structure of generating series. Our methods heavily rely on tropical geometry. This talk is based on a joint work in progress with Danilo Lewanski and Jonas Wahl.