L’après-midi de l’équipe, le 09/03/2023

Programme de l’après-midi de l’équipe AA, le 09/03/2023

En salle 15.16.413

15h30 – 18h00 : Exposés des post-doctorants de l’équipe (titres et résumés plus bas)

15h30 – 16h10 Mura Yakerson

16h20 – 17h00 Xiaohan Yan       

17h10 – 17h50 Dustin Connery-Grigg

18h00 – 18h15 : Réunion de l’équipe

À partir de 18h15 : pot dans la salle 15.25.502

Résumés des exposés :

Mura Yakerson

Title : Universality of cohomology theories in algebraic geometry

Abstract : Motivic homotopy theory provides a framework for studying various cohomology theories of algebraic varieties. In this talk, we will discuss how many interesting examples of these cohomology theories, such as algebraic K-theory or algebraic cobordism, acquire universality properties, which are based on certain covariance structures of these cohomology theories. This is a summary of joint projects with Tom Bachmann, Elden Elmanto, Marc Hoyois, Joachim Jelisiejew, Adeel Khan, Denis Nardin, Vladimir Sosnilo and Burt Totaro.

Xiaohan Yan

Title : Level structures in quantum K-theory 

Abstract : Quantum K-theory studies a K-theoretic analogue of Gromov-Witten invariants defined as holomorphic Euler characteristics over the moduli spaces of stable maps. Inspired by Verlinde/Grassmannian correspondence, level structures are introduced into quantum K-theory as determinant-type twistings. Such structures admit various symmetries, reveal surprising connections of quantum K-theory to mock theta functions, and appear in the so-called quantum Serre duality as well.

Dustin Connery-Grigg

Title : Symplectic dynamics and Hamiltonian Floer theory

Abstract : Given a Hamiltonian dynamical system on a symplectic manifold, what is the relationship between the dynamical features which the system may (or must) exhibit, and the (symplectic) topology of the underlying manifold? In 1989, Andreas Floer introduced an approach to doing relative Morse theory for the Hamiltonian action functional which provided a lower bound for the number of 1-periodic orbits of the associated Hamiltonian system in terms of the (quantum) homology of the underlying symplectic manifold, answering a version of Arnold’s conjecture in the process. The tools introduced by Floer in this work have since become a cornerstone of modern symplectic geometric research, but the finer-grained relationship between the objects appearing in this theory and the underlying dynamics remain somewhat mysterious. In this talk, I will give an brief introduction to Floer’s theory, and discuss some results in low-dimensions which provide links between the qualitative dynamics of low-dimensional Hamiltonian systems and their Floer theory.

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.