Responsable d’équipe : François Loeser
Responsables adjoints : Penka GEORGIEVA, Maxime ZAVIDOVIQUE
Gestionnaire : Julienne PASSAVE
Adresse postale :
|IMJ-PRG – UMR7586|
Université Pierre et Marie Curie
Boite courrier 247
Couloir 15-25 5e étage
4 place Jussieu, 75252 Paris Cedex 05
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 :
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.
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.
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.
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 :
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.
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.
Title : Coarse nodal geometry and topological persistence