TOUTELIA 2021 : Geometry, Topology and AI

Amphiteatre Schwartz

Amphiteatre Schwartz

IMT, Université Paul Sabatier
Francesco Costantino (Université Paul Sabatier, Toulouse)

Date : September the 10th 2021

Place : Institut de Mathématiques de Toulouse (IMT), Amphithéatre Schwartz



This conference will be part of a series of four similar sessions dedicated to interactions of AI with other branches of mathematics.

The goal of the session will be to outline some interactions between topological and geometrical ideas and AI.

On one side we will see how AI performs in studying geometric or topological objects. On the otherside we will hear about techniques using geometric or topological ideas to improve machine learning.

If the sanitary conditions will allow it, the conference will be held in presence.

The speakers will be :

The titles of the talks will be announced directly on the schedule.

    • 09:00 09:30
    • 09:30 10:30
      Yang Hui He - Machine-Learning Mathematical Structures

      We report and summarize some of the recent experiments in supervised machine-learning of various structures from different fields of mathematics, ranging from geometry, to representation theory, to combinatorics, to number theory. We speculate on a hierarchy of inherent difficulty and where geometric and compbinatorial problems tend to reside.

    • 10:30 11:00
      Coffee Break
    • 11:00 12:00
      Steve Oudot - Toward Explainable Topological Features for AI

      This talk will be a review of the efforts of the Topological Data Analysis (TDA) community to design effective features for data, to be used in applications, and to make these features explainable. After a general introduction on TDA, the main focus will be on recent attempts to invert the TDA operator. While this line of work is still in its infancy, the hope on the long run is to use inverses for feature interpretation. The mathematical tools involved in the analysis come mainly from metric geometry, spectral theory, and the theory of constructible functions---specific pointers will be given in the course of the exposition.

    • 12:00 14:00
      Lunch 2h
    • 14:00 15:00
      Piotr Sułkowski: Knots and AI - Learning to Unknot

      I will discuss various features of knot theory that make it
      a particularly interesting playground from the viewpoint of machine
      learning. In particular, I will focus on the unknot recognition
      problem, and show how successfully it can be solved combining
      techniques from machine learning and natural language processing.

    • 15:00 15:30
      Coffee Break
    • 15:30 16:30
      Fabian Ruehle - Moduli-dependent Calabi-Yau and SU(3)-structure metrics from Machine Learning: Moduli-dependent Calabi-Yau and SU(3)-structure metrics from Machine Learning

      Calabi-Yau manifolds play a crucial role in string compactifications. Yau's theorem guarantees the existence of a metric that satisfies the string's equation of motion. However, Yau's proof is non-constructive, and no analytic expressions for metrics on Calabi-Yau threefolds are known. We use machine learning, more precisely neural networks, to learn Calabi-Yau metrics and their Kahler and complex structure moduli dependence.
      I will start with an introduction to Calabi-Yau manifolds and their moduli. I will then illustrate in an example how we train neural networks to find Calabi-Yau metrics by solving a Monge-Ampere type partial differential equation. The approach generalizes to manifolds with reduced structure, such as SU(3) structure or G2 manifolds, which feature in string compactifications with flux and in the M-theory formulation of string theory, respectively. I will illustrate this generalization for a particular SU(3) structure metric and compare the machine learning result to the known, analytic expression.