MLMath projects

Resolution of singularities: Github repo , ICML 2023 paper: arXiv:2307.00252 

In this joint project with Honglu Fan (Geneva) and Mingcong Zeng (Bonn) we search for optimal resolutions of singularities using reinforcement learning, through the Hironaka game. In 1964 Hironaka proved that it was possible to resolve singularities of varieties over fields of characteristic 0 by repeatedly blowing up along non-singular subvarieties, using a very complicated argument by induction on the dimension. 

The process can be translated into various versions of a 2-player game, the so-called Hironaka game, and a winning strategy for the first  player provides a solution to the resolution problem. In this paper we introduce a new approach to the Hironaka game that uses reinforcement learning agents to find optimal resolutions of singularities. 

As application, we derive new formulas for Thom polynomials and integrals over Hilbert scheme of points on manifolds based on blow-up trees generated from different versions of the Hironaka resolution game. 

Stanley chromatic positivity conjecture, in progress: Github repo

Joint project with Jonas Klüver.  We develop an ML assisted proof to a central problem in combinatorics, the Stanley positivity conjecture on chromatic polynomials of graphs. Based on recent work of  Szenes, Paunov and Rok we develop a graph reinforcement learning agent to find optimal combinatorial data encoding the Stanley coefficients. We build a small GPT model to study a reconstruction conjecture. 

Voice cloning: Github repo 

Joint project with Jonas Klüver (Aarhus). We use MFCC technology and PixtoPix deep network for voice adapting.

Regularization via complete collineations:  https://arxiv.org/abs/2311.03329

In this joint work with with E. Hamilton, P. Reichenbach and A. Seigal we uncover connections between maximum likelihood estimation in Gaussian graphical models described by directed acyclic graphs and GIT stability properties of sample points. As a result we develop a new regularization method in ML and statistics, extending Tikhonov regularization.