Go top
Project information

Applications of computational methods and artificial intelligence to the study of moduli spaces

D. Alfaya J. Portela

September 2023 - August 2026

Funding entity Convocatoria de Financiación de Proyectos de Investigación Propios 2023

It is common to find applications of mathematics to AI and computation. Recent breakthroughs like ChatGPT have increased the interest on the opposite process: the ability of computers to help proving new theorems. This project deepens in this question, exploring the following lines of research in which AI and computation are used to study open problems related with moduli spaces which are very relevant in the current mathematical landscape.

Analysis of the stability chambers of the moduli space of parabolic bundles:
The geometry of these moduli spaces depends on certain weights which are chosen for their construction. These are grouped in stability chambers in which the moduli space remains constant, represented by regions in a hypercube and there exist transformations identifying moduli from certain different chambers. Our goal is to estimate the number of different moduli spaces that there exist and to study their birational geometry through the study of the geometry of these chambers with computer assisted techniques.

Decompositions of motives of moduli:
Motives are invariants which provide a lot of geometric information. Manipulating motivic formulas and understanding when can two different expressions represent the same variety is a problem of great interest. We will develop a software package for manipulating and comparing motives efficiently and we will apply it to Mozgovoy’s conjecture on the L-Higgs moduli and to obtain formulas with positive coefficients for the moduli of vector bundles.

Study of Markoff m-triples and its relation to Higgs moduli:
Markoff triples are integral solutions to x^2+y^2+z^2=3xyz. They are structured in trees, subject of the famous Markoff Conjecture. m-triples are solutions to x^2+y^2+z^2=3xyz+m and their theory is jet to be developed. We will use computational techniques to research conjectures about its structure and, starting from the relations between triples and points in the moduli of representations, we will explore the relations between m-triples and the Higgs moduli.

Layman's summary: This project explores the usage of artificial intelligence and multiple computational techniques for studying several open problems related with the geometry of some moduli spaces which are very relevant in the current mathematical landscape.