Speaker: Felipe Ramos
Date: 10:30 23/02/2024
Title: Modularity and Enumerative Geometry of Physical Invariants

Abstract: In Theoretical Physics, more specifically on String Theory, it is predicted that the so called Calabi-Yau manifolds play an important role in the structure of our universe. Of importance in this Physical theory are the enumerative invariants associated to such manifolds. Among those, we focus on the study of Gromov-Witten invariants (counts of stable maps) and Donaldson-Thomas invariants (counts of ideal sheaves). Our main contribution is, on the first part of this Thesis, to introduce an algebraic framework for the computation of open string (or real) Gromov-Witten invariants for the quintic threefold (first achieved by Walcher). We show, using Hodge theory, that there is a possible modular interpretation to some generating series, based on previous work by Movasati for the closed string case. On the second part, we introduce, from the motivic invariants defined in the work of Kontsevich and Soibelman, new refinements of Donaldson-Thomas invariants based on a version of enumerative geometry over arbitrary fields developed and studied by Kass-Wickelgren, Levine, and others, in which the counts obtained are not integers but quadratic forms. We also give examples and applications. We finish the thesis posing some questions for further research and on possible relations between the two parts of this work.