I am Miguel Gonzalez, a second year mathematics PhD student at Instituto de Ciencias Matematicas (ICMAT) in Madrid, Spain. My mathematical interests lie in the area of algebraic and complex geometry. I am interested in the study of moduli spaces of geometric structures on Riemann surfaces (mainly Higgs bundles), Langlands duality and mirror symmetry.

I want to join the Forum to be able to interact with other scientists both in my field and others, as well as to be able to benefit from the expertise of the participating laureates and distinguished scientists. In particular, the participating laureates in mathematics in 2025 have made numerous central contributions to the areas of mathematics I am interested in: Nigel Hitchin started the study of Higgs bundles and has made many advances within this topic as well as many others within geometry (such as generalised geometry). George Lustzig is a key figure in the area of representation theory, which is very closely related to my research interests and to my current PhD project. I strongly believe that I would benefit from interacting with them as well as the other scientists attending the event.

My current research, set within the area of algebraic and complex geometry, focuses primarily on the mirror symmetry aspects of the study of moduli spaces M_G of G-Higgs bundles over a projective curve C, where G is a reductive group, in relation to the circle of ideas introduced for the case of G=GL_n by Hausel and Hitchin in 2022. The main goal of this project is to study the situation for an arbitrary choice of group G seeking, among others, connections to the theory of Kirillov algebras and Hausel algebras introduced by Hausel (2023). In particular, we want to better understand the geometry of the moduli space by studying certain complex lagrangian subvarieties (the ones invariant by the natural scaling action, called upward flows) and their relation with the nilpotent cone. This has been achieved so far in the case where the limit point under the scaling action of these subvarieties is a regular Higgs bundle, and partially in other cases. I am also interested in the relation of these moduli spaces with higher Teichmuller theory and representations of the fundamental group of C.