Physics and Mathematics
By upbringing, I am a theoretical high energy physicist with a focus on string theory. In the past years, I have been working on a framework of string theory, called F-theory. It strongly utilizes geometric tools to investigate solutions to string theory.
Of ample importance for string phenomenology is the number of Higgs fields in a particular compactifications. This is because the existence or absence of these fields determines if a Higgs mechanism can be employed to give masses to the other massless fields. As simple as this may sound, it leads to rich mathematics such as Freyd categories, Chow groups and coherent sheaves.
This analogy extends beyond F-theory. Namely, also in type IIB string theory, the topological B-model and heterotic string theory, such zero mode countings can be phrased in terms of cohomologies of coherent sheaves. Even more, homological mirror symmetry is a categorical equivalence of the category of coherent sheaves (on a certain Calabi-Yau manifold) and the Fukaya category (on the dual Calabi-Yau manifold). This underlines the geometric importance of coherent sheaves, and is why I am very interested in studying these objects.
In a very harsh first order approximation, merely counting sheaf cohomologies allows to tell good and bad string theory solutions apart. To facilitate such types of computations for a large number of geometries, I have developped software packages ranging from toric geometry to category theory. This software is part of the homalg_project of Mohamed Barakat.
My current research focuses on understanding how the geometry of the compactification space and the choice of further internal structures (G-fluxes) can be used, to modify the number of vector-like modes. Among others, this includes theoretical approaches, approaches by machine learning and data science as well as necessary software extensions. Here you can learn more about the latter.