I generally like abstraction and geometric thinking, but am also very happy to use combinatorial arguments. In general, I enjoy using tools from many different parts of mathematics.
Physical mathematics in one complex dimension
My main interests are on the mathematical side of mathematical physics, or more aptly in physical mathematics, specifically in one complex dimension. These include, but are not necessarily limited to, topological recursion, moduli spaces of curves, Hurwitz numbers, integrable hierarchies, and their interconnections and relations to representation theory. As an important tool, I use Fock space formalism.
For a concise overview of the relations between these topics, print and assemble this simplicial space, an updated version of one I made for my PhD defense.
Hurwitz theory
Classically, Hurwitz theory aims to calculate the number of (ramified) covers of a given Riemann surface with given ramification conditions. By the monodromy representation of the fundamental group, this can be stated equivalently as counting compositions in symmetric groups. And this itself is usually reformulated through representation theory, in terms of characters or in terms of symmetric functions. But it can also often be interpreted using different kinds of combinatorial tools. We (or I) often do not care so much about individual numbers, but rather about the way they are related. For example, what happens if, in the composition picture, we add one more ramification profile?
In modern mathematics, Hurwitz theory turns out to be related to many more interesting topics, especially through the behaviour of different kinds of generating series. These relations also yield many different generalisations, some of which are only tangentially related to the original definition. Adding an extra ramification profile translate on generating functions as actions by different kinds of operators, called cut-and-join operators.
Topological recursion
Topological recursion is a procedure that assigns to an analytic curve with two differentials and a bidifferential (a spectral curve), a collection of symmetric multidifferentials on this curve. These can have many interpretations, but for Hurwitz theory they are the generating series of certain Hurwitz numbers. The spectral curve itself is obtained by flowing with the cut-and-join operator.
The theory of topological recursion in itself, e.g. how this assignment works, how it depends on the initial data, and what the initial data can even be, is also interesting to me.
Integrable hierarchies
Integrable hierarchies are infinite collections of compatible partial differential equations, usually all evolutionary. Prime examples are the Korteweg–de Vries hierarchy, the Kadomtsev–Petviashvili hierarchy, and the Toda lattice hierarchy, but there are many other related hierarchies. Many of my favourite enumerative problems have generating series which are solutions (or tau-functions) of these hierarchies, and this also generally holds for the output of topological recursion. Furthermore, the space of solutions of an integrable hierarchy is often very well-behaved: they are infinite Grasmannian spaces or similar. This yields a lot of fun tools to work with them.
Gromov–Witten theory
Gromov–Witten theory is the study of of maps from algebraic curves to a target. It uses heavy algebro-geometric machinery to construct moduli spaces of the maps to study. Afterwards, numbers are calculated by intersection theory on these spaces. This theory was originally motivated by string theory as counting world sheets in a space.
Despite the construction being far more involved, Gromov–Witten theory and its enumerative cousins of Donaldson–Thomas, Pandharipande–Thomas, and Gopakumar–Vafa theory have many relations to the more combinatorial model of Hurwitz theory and to topological recursion, both on a precise mathematical level, and as an extended analogy. As such, Hurwitz theory is a good playing ground to understand these more complicated enumerative geometry theories.
Side interests
I have an ongoing fascination with what some might call abstract nonsense, such as (higher) category theory, and especially its applications to algebraic geometry and algebraic topology.
I also did a fun side project on transmission of quantum information over bipartite and semi-bipartite graphs.
My coauthors
I very much enjoy collaborating on mathematical projects, as can be seen at my list of papers. I have written papers with the following people.
Nezhla Aghaei, Raphaël Belliard, Gaëtan Borot, Vincent Bouchard, Guido Carlet, Nitin Kumar Chidambaram, Petr Dunin-Barkowski, Elba Garcia-Failde, Alessandro Giacchetto, Carla Groenland, Koen Groenland, Marvin Anas Hahn, Farrokh Labib, Danilo Lewański, Tanner Nelson, Nicolas Orantin, Kento Osuga, Alexandr Popolitov, Adrien Sauvaget, Yannik Schüler, Sergey Shadrin, Quinten Weller
