Researcher in mathematics
I am a postdoctoral fellow at the Max Planck Institute for Mathematics (Bonn, Germany). My mentor is Catharina Stroppel.
Previously, I was a graduate student at the UCLouvain (Belgium), working under the supervision of Pedro Vaz.
You can find a list of my arXiv preprints here.
Contact: lastname(at)mpim-bonn.mpg.de (lastname=schelstraete)
My research lies in low-dimensional topology, representation theory and rewriting theory.
On the one hand, I study quantum invariants of knots related to the representation theory of quantum groups. Both sides can be categorified: as homological invariants for the former (e.g. Khovanov homology) and as diagrammatic algebras for the latter (e.g. KLR algebras). I am interested in their interaction as well as their connection with various other fields. My recent research has focused on the "oddification" of this story; intuitively, moving form commutativity to anti-commutativity. This features odd Khovanov homology on the topological side, and super algebraic structures on the representation theoretic side.
On the other hand, I am interested in rewriting theory (the algorithmic study of presented algebraic structures) and its applications to diagrammatic algebras, categorification and quantum topology. My recent research has focused on developing a general rewriting toolkit for the higher algebraist, as well as applications where ad-hoc techniques to find basis have failed.
A list of my work and projects:Abstract. We develop a rewriting theory suitable for diagrammatic algebras and lay down the foundations of a systematic study of their higher structures. In this paper, we focus on the question of finding bases. As an application, we give the first proof of a basis theorem for graded $\mathfrak{gl}_2$-foams, a certain diagrammatic algebra appearing in categorification and quantum topology. Our approach is algorithmic, combining linear rewriting, higher rewriting and rewriting modulo another set of rules—for diagrammatic algebras, the modulo rules typically capture a categorical property, such as pivotality. In the process, we give novel approaches to the foundations of these theories, including to the notion of confluence. Other important tools include termination rules that depend on contexts, rewriting modulo invertible scalars, and a practical guide to classifying branchings modulo. This article is written to be accessible to experts on diagrammatic algebras with no prior knowledge on rewriting theory, and vice-versa.
This thesis is devoted to the fields of quantum topology and rewriting theory, and their surprising interconnections. In the first part of the thesis, we develop a higher representation theoretic approach to odd Khovanov homology; this is the content of arXiv:2311.14394. One of the essential ingredients is a certain graded-2-category of graded $\mathfrak{gl}_2$-foams. In the second part of the thesis, we develop a rewriting theory suitable for higher algebras and their super or graded analogues, and use it to show a basis theorem for graded $\mathfrak{gl}_2$-foams. These techniques have the potential to be applied to a wide variety of contexts. Both parts of the thesis can be read independently. Each has its own comprehensive introduction, allowing experts from one field to get acquainted with the other field.
Abstract. We define a supercategorification of the $q$-Schur algebra of level two and an odd analogue to $\mathfrak{gl}_2$-foams. Using these constructions, we define an homological invariant of tangles, and show that it coincides with odd Khovanov homology when restricted to links. This gives a representation theoretic construction of odd Khovanov homology. In the process, we define a tensor product on the category of chain complexes in super-2-categories which is compatible with homotopies. This could be of independent interest.