I am interested in geometry, quantization and their relationship. On the geometric side, I focus on complex manifolds or algebraic varieties and their derived categories of coherent sheaves. In particular, I want to understand how the derived category encodes the geometry of the underlying space. String Theory motivates many interesting structures on derived categories, for example stability conditions, moduli spaces of semistable objects, and so on, and predicts a deep connection between Complex and Symplectic Geometry, known as Mirror Symmetry, which can be expressed in terms of derived categories.

An important problem in String Theory is the count of BPS states. In mathematical terms, this corresponds to the count of semistable objects in the derived category. To find a proper way to count such objects is a nontrivial problem but there are several (partial) solutions given by Joyce, Thomas, Behrend, Kontsevich and Soibelman. Since my Postdoc period in Oxford, I am fascinated by this area of mathematics and I have done a lot of research in Donaldson-Thomas theory throughout the past years. Donaldson-Thomas theory has connections with many branches of mathematics and physics including representation theory, Chern–Simons theory and (derived) algebraic geometry, to name a few.

In recent years I was studying cohomological Hall algebras and their relationship to Donaldson-Thomas theory. It is the framework of cohomological Hall algebras which allows a categorification of many results in Donaldson-Thomas theory leading to new algebraic structures and possible applications in geometric representation theory.

Another project of mine is the construction of field and vertex algebra structures in algebra and geometry following ideas of Joyce. It turns out that moduli spaces as described above give rise to interesting field and vertex algebra structures.

As my research interest is mostly influenced by physics, I was always trying to establish some joined work with physicists. In particular, I have given a few mathematical lectures for the string theory group of the physics department in Bonn, and I was also the leader of a research group on BPS states at the Hausdorff Research Institute for Mathematics involving both mathematicians and physicists.