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.
Here is a list of my publications. Click on the name for more information!