The Langlands Program is an influential research direction connecting Number Theory to Representation Theory. Based on the pioneering works of Lusztig (and many others), I study the geometry of affine Deligne--Lusztig varieties and their applications to the Langlands Program. This research direction comes with ample connections to many areas of mathematics, such as geometry, Lie theory, quantum algebra, number theory and representation theory.

An event like the Hong Kong Laureate Forum allows to connect to many like-minded minds from other areas of mathematics and the world, which is very fruitful for this kind of research. Moreover, the opportunity to learn first-hand from the Shaw Laureates themselves is invaluable for any aspiring researcher like me.

Lastly, being an expatriate working in Hong Kong for over two years now, I would love the opportunity to share my enthusiasm on Hong Kong as an international hub for science, innovation and technology at such a prestigious event.

Promotionsurkunde.pdf

Highest award in the "Bundeswettbewerb Mathematik"

Scholarships for studying and PhD by the German Academic Scholarship Foundation

Scholarship of the Marianne-Plehn Program for my PhD

A linear algebraic group G defined over a field F is, by definition, a group object in the category of affine F-varieties. As such, G can be studied using methods from algebraic geometry and group theory. If F is a local field, e.g. the p-adic numbers, the geometry of G has number-theoretic significance. Such groups are used frequently in the Langlands program, a very influential research direction linking number theory with representation theory.

My main object of interest are affine Deligne-Lusztig varieties. These are finite-dimensional schemes over the residue field of F, and are used primarily to study the special fibre of Rapoport-Zink moduli spaces. Via these Rapoport-Zink spaces, the geometry of affine Deligne-Lusztig varieties is related to Shimura varieties.

The definition of affine Deligne-Lusztig varieties is motivated by a classical paper of Deligne and Lusztig. In a non-affine setting, they construct certain varieties, whose cohomology provides important representations of finite groups of Lie type. The geometry of these Deligne-Lusztig varieties is well-understood. This connection is not only motivational – sometimes the nice geometric properties of classical Deligne-Lusztig varieties come up when studying the affine case.