Dr Christian Korff

What's this? Above you see the a graphical depiction how to embed the non-crystallographic Coxeter group H₄ (the symmetry group of the 120-cell) into the Weyl group E₈. Namely, map each simple reflection of H₄ labelled by an index i = 1, ..., 4 onto a product of two simple E₈ reflections labelled by the pair (i,i+4) as indicated by the folding of the Coxeter-Dynkin diagram: the pair (1,5) in E₈ corresponds to 1 in H₄, the pair (2,6) to 2 and so on. You can check that this map preserves the Coxeter group relations.

Don't understand a word or want to know more? If you are a student you can do a 4h or 5h project with me and learn about Coxeter and Weyl groups and their applications in mathematical physics. If you already know what these groups are, you can find more information in my joint paper with Andreas Fring. It has featured in week270 of John Baez's online column "This Weeks Finds in Mathematical Physics".

My research in key words

My main interests are quantum integrable systems which are described in terms of solutions to the quantum Yang-Baxter equation. More specifically:

I am particular fond of the algebraic and representation theoretic aspects which arise when trying to solve these models. Some of the algebras involved are: quantum and Hecke algebras, the Temperley-Lieb algebra, the Virasoro algebra and Kac-Moody algebras.

More recently I have also looked at combinatorial and algebro geometric aspects of integrable models. In my joint work with Catharina Stroppel it is discussed how the Bethe ansatz from integrable systems can be used to compute Gromov-Witten invariants and dimensions of moduli spaces of generalized theta-functions.

What is an integrable system?

In modern terminology the adjective "integrable" is usually attached to physical systems which can be solved exactly because they possess sufficiently many conserved quantities, sometimes infinitely many. Nonlinear partial differential equations which describe solitons are a particular example.

Quantum integrable systems are physical models which describe microscopic many-particle systems and underlie the laws of quantum mechanics. They are the exception rather than the rule and are usually confined to lower dimensions, such as particles moving on a line.

Why is this interesting?

From a mathematical point of view, integrable systems are interesting since they are usually associated with rich geometric or algebraic structures. Examples are Coxeter groups and Lie algebras, Riemann surfaces, quantum groups. Integrable systems can shed new light on these mathematical structures by showing new surprising connections between them. They have also led to new mathematical discoveries and inspired new mathematical research subjects.

There is also physical interest. Integrable systems can nowadays be `manufactured' in laboratories using for instance optical lattices or new compound materials. By providing exact and accurate data integrable systems can help to calibrate new technological equipment through high precision measurements. This technological advancement will benefit society. The interplay between concrete physical applications and mathematical theories makes this a fascinating and evolving research field.