**Presenter:** Dr Leonardo Patimo, Max Planck Institute for Mathematics

A central problem in representation theory is determining the characters of irreducible representations of modules. A complete answer in the case of highest weight modules of complex semisimple Lee algebras is given by the Kazhdan-Lusztig conjecture (a theorem!), which provides an explicit formula to calculate their characters. The original proof of this theorem relies on deep geometrical tools such as the decomposition theorem, which in turn ultimately relies on the Hodge theory of complex varieties. Soergel bimodules give an alternative, completely algebraic approach to the Kazhdan-Lusztig problem. This algebraic approach mimics the geometric one by proving an analogue of the decomposition theorem, where the key step is to show the existance of an algebraic Hodge theory. It is also interesting to look at what happens in the positive characteristic world. Here there is no Hodge theory, yet investigating when certain hodge theoretic properties still hold could help to understand the Lusztig's conjecture on characters of simple representations of algebraic groups

### About Pure mathematics seminars

We present regular seminars on a range of pure mathematics interests. Students, staff and visitors to UQ are welcome to attend, and to suggest speakers and topics.

Seminars are usually held on Tuesdays from 3pm to 3.50pm.

Talks comprise 45 minutes of speaking time plus five minutes for questions and discussion.

#### Information for speakers

Researchers in all pure mathematics fields attend our seminars, so please aim your presentation at a general mathematical audience.

#### Contact us

To volunteer to talk or to suggest a speaker, email Ole Warnaar or Travis Scrimshaw.