##
Kolloquium

Algebra - Geometrie - Kombinatorik

an der Fakultät für Mathematik der Ruhr-Universität Bochum

Markus Reineke, Gerhard Röhrle, Christian Stump

## Vorträge Wintersemester 2019/20

#### Mittwoch, 16. Oktober 2019

**Bernd Sturmfels**

Max-Planck-Institut Leipzig

16:00-17:00

Gebäude IA, Raum 01/473

**The geometry of Gaussoids**

**Abstract**: Gaussoids offer a new link between combinatorics, statistics and algebraic geometry. Introduced by Lnenicka and Matus in 2007, their axioms describe conditional independence for Gaussian random variables. We explain this theory and how it relates to matroids. The role of the Grassmannian for matroids is now played by a projection of the Lagrangian Grassmannian. We discuss the classification and realizability of gaussoids, and we explore oriented gaussoids, valuated gaussoids, and the analogue to positroids. This is joint work with Tobias Boege, Alessio D'Ali and Thomas Kahle.

#### Mittwoch, 13. November 2019

**Peter Littelmann**

Universität zu Köln

16:00-17:00

Gebäude IA, Raum 01/473

**Towards a Riemann-Roch type character formula**

**Abstract**: Standard monomial theory has its origin in the work of Hodge, where he constructed a nice basis for the coordinate of the Grassmann variety, embedded via the canonical Plücker embedding in a projective space. Seshadri, Musili, Lakshmibai (and many others) tried to generalize this approach to the setting of Schubert varieties in flag varieties associated to arbitrary semisimple algebraic groups. They came up with a remarkable conjectural combinatorial character formula for the homogeneous coordinate ring of an embedded Schubert variety. By combining combinatorial tools with methods from Newton-Okounkov theory and ideas developed by Rees and Samuel, we present a new approach towards a general construction of standard monomials. The classical Riemann-Roch theorem calculates the dimension of the space of meromorphic functions with prescribed zeroes and allowed poles on a compact Riemann surface. Our aim is to give a similar interpretation of the character formula conjectured by Lakshmibai: it counts functions with prescribed vanishing multiplicities along a sequence of Schubert varieties, subsequently contained in each other. This is ongoing joint work with Rocco Chirivì and Xin Fang.

#### Mittwoch, 29. Januar 2020

**Eric Opdam**

Universiteit van Amsterdam

16:00-17:00

Gebäude IA, Raum 01/473

**Square integrable hypergeometric functions for root systems**

**Abstract**: Classifying the square integrable solutions of the system of hypergeometric equations for root systems is relevant to understanding the discrete series for real symmetric spaces. We will discuss this connection and explain some aspects of this classification.