# CMU Algebra and Combinatorics Seminar

## Fall 2014

### Meeting Times

Wednesdays, 4:00–5:00pm, in Pearce 328.

### Schedule

 Date Speaker Affiliation Title 9/3 Meera Mainkar Central Michigan University Anosov Lie algebras and algebraic units in number fields 9/10 Ben Salisbury Central Michigan University The Gindikin-Karpelevich formula and combinatorics of crystals 9/17 9/24 Travis Scrimshaw University of California, Davis TBD 10/1 10/8 10/15 Gabriel Feinberg Haverford College TBD 10/22 10/29 11/5 11/12 11/19 11/26 12/3

### Abstracts

Speaker: Meera Mainkar (Central Michigan University)
Title: Anosov Lie algebras and algebraic units in number fields
Abstract: In the theory of dynamical systems, Anosov diffeomorphisms are an important class of dynamically interesting diffeomorphisms of Riemannian manifolds. The study of Anosov diffeomorphisms on nilmanifolds (compact quotients of simply connected nilpotent Lie groups) leads to interesting algebraic and arithmetic problems related to Lie algebras. In particular, this relates to very special algebraic units in number fields. We study the classification of Anosov Lie algebras by studying the properties of these algebraic units.

Speaker: Ben Salisbury (Central Michigan University)
Title: The Gindikin-Karpelevich formula and combinatorics of crystals
Abstract: The Gindikin-Karpelevich formula computes the constant of proportionality for the intertwining integral between two induced spherical representation of a $p$-adic reductive group $G$. The right-hand side of this formula is a product over positive roots (with respect to the Langlands dual of $G$) which may also be interpreted as a sum over a crystal graph. The original interpretation as a sum is due to Brubaker-Bump-Friedberg and Bump-Nakasuji where $G = \mathrm{GL}_{r+1}$, in which vertices of the crystal graph are parametrized by paths to a specific vector in the graph according to a pattern prescribed by a reduced expression of the longest element of the Weyl group of $G$. I will explain this rule, explain how it may be translated into a statistic on the Young tableaux realization of the same crystal graph, and discuss its generalization to the affine Kac-Moody setting where the notion of the longest element of the Weyl group no longer makes sense. This is joint work with Kyu-Hwan Lee, Seok-Jin Kang, and Hansol Ryu.

Speaker: Travis Scrimshaw (University of California, Davis)
Title: TBD
Abstract: Coming soon …

Speaker: Gabriel Feinberg (Haverford College)
Title: TBD
Abstract: Coming soon …