# Mathematics Colloquium Winter 2018

For further information please contact Professor Torsten Ehrhardt or call the Mathematics Department at 459-2969

**Tuesday, January 9, 2018**

**No Colloquium**

**Tuesday, January 16, 2018**

**No Colloquium**

**Tuesday, January 23, 2018**

**ZongZhu Lin, Kansas State University**

**Comparing of Highest Weight Representations of Various Forms of Quantum Groups**

In 1979, Kazhdan and Lusztig conjectured a formula computing the characters of irreducible highest weight representations of a finite dimensional semisimple complex Lie algebra in terms of combinatorial data of the corresponding Weyl group. This conjecture and many other analogs, in fact, imply that many different categories, being from geometry, or representations of Lie algebras, algebraic groups, quantum groups are equivalent. The approach in proving the conjectures is to formulate certain categories that can link different representation theories and to establish the categorical equivalences. In the talk, I will first survey some of the main approaches, in particular, of Lusztig’s series of conjectures comparing representations of quantum groups, affine Lie algebras, Lie super algebras, and algebraic groups. Then I will introduce a new frame work to compare certain classes of weighted representations of Lie algebras and quantum groups as well as their variations and prove that, despite the algebras are different, they share a common weighted representation theory. This new framework follows from Lusztig’s constructions of modified quantum groups. This work is joint with Zhaobing Fan and Yigiang Li.

**Tuesday, January 30, 2018**

**Pham Tiep, Rutger's University**

**Representations of Finite Groups and Applications**

In the first part of the talk we will survey some recent results on representations of finite groups. In the second part we will discuss applications of these results to various problems in group theory, number theory, and algebraic geometry.

**Tuesday, February 6, 2018**

**No Colloquium**

**Tuesday, February 13, 2018**

**Gunter Malle, University of Kaiserslautern**

**Counting characters of finite groups**

More than 60 years ago Richard Brauer developed the theory of representations of finite groups over arbitrary fields. It showed a strong connection between the representation theory of a finite group and that of its p-local subgroups, for p a prime. Many more such connections have been observed in the meantime, but most of these are still conjectural. These ``local-global'', or ``counting conjectures'', have guided the development of character theory of finite groups in the past decades.

Recently, a new reduction approach has offered the hope to solve all of these fundamental conjectures by using the classification of finite simple groups. In our talk we will try and explain the nature of these problems and will report on recent progress which might eventually lead to a solution of these long standing fundamental questions.

**Mathematics Colloquium/Geometry and Analysis Seminar**

**Tuesday, February 15, 2018**

**Rafe Mazzeo, Stanford University**

**The Extended Bogomolny and Kapustin-Witten Equations**

**Tuesday, February 20, 2018**

**Edmund Karasiewicz, University of California Santa Cruz**

**The Fourier Coefficients of a Minimal Parabolic Eisenstein Series on the Double Cover of GL(3) over Q.**

Eisenstein series are special types of automorphic forms that have proved useful in the study of L-functions. For example, Shimura considered a Rankin-Selberg construction using an Eisenstein series on the double cover of SL(2) to study the symmetric square L-function of a modular form. We will discuss the Fourier coefficients of the titular Eisenstein series and describe some potential applications to the Archimedean theory of GL(3) symmetric square L-functions and the study of moments of quadratic Dirichlet L-functions.

**Tuesday, February 27, 2018**

**Michael Beeson, San Jose State University**

**Proof-checking Euclid**

We used computer proof-checking methods to verify the correctness of our proofs of the propositions in Euclid Book I. We used axioms as close as possible to those of Euclid, in a language closely related to that used in Tarski's formal geometry. We used proofs as close as possible to those given by Euclid, but filling Euclid's gaps and correcting errors. Then we checked those proofs in the well-known and trusted proof checkers HOL Light and Coq. The talk will contain many geometrical diagrams and discuss both the geometry and the proof-checking.

**Tuesday, March 6, 2018**

**Sacha Klechev, University of Oregon**

**Generalized Schur algebras and RoCK blocks of symmetric groups**

A highest weight category is a notion (introduced by Cline, Parshall and Scott) which captures many "typical situations in Lie representation theory". The category of modules over a finite dimensional algebra is a highest weight category if and only if the algebra is quasi-hereditary—a purely structural property described in terms of certain chains of ideals. In this talk, we discuss a general procedure which allows one to associated a new quasihereditary algebra S(A,n,d) to any quasihereditary (super)algebra A. When A is a field, the procedure returns the classical Schur algebra S(n,d). The motivation comes from the following fact: when A is a truncated zigzag algebra, the “schurification” procedure returns the so-called Turner's double algebra, which is Morita equivalent to a “generic" block of a symmetric group. This was conjectured by Turner and proved recently jointly with Anton Evseev.

**Tuesday, March 13, 2018**

**Postponed**

**Gabriel Navarro, MSRI**