# Mathematics Colloquium Spring 2010

Jack Baskin Engineering Room 301A

Refreshments served at 3:40

For further information please contact the Mathematics Department at 459-2969

**April 16, 2010**

**Cohomology of GLn(Q) and the p-adic zeta functions for totally real number fields**

**Cohomology of GLn(Q) and the p-adic zeta functions for totally real number fields**

**Pierre Charollois, Paris VII**

The zeta function attached to a totally real number field takes rational values at negative integers. This is a result obtained by Siegel-Klingen, and by Shintani using other means. Both of these method can be interpolated p-adically to construct a corresponding p-adic zeta function, as was shown by Deligne-Ribet and Cassou-Nogus respectively. We propose here a third method, based on the cohomology of GLn(Q) and formulae obtained by Sczech. This is joint work with Samit Dasgupta.

**April 23, 2010**

**Modular forms for noncongruence subgroups**

**Modular forms for noncongruence subgroups**

**Ling Long, Iowa State University**

The theory of modular forms for congruence subgroups is well-developed mainly due to the role of Hecke operators. Though Hecke operators act ineffectively on modular forms for genuine noncongruence subgroups, rapid progress has been made on modular forms for noncongruence subgroups. It reveals some intriguing links between congruence and noncongruence modular forms. In this talk, we will discuss some results and open questions in the study of noncongruence modular forms.

**May 14, 2010**

**On invariants of pivotal categories**

**On invariants of pivotal categories**

**Siu-Hung Ng, Iowa State University**

The dimension of an irreducible representation of a finite group G and the exponent of G are not preserved by a general linear equivalence between the representation categories of finite groups. However, they are invariant under monoidal equivalences. These invariants can be obtained from a more general invariant, called Frobenius-Schur indicator, which can be defined for each object in a pivotal category. In this talk, we discuss dimensions, exponents and Frobenius-Schur indicators for the representations of finite groups, and their application to modular categories.

**May 21, 2010**

**Schur Polynomials and the Yang-Baxter Equation**

**Schur Polynomials and the Yang-Baxter Equation**

**Dan Bump, Stanford University**

Schur polynomials are the characters of irreducible representations of GL(n). They are given by the Weyl character formula. Tokuyama, Hamel and King and Brubaker, Bump and Friedberg gave expressions for these that are deformations of the Weyl character formula. The most interesting and general involves expressing the Schur polynomial as the partition function of a statistical mechanical system (six vertex model). This involves interesting applications of the Yang-Baxter equation.