Algebra and Number Theory Seminar Spring 2014

Wednesdays from 2:30-3:30pm

McHenry Library room 4130
For more information, contact Cameron Franc

April 9, 2014

Rank and exponent of p-groups and applications

Benjamin Sambale, University of Jena

I present a bound on the order of a finite p-group in terms of its rank and exponent. Here the rank is the maximal rank of an abelian subgroup. By a joint work with Koshitani and Külshammer this leads to a lower bound on the Loewy length of a p-block of a finite group in terms of its defect. In particular, one can give a list of possible defect groups for p-blocks with Loewy length at most 4.

April 23, 2014

Algoriths for Mumford curves over the p-adics

Ralph Morrison, UC Berkeley

Let K be a non-Archimeadean field, such as the field of p-adic numbers, In the 1970's David Mumford considered the action of certain matrix groups on K, and showed that this action give rise to algebraic curves over K, now called "Mumford curves". I'll begin with an introduction to these curves, and then discuss how to compute with them algorithmically. The algorithms include finding Berkovich skeleta, approximating Jacobian, and computing polynomial representations (and the corresponding tropicalizations). This talk is based on the joint work with Qingchun Ren

Bay Area Algebraic Number Theory and Arithmetic Geometry Day 8

Saturday, April 26, 2014

Stanford University Mathematics Department (Building 380)
The speakers are:
Henri Darmon, McGill University
Michael Larsen, Indiana University
Michael Magee, University of California, Santa Cruz
Alice Silverberg, University of California, Irvine
Ander Steele, University of Calgary

April  30, 2014

Frobenius Green Functors

Andrew Baker, University of Glasgow

Frobenius Green functors are Green functors taking values in Frobenius algebras over positive characteristic fields. They extend to the category of groups and homomorphisms whose kernels have order prime to the characteristic, and also to saturated fusion systems.
The original examples are obtained from Morava K-theory of classifying spaces but they may be of interest as purely algebraic objects.

May 7, 2014

Analytic torsion and torsion in the cohomology of arithmetic groups

Jonathan Pfaff, Stanford

We will recall the definition of the analytic torsion, which is a special invariant originally defined for closed manifolds. Then we will describe some recent results in which the growth of torsion in the cohomology of arithmetic groups was studied by using the analytic torsion.

May 21, 2014

Burnside rings and fusion systems

Sune Precht Reeh, University of Copenhagen

The Burnside ring of a finite p-group S consists of isomorphism classes of finite S-sets, with disjoint union as addition, and formal additive inverses. We will consider in particular those S-sets where the action of S respects the additional structure of a saturated fusion system F over S. The ring generated by these F-stable sets is the Burnside ring of F. We will discuss transfer maps constructing F-stable sets from general S-sets, and we will see how all of this can be sued to construct and describe the unique so-called F-characteristic idempotent in the double Burnside ring of S.

May 28, 2014

A p-adic interpretation of some integral identities for Hall-Littlewood polynomials

Vidya Venkateswaran, UC Davis

If one restricts an irreducible representation of GL n to the orthogonal subgroup (respectively, the symplectic subgroup), classical branching rules tell us when the trivial representation is contained in the restricted representation. In both cases, the partition λ that indexes the original representation must satisfy a particular condition: in the orthogonal (respectively, symplectic) case, λ (resp. λ ) must have all even parts. Using character theory, these results may be rephrased in terms of integrals involving the Schur functions. Since Hall-Littlewood polynomials are t-generalizations of Schur functions, one may consider t-analogs of these results. We will discuss these identities, focusing on an interpretation using p-adic representation theory that parallels the Schur case.