Combinatorics, Algebra,
Number Theory Seminar
Spring 2011 Archive:

Abstract: Following S. Kudla, J. Bruinier and T. Yang, we compute the integral of an automorphic Greene function coming from vector-valued harmonic Maass forms for the dual pair (O(n);Sp(1)) over the negative 2-planes with signature (r,2) for 0 < r < n. The integral is of interest in Arakelov geometry. We connect this integral of Borcherds forms with the derivative of a Rankin-Selberg L-function.

Abstract: We introduce some hypergeometric functions and discuss their role in supercongruences. In particular, we introduce Greene's hypergeometric function over finite fields which features in the proofs of many supercongruence results. We highlight a limitation of this function and explain how this may be overcome by extending it to the  p-adic setting. We will outline recent work on using this extended function to prove a supercongruence conjecture of Rodriguez-Villegas.

Abstract: Let  w  be a third root of unity. A complex reflection of order 3 is a matrix with one eigenvalue  w  and all other eigenvalues equal to 1. We shall study an infinite group  R  of 14 by 14 matrices with entries in  Z[w], generated by complex reflections of order 3. These matrices preserve a Hermitian form of signature (1,13) obtained from the lattice that gives the densest sphere packing in dimension 24. Our interest in the group  R  stems from a conjecture by Daniel Allcock relating it to the monster simple group. The reflection group  R  naturally acts on the 13-dimensional complex ball preserving a metric of negative curvature. Let  X  be the complement of a collection of of hyperplanes (called mirrors of reflection) in this 13-dimensional complex ball. Allcock's conjecture implies that the monster is a quotient of the fundamental group of  X/R. We shall explain the context of this conjecture, its possible consequences, and its present status. Familiarity with anything monstrous or with complex reflection groups or geometry of negative curvature will not be assumed.

Abstract: We will first recall the basics of quiver representations and co-representations, path algebras and path coalgebras, and present a series of examples. We then introduce an important property of categories, the Frobenius property, and give its interpretation for algebras, coalgebras and Hopf algebras. We then present recent results on the classification of infinite-dimensional pointed (co)Frobenius coalgebras, and pointed Hopf algebras having a nonzero integral. Finally, we examine a conjecture regarding a finiteness property of such infinite-dimensional Hopf algebras, from the point of view of tensor categories and also of quiver coalgebras. We will introduce most definitions and notions, and familiarity with the subjects is not necessarily assumed.

Abstract: In this talk I will discuss some questions using quiver representations to describe the canonical bases introduced by Lusztig in terms of simple perverse sheaves. There is a correspondence between isomorphism classes of representations of a quiver and orbits of a collection of algebraic varieties with algebraic group actions. Some of the orbits correspond to simple perverse sheaves which give canonical basis elements. I will describe some of these orbits, and other perverse sheaves that do not arise from a single orbit.

Abstract: In this talk, we introduce the zeta function of a finite complex arising from a p-adic reductive Lie group. This zeta function is an analogue of the local zeta function of a variety, such that it is a rational function and has a homological interpretation. We will study this zeta function from both combinatoric and representation-theoretic viewpoints. Finally, we will discuss its relation to local L-factors of automorphic representations.

Abstract: We will discuss the generalization of the concept of a convex set over the field of real numbers, first to subfields of this field, and then to some of its subrings. We will then introduce a new algebraic closure condition for convex sets. In the "classical" situation this algebraic closure coincides with the usual topological closure.

Abstract: Vertex algebras, and their conformal algebra fragments, play an important role in quantum field theory, group theory, and other areas. Hitherto, they have not appeared to be amenable to universal algebra techniques, such as the construction of free algebras. In this talk, we will give a quick introduction to these algebras, and show how they may be studied using universal algebra by regarding them as two-sorted algebras, with a second sort that encodes the logic of the underlying spacetime of the quantum field theory.