Combinatorics, Algebra,
Number Theory Seminar
Spring 2009 Archive:

Abstract: In the previous talk (January 26), two computational constructions of lower-dimensional Hopf algebras were given. This talk will discuss an idea for the classification of nonsemisimple Hopf algebras of dimensions 12 and 20, and some of its limitations .

Abstract: Several systems of interest in applications can be modeled via right invariant systems whose state varies on a compact Lie group. An important example is given by finite-dimensional quantum mechanical systems where the dynamics, governed by the Schrödinger equation, is controlled by modifying the Hamiltonian. Techniques for control of these systems have typically been tailored to the specific case at hand. In this talk I will present three different but related methods which allow one to control any system with this structure. I will also establish the connection of this problem with the Dirichlet approximations studied in number theory.

Abstract: The zeros of classical Eisenstein series satisfy many intriguing properties. Work of F. Rankin and Swinnerton-Dyer pinpoints their location to a certain arc of the fundamental domain, and recent work by Nozaki explores their interlacing property. In this talk we will discuss these distribution properties for a particular family of level 2 Eisenstein series. This is joint work with Sharon Garthwaite, Holly Swisher, and Stephanie Treneer.

Abstract: An integral domain is a valuation domain if for each element  r  of the field of quotients, either  r  or its inverse is in the integral domain itself. These integral domains have some very interesting factorization properties, and can be classified up to isomorphism by their group of divisibility. In this talk I will discuss some properties of valuation domains, relate them to pseudovaluation domains, and discuss my attempt to classify pseudovaluation domains via their groups of divisibility.

Abstract: Ramanujan's 1914 paper "On certain arithmetic functions" contains a derivation for a coupled system of nonlinear differential equations involving the classical Eisenstein series. The derivation depends on two identities involving elliptic functions. More recently, R. Maier has shown that analogous differential equations hold for Eisenstein series on  Γ₀(n), where  n = 2, 3, 4. We will show that Maier's differential relations can be deduced from Ramanujan's original identity. We indicate a method for deriving differential equations for Eisenstein series on  SL(2,Z)-conjugate subgroups of  Γ₀(n). Our method can be extended to obtain analogous systems for higher level subgroups such as  Γ₀(5). In summary, we will observe that Ramanujan's original elliptic function identities encode a great deal of information about differential equations for Eisenstein series on subgroups of the modular group. Applications of these differential equations will be presented as time permits.

Abstract: Let  e  be a nilpotent (or topologically nilpotent) element of a commutative ring  R. Let  E  be the annihilator of  e. A product is defined on the direct square  T  of the quotient  R/E  by

( x, x' )( y, y' ) = ( xλ² + yλ, x'λ^-1 + y' )

with

λ = 1 + e²yx' .

Then  T  forms a loop, known as the Catalan loop. While the multiplication and right division are rational, the left division is given by a quadratic irrationality involving the generating function for the Catalan numbers. The whole construction is motivated by an example from number theory.

Abstract: The talk will first give an introductory survey of automorphic  L-functions on  GL(n), and then end by mentioning a prime-number theorem for Rankin-Selberg  L-functions.

Abstract: We consider convex figures in the dyadic plane  D², where  D  is the set of rationals of the form  m2^-n with integral  m  and n. These figures carry the algebraic structure of an idempotent, entropic, and commutative groupoid. We compare their properties with those of convex figures in the real plane. In particular, we characterize all dyadic convex line segments, and analyze the isomorphism types of dyadic triangles.

Abstract: Let  k  be an algebraically closed field. Hopf algebras over  k  of even dimension are sometimes more difficult to classify due to some relation and known formulas. It has been proved that for  p  an odd prime, Hopf algebras of dimension  2p  are semisimple. Also, if  H  is a nonsemisimple Hopf algebra of dimension  2p², either  H  or  H*  is pointed. In this talk, I will present some examples of Hopf algebras of such even dimensions as 2, 4, 6, 12, and talk about the current limitation in the case of dimension  4p.