Combinatorics, Algebra,
Number Theory Seminar
Spring 2009 Archive:

April 27
Yi Lin Cheng: On some Hopf algebras of even dimension II.
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 .
April 20
Domenico D'Alessandro: General methods for control of systems on compact Lie groups.
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.
April 13
Ling Long: Zeros of some level 2 Eisenstein series.
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.
April 6
Elijah Stines: Valuation domains, pseudovaluation domains, and the group of divisibility.
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.
March 30
Tim Huber and Jonathan Smith: Report from the AMS Sectional Meeting, Urbana-Champaign, Illinois.

March 23
Tim Huber: Differential equations for Eisenstein series on subgroups of the modular group.
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.
March 16
Spring break: no seminar.

March 2, 9
Jonathan Smith: Catalan loops.
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' )
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.
February 23
Tim Gillespie (University of Iowa): Automorphic  L-functions and a prime number theorem.
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.
February 16
Anna Romanowska (ISU/Warsaw University of Technology): Dyadic geometry.
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.
February 9
Faculty meeting: no seminar.

February 2
Candidate colloquium: no seminar.

January 26
Yi Lin Cheng: On some Hopf algebras of even dimension.
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.
January 19
King Holiday: no seminar.

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