John Gillespie/Christian Roettger: The Galois connection between sequences and ideals.
Number Theory Seminar
Fall 2009 Archive:
Geoff Tims: The Congruent Number Problem: Its solution using elliptic curves.
Abstract: A rational number is called a congruent number if it is equal to the area of a right triangle with rational side lengths. The problem of finding congruent numbers was studied beginning more than 1000 years before mathematicians were finally able to determine a simple criterion to determine whether or not any rational is congruent (although its validity relies on the Birch/Swinnerton-Dyer conjecture). In this talk, I will give a brief history of the problem and results, culminating in 1983 with Tunnell's Theorem. Tunnell's Theorem, and other recent results on the problem, rely on elliptic curves, so a brief introduction to elliptic curves and their group structure will be given as well.
Thanksgiving Break: no seminar.
Jonathan Smith: Lambda-rings of automorphisms.
Abstract: The set of automorphisms of a single algebra forms a group under composition. Attention now shifts to the study of the full set of all automorphisms of all algebra structures in a given variety on a fixed finite set. A ring (with the added structure of a lambda-ring) is associated with this full set of automorphisms.
Eli Stines: Lattices of ideals of restricted power series.
Abstract: Formal power series over fields are a well cited example of commutative unital rings. They tend to have very nice factorization properties, hence their congruence lattices behave quite predictably. In this talk I will show how a simple restriction of the constant term of a formal power series can affect the congruence lattice and the divisibility structure of a power series ring. I will characterize the groups of divisibility of pseudo valuation domains, and use that to characterize the congruence lattices of rings of the form K + XF[[X]] for a field extension [F:K].
Yilin Cheng: p-dimensional Hopf algebras in the Yetter-Drinfeld category.
Abstract: I will talk about some properties for a semisimple p-dimensional
braided Hopf algebra with p an odd prime in the Yetter-Drinfeld category
over a 4-dimensional Taft algebra, and some results on Radford biproducts of
this kind of braided Hopf algebra and the Taft algebra.
Oktay Olmez: Directed strongly regular graphs.
Abstract: The concept of a strongly regular graph is one of the central objects in modern algebraic graph theory which was introduced by Bose. Looking at the algebraic characterization of strongly regular graphs, it seems to be natural to look for directed graphs whose adjacency algebras have similar properties. The notion of a directed strongly regular graph was introduced by A. Duval in 1988. Infinite series of directed strongly regular graphs were constructed after Duval's first paper. In this talk, we will consider some specific constructions such as directed strongly regular graphs arising from Cayley graphs, flag-algebras of Balanced Incomplete Block Designs with lambda equal to 1, tournaments, and block matrices.
Ling Long: A supercongruence motivated by the Legendre family of elliptic curves.
Abstract: We will discuss the background and proof of a new supercongruence related to the Legendre family of elliptic curves
Andrew Wells: Work on the subloop lattice of the Moufang loop of Zorn vector matrices over the integers modulo 4.
y2 = x ( x - 1 ) ( x - a ) .
This is joint work with Heng Huat Chan and Wadim Zudilin.
Abstract: The construction used to create simple Moufang loops from fields can also be used to make (non-simple) Moufang loops from commutative rings. In order to understand the structure of these loops, it is important to examine the structure of the loops arising from the ring of integers modulo 4. This talk examines the subloop lattice structure of the Moufang loop arising from the application of this construction to the ring of integers modulo 4. This subloop structure is related to the subloop structure of the simple Moufang loop which arises using Paige's construction on the ring of integers modulo 2. The subloop lattice of this "smallest simple Moufang loop" has been detailed by other authors, including Giuliani and Milies, and Vojtechovsky. The talk will include some key portions of the subloop lattice of the loop over the ring of integers modulo 4, illuminating the connections it has to the already established lattice of the smallest simple Moufang loop.
Mike Hilgemann: Frobenius-Schur indicators of Hopf algebras constructed from factorizable groups.
Abstract: The classical Frobenius-Schur theorem gives a simple indicator function which divides the irreducible representations of a finite group into three types, depending on the existence of certain bilinear forms. This result was extended by Linchenko and Montgomery to finite-dimensional semisimple Hopf algebras over C. In this talk, we examine the indicators of some semisimple Hopf algebras constructed from finite groups L which are factorizable, in the sense that there exist subgroups F and G with ttrivial intersection such that L = FG. We will consider some specific examples studied by Jedwab and Montgomery involving groups which admit only positive indicators, and discuss a conjecture that Hopf algebras arising from such groups will admit only non-negative indicators.
Eli Stines: Valuation domains.
Abstract: An integral domain is a valuation domain if in its quotient field, each element or its inverse lies back in the integral domain. A related structure is the Pseudo Valuation Domain (PVD). It turns out that these two structures have a rich interconnected relationship governed by the divisibility partial ordering.
Anna Romanowska (ISU/Warsaw University of Technology): Dyadic polygons.
The construction of the field of fractions for a given commutative unital ring is obviously a useful tool. This construction is a specific instance of a more general tool known as localization. In this talk we will look at localization in integral domains, specifically geared toward the construction of PVDs from valuation domains. We will investigate several examples, determining their groups of divisibility and lattices of ideals.
Abstract: Dyadic rationals are rationals whose denominator is a power of 2.
Dyadic triangles and dyadic polygons are respectively
defined as the intersections with the dyadic plane of a triangle or
polygon in the real plane whose vertices lie in the dyadic plane.
The one-dimensional analogues are dyadic intervals.
Algebraically, dyadic polygons carry the structure of a commutative,
entropic and idempotent algebra under the binary operation of
arithmetic mean. In this talk, dyadic intervals are classified up to algebraic isomorphism, and dyadic
polygons are shown to be finitely generated as algebras.
Jonathan Smith: Commutators and associators in groups and Catalan loops.
Abstract: Commutators and associators measure the deviation from commutativity and associativity in right and left loops. This talk will discuss some general aspects, and then present Jan Raasch's recent work approximating commutators and associators in Catalan loops.
Labor Day: no seminar.
Anna Romanowska (ISU/Warsaw University of Technology): Topology, Algebra and Categories in Logic.
Abstract: The conference with this title was held in Amsterdam in July. We report on some of the highlights.
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