
Combinatorics, Algebra, Number Theory Seminar 2013 Archive:

December 2, 9

Jolie Roat: Hopf algebras: An introduction, I, II.
Abstract:
Hopf algebras, named for German mathematician Heinz Hopf, are bialgebra structures together with an antiautomorphism, called the antipode. These talks will serve as an introduction to Hopf algebras, including their definition and some properties. Additionally, we will introduce the notion of an integral in a Hopf algebra, as well as prove its existence and uniqueness.

November 4, 11, 18

Ryan Johnson: The FrobeniusSchur indicator of TambaraYamagami categories, I, II, III.
Abstract:
Fusion categories lie in the intersection of group theory, knot theory, and quantum physics. If one is given a fusion category, a sequence of complex numbers can be computed which are called the FrobeniusSchur indicator. In these talks I will consider a particular subclass of fusion categories whose data is defined using a finite abelian group and a bilinear form on that group. Computing the indicator of these categories requires the use of quadratic Gauss sums. The aim of my research is to show that the FrobeniusSchur Indicator of TambaraYamagami categories is unique up to equivalence.

October 21, 28

Michael Munywoki: The upper triangular algebra loop of degree 4, I, II.

October 14

Paul Hertz: Representing quasigroups with proper graph colorings.
Abstract:
It is a wellknown fact in combinatorics that every quasigroup of finite order n can be represented by a proper edge coloring of the complete bipartite graph K_{n,n} , and vice versa. I will discuss generalizations of this theorem, including representations of commutative quasigroups, idempotent quasigroups, loops (quasigroups with identity) and unipotent quasigroups by proper edge colorings and proper total colorings of various graphs. Depending on time available, I may also discuss some other work I've been doing on quasigroups.

September 30

John Gillespie: Computing with the lambda calculus.

September 16

Anna Romanowska (Warsaw University of Technology): Lattices without absorption.
Abstract:
What will happen with lattices, and in particular with distributive
lattices, when we drop the absorption laws?
We will discuss the properties of algebras obtained by
such an axiomatization, in particular algebras satisfying one or two
distributive laws. Some structural characterizations will be provided,
along with some examples and applications, based on various older and newer
results.

September 9, 23; October 7

Jonathan Smith: Entropic Hopf algebras, I, II, III.
Abstract:
Hopf algebras embody algebra and coalgebra structures that provide a common framework for both groups and Lie algebras. They are traditionally defined with an underlying vector space or module structure. In these talks, we will begin to consider Hopf algebras from the standpoint of universal algebra, based on the general concept of entropic algebras that includes modules, sets, semilattices, and convex sets.

September 2

Labor Day: no seminar.

January 17, 3:104pm, Carver 401

Rachel Davis (University of WisconsinMadison): On the images of outer Galois representations associated to elliptic curves
Abstract:
For ladic representations of the absolute Galois group associated to elliptic curves,
much is understood about the sizes of the images and about the conjugacy invariants of the images
of Frobenius elements. On the other hand, much less is known about the outer Galois representations
associated to elliptic curves. These are representations from the absolute Galois group to an outer
automorphism group of a free prol group.
The goal of this research is to take a first step in understanding more concretely Galois representations to automorphism groups of nonabelian groups. Let E be a semistable elliptic curve over Q
of negative discriminant with good supersingular reduction at 2. Associated to E, there is a Galois representation to a certain subgroup of the automorphism group of a metabelian group. I show that there is a Galois representation surjecting to this subgroup (with the right ramification). Then, I compute some conjugacy invariants for the images of the Frobenius elements. This will give rise to
new arithmetic information analogous to traces of Frobenius for the ladic representation.
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