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Working Seminar in Algebra

Spring 2009

Fridays, 1:50–2:40 p.m., Milner 317

Seminar History

Our main focus for this semester is to work through parts of the new manuscript of Aguiar and Mahajan. (Other topics as interest and time dictate.) The main sources are:

January 30
No seminar

February 6
Matt Papanikolas
Tannakian categories and transcendence

We will discuss the basics of Tannakian categories and their applications to transcendental number theory. Via Tannakian duality, neutral Tannakian categories are categories of finite dimensional representations of algebraic groups over fields, or more generally of affine group schemes. The main application to transcendence that we will discuss is the Siegel-Shidlovskii Theorem, which relates the dimension of the differential Galois group of a system of linear ODE's to the transcendence degree of the special values of its solutions. This theorem provides insight for a conjecture of Grothendieck on periods of algebraic varieties.

February 13
Marcelo Aguiar
Overview of the Monograph with Mahajan

We explore possible directions to take the working seminar this semester. The monograph is divided into parts as follows.

  1. Monoidal categories and functors
  2. Hopf monoids in species
  3. Fock functors

We will give an indication of the main topics that make up each part, mentioning questions or possible extensions people may want to get involved with. Depending on people's interests, we may get started in one of various possible directions, perhaps by going to part II with minimal supplements from part I.

February 20
No seminar

February 27
Marcelo Aguiar
Overview of the Monograph with Mahajan, part II

I'll talk about Hopf monoids, Takeuchi's antipode formula, and two basic examples: the exponential species and the linear order species.

March 6
Svetlana Poznanovik
More examples of Hopf monoids in the category of species

We give several examples of Hopf monoids in species that arise naturally in combinatorics and morphisms between them.

March 13
Aaron Lauve
Universal constructions in the category of species

We describe the construction of free monoids and cofree comonoids in the category of species.

March 20
No seminar (Spring Break)

March 27
Marcelo Aguiar
The Coxeter complex of type A

The Coxeter complex of type A is the triangulation of the unit sphere induced by the braid hyperplane arrangement. We will describe this simplicial complex in combinatorial terms: chambers, faces, flats, the gallery metric, the structure of stars, and Tits projection maps.

April 3
Deepak Naidu
Bilax monoidal functors

The goal of this talk is to define and study an appropriate notion of a morphism between braided monoidal categories. We begin by defining appropriate notions of morphisms between monoidal categories: lax and colax monoidal functors. These two notions combine to give the notion of a bilax monoidal functor between braided monoidal categories. We will study two connections between bimonoids and bilax functors. We will end by defining and studying a certain property called normality for bilax functors.

April 10 (11 a.m.)
Sarah Witherspoon
The bilax structure of the chain complex functor

April 16 (Thursday, 1 p.m.)
Marcelo Aguiar
Fock functors

We define bilax monoidal functors from species to graded vector spaces and discuss their interrelationships.

April 24
Frank Sottile
Hopf monoids associated to the Coxeter complex and antipode formulas

The organizers this semester are Marcelo Aguiar and Aaron Lauve.