# Working Seminar in Algebra

## Spring 2009

Fridays, 1:50–2:40 p.m., Milner 317

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:

- Monoidal functors, species and Hopf algebras (draft), Marcelo Aguiar and Swapneel Mahajan.
- Introduction to the Theory of Species of Structures, F. Bergeron, G. Labelle, P. Leroux.

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.

- Monoidal categories and functors
- Hopf monoids in species
- 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.