The seminar will meet Tuesdays at 4:00pm in 2-132 unless otherwise noted.

This talk will discuss what topological quantum field theories are, and
how to construct them. A "folklore" theorem states that a 2-dimensional
TQFT is equivalent to a Frobenius algebra. 3-dimensional TQFTs are harder
to construct, and the main point of the talk will be to show that the
corresponding algebraic object that produces one is a modular tensor
category. I'll discuss tensor category basics +additional structures
(braiding, traces) that may occur in a tensor category, with examples.
I have been instructed to make this accessible to first years.

Ordinary morse theory allows us to give cellular constructions of
manifolds by studying critical points of vector fields. In this talk,
I'll describe a discrete generalization of this due to Robin
Forman. We'll learn about Morse functions and gradient vector fields on
simplicial sets, and how these give rise to simplified computations of
homology.

09.26.06:

Matt Gelvin (

MIT).

Homology and Cohomology of Discrete Groups.

This talk will focus on some of the basics of the theory of group (co)homology. After discussing the general setup, I'll give some applications, such as the transfer map and group extensions. While the initial definitions will be made in both topological and purely algebraic contexts, much of what follows will deal more with the algebraic side of things.

10.03.06:

Ricardo Andrade (

MIT).

Simplicial (homotopy) groups.

In this talk, the focus will be on simplicial sets and how to compute
homotopy groups of simplicial sets without reference to topological
spaces. For this purpose, it will be useful to analyze simplicial groups
and to introduce special ones that mimic such familiar objects as loop
spaces in the topological category.

10.17.06:

Nick Rozenblyum (

MIT).

Localization and Completion of Spaces.

In his work on the spectral sequence for the cohomology of fibrations,
Serre noticed that one can carry out many of the same arguments looking at
homology and homotopy mod p. This allowed him to prove that the stable
homotopy groups for spheres are finite (except in dimension 0). I will
discuss a number of constructions of completions and localizations of
spaces which rigidify this notion and are the topological analogues of
localizations and completions in algebra. If time permits, I'll
discuss some applications of these ideas to H-spaces.

10.24.06:

Veronique Godin (

Harvard).

Hyperbolic geometry, moduli spaces of Riemann surfaces, mapping class groups and a PROP.

A smorgasbord of not-so-algebraic topology. May contain traces of
connections and geodesics. Bring your own wine

Can the model structure of a stable model
category be recovered just from the triangulated structure of its
homotopy category? In this talk we will discuss examples such as the
K-local stable homotopy category.

Let M be a topological manifold. Under what conditions can M be given the structure of a smooth manifold, and how can such structures be enumerated? I will explain how this question (in suitable dimensions) reduces to a problem in homotopy theory.

11.14.06:

Sam Isaacson (

Harvard).

Homotopy algebras in model categories.

Operads and PROPs are devices for collating multiplication maps.
Boardman, Vogt, and May introduced them to describe homotopy-invariant
algebraic structures on spaces. In this talk, I'll discuss a recent
paper of Berger and Moerdijk's that generalizes homotopy-invariance to
operads and algebras in monoidal model categories.

11.28.06:

Josh Nichols-Barrer (

MIT).

Quasi-Categories with an Application or Two.

We recall the basic facts of quasi-category theory, with a
review of the extant Quillen equivalences relating the theory of
quasi-categories to other homotopy theories of homotopy theory. We will
then try to present some situations for which quasi-categories provide a
natural language, including (pseudo)functors and a notion of p-adic Euler
characteristic for (some) Kan complexes.

12.05.06:

Peter Lee (

MIT).

NP-hard problems in rational homotopy theory.

After a quick overview of complexity theory and
rational homotopy theory, I'll talk about some very
simple problems that turn out to be very hard.

12.12.06:

Grace Lyo (

UC Berkeley).

Semilinear Actions of Galois Groups and the Algebraic $K$-theory of Fields.

We will discuss a conjecture of Carlsson's that describes the completed $K$-theory spectrum of a field in terms of the semilinear representation theory of its absolute Galois group. The completion involved in this conjecture is called the derived completion, which is a construction of Carlsson's that is modeled after the completion in algebra and, in addition to possessing many naturality properties, agrees with the Bousfield completion on $K$-theory spectra of fields. We will sketch a proof that Carlsson's conjecture holds in a special case.

Please send mailing list requests and questions to Teena Gerhardt.

Other useful links: