University of Chicago Algebraic Topology Seminar

4:30 pm   Tuesday, January 4, 2005   in Eckart 203

Algebraic Topology: Tropical geometry and the Bergman complex
by Caroline Klivans (Chicago)



4:30 pm   Tuesday, January 11, 2005   in Eckart 203

Algebraic Topology: Localization of Mackey functors
by Gaunce Lewis (Syracuse)


The category of Mackey functors for a finite group $G$ is a symmetric monoidal closed abelian category. The unit for the tensor product on this category is the Burnside Mackey functor $B$. Purely formal results about symmetric monoidal closed abelian categories allow us to localize the category of Mackey functors at any Mackey functor prime ideal $P$ of $B$. These localizations ought to be valuable tools for studying the contribution of each subgroup $H$ of $G$ to any given Mackey functor, such as the representation ring. Unfortunately, the formal approach to localization is so abstract that it provides very little insight into the behavior of these localizations of the category of Mackey functors. In this talk, we provide a much more concrete alternative description of these localizations. This approach is far more tractible and more easily related to geometry.


4:30 pm   Tuesday, January 18, 2005   in Eckart 203

Algebraic Topology: Open-closed string topology
by Antonio Ramirez (Stanford)


The area of string topology began with a construction by Chas and Sullivan of previously undiscovered algebraic structure on the homology H_*(LM) of the free loop space of an oriented manifold M. Among other results, Chas and Sullivan showed that H_*(LM), suitably regraded, carries the structure of a graded-commutative algebra. The product pairing was subsequently extended by Cohen and Godin into a form of topological quantum field theory (TQFT). Open-closed string topology, first sketched by Sullivan, arises when considering spaces of paths in M with endpoints constrained to lie on given submanifolds (the so-called D-branes). In this talk, I describe a way to extend the TQFT structure of string topology into an analogue of TQFT which incorporates open strings. The method of construction is homotopy theoretic, and it makes use of constrained mapping spaces from fat B-graphs (which I define) into the ground manifold M. I also discuss how certain spaces of fat B-graphs model the classifying space of constrained diffeomorphism groups of an open-closed cobordism.


4:30 pm   Tuesday, January 25, 2005   in Eckart 203

Algebraic Topology: The Thom Isomorphism in Equivariant Elliptic Cohomology
by David Gepner (UIUC)


Inspired by work of Grojnowski and Greenlees on Rational Circle- Equivariant Elliptic Cohomology, we give a new construction of an equivariant theory valid for any compact Lie group which agrees with Greenlees' when C is an elliptic curve over the rationals and Grojnowski's in the case of a complex- analytic curve (both of which are only defined in case G=S^1). We then go on to construct a natural Thom isomorphism in elliptic (co)homology for complex vector bundles with certain vanishing conditions on their Borel-equivariant Chern classes. Applications include a short and elegant proof of the Witten rigidity theorem.


4:30 pm   Tuesday, February 1, 2005   in Eckart 203

Algebraic Topology: A Generalization of the Character Theory of Hopkins, Kuhn and Ravenel
by Jacob Lurie (AIM)



4:30 pm   Tuesday, February 8, 2005   in Eckart 203

Algebraic Topology: Orbispaces as stratified fibrations
by Andre Henriques (MIT)


I'll explain why orbispaces can be viewed as a special case of stratified fibrations. Then I'll construct a stratified classifying space for orbispace structures. If time permits, I'll show how to use it in order to prove that compact orbispaces have enough vector bundles.


4:30 pm   Tuesday, February 15, 2005   in Eckart 203

Algebraic Topology: Topological Hochschild cohomology of some ring spectra
by Vigleik Angeltveit (MIT)



4:00 pm   Monday, February 21, 2005   in 636 SEO (the math building) at UIC

AllChicago Algebraic Topology: Interchange of operad actions, E_n-structures and applications
by Rainer Vogt (University of Osnabruck)


An E_n-structure is essentially the algebraic structure of an n-fold loop space Omega^n X. More precisely, an n-fold loop space has an E_n-structure, and a connected E_n-space is of the weak homotopy type of an n-fold loop space. Now Omega^n X has n interchanging single loop space strucures. So a space having an E_k-structure and an E_l-structure which interchange morally should be an E_k+l-space. We discuss this question. The talk recalls the notions of an E_n-structure and of the interchange of two algebraic structures, before it treats the question mentioned above. We apply the results obtained to show that the topological Hochschild homology THH(R) of an E_n+1 ring spectrum is an E_n ring spectrum. Remarks: (1) The last result has also been obtained by Maria Basterra and Mike Mandell using different methods. (2) The talk is about joint work with Zig Fiedorowicz.

PRACTICAL INFO:This is the third AllChicago topology seminar this time at UIC. The talk will be preceded by a "pre-talk" at 3pm (in 512 SEO) and tea at 3:45 in 636 SEO. This will give graduate students and others time to learn about a bit of the background for this talk. After the talk we will be going out for dinner at a local restaurant. Directions: There is visitor parking on the northeast corner of Taylor and Halsted. SEO is a few blocks west, just north of Taylor on Morgan; it is the second tallest building on this UIC campus. Contact Brooke Shipley (bshipley at math.uic.edu) with any further questions. Contact anyone from the UC topology group if you want to share a ride up. Peter leaves at 2:00; meet in front of his office. Nathalie leaves at the break in Benson's course ca 2:20; meet in front of E202.


4:30 pm   Tuesday, February 22, 2005   in Eckart 203

Algebraic Topology: Homotopical group theory
by Jesper Grodal (Chicago)


I'll describe how to view a finite group from the point of view of homotopy theory, and explain how this leads to a good notion of a p-local finite group. I'll then go on to discuss their extension theory and speculate on the possibility for a classification. How would this relate to the classification of finite simple groups??
This talk is in part meant to be a teaser for my course next quarter with Daniel Biss entitled "Cohomology of Groups".


4:30 pm   Tuesday, March 1, 2005   in Eckart 203

Algebraic Topology: (infinity,n)-Stacks and Algebraic Geometry over Structured Ring Spectra
by Clark Barwick (UPenn)


There are two different ways --- due, evidently independently, to C. Simpson and C. Rezk --- to use G. Segal's delooping machine to produce, for any n, a theory of (infinity,n)-categories such that the i-morphisms are isomorphisms for i>n. As I shall explain, the category of these ``(infinity,n)-categories'' carries a number of model structures of interest, and one can use the homotopy theory of these (infinity,n)-categories to formulate the theory of symmetric monoidal (infinity,n)-categories and that of (infinity,n)-stacks. Using these ideas, I shall discuss some surprising applications of homotopy theory to algebraic geometry. In particular, I sketch the proof of a powerful new local-to-global principle for complexes of modules, and for an algebraic variety X, I describe a kind of motivic homotopy type X_M, the representations of the loop space of which correspond naturally to cohomology theories on X.


4:30 pm   Tuesday, March 29, 2005   in Eckart 203

Algebraic Topology: Iterated THH and obstruction theory for E_n ring spectra
by Mike Mandell (Cambridge University)