University of Chicago Algebraic Topology Seminar

Algebraic Topology Seminar
Location: Room 203 in Eckhart Hall (Directions)
Day: Tuesdays
Time: Pretalk at 3:00, Talk at 4:30 (unless otherwise specified)
Practical Information for speakers.

To request a future speaker or for questions, please contact Vigleik Angeltveit (vigleik AT math.uchicago.edu).
To sign up for electronic announcements of talks, please visit http://zaphod.uchicago.edu:8080/mailman/listinfo/topology.
Here is a link to the proseminar.

Winter 2010 Seminars

4:30 pm Tuesday, January 12, 2010 in E203

Algebraic Topology

The K(2)-local Picard group

Paul Goerss

The Picard group of a symmetric monoidal category is the isomorphism classes of invertible objects. For the stable homotopy category itself, the only invertible objects are the spheres. However, Mike Hopkins has observed that, after localizing the stable category at a homology theory, the Picard group can contain exotic elements and, as a result, this group is a sensitive invariant of the new category. This is especially interesting when the homology theory is complex oriented, as we can then compare homotopy theory and algebraic geometry computations. In this talk, I will show how to compute the Picard group at chromatic level 2 and primes bigger than 2. In particular, at the prime 3 there is a significant group of exotic elements. This is joint work with Hans-Werner Henn and builds on work of Shimomura, Karamanov, Hopkins, and others.

1:30 pm Tuesday, January 26, 2010 in E203

Algebraic Topology

The universal loop space operad and generalisations

Eugenia Cheng

The problem of modelling homotopy types by omega-groupoids has been approached in many different ways in recent years. I will discuss the theory proposed by Batanin/Leinster using globular operads. There are various analogies with the use of operads to study loop spaces; I will recall the loop space operad E called "universal" by Salvatore, and the analogous globular operad G which can be used to recognise the fundamental ω-groupoid of a space. I will give a categorical proof of the universal property of E, enabling generalisation to prove a universal property of G. The helps us identify other, more tractable, operads for fundamental ω-groupoids.

4:30 pm Tuesday, February 2, 2010 in E203

Algebraic Topology

Mapping spaces of commutative ring spectra and localizations of spaces

Jennifer French

We can study the homotopy groups of a space by studying it locally---rationally and p-adically for all primes. We can study the p-adic homotopy further by considering the K(n)-localizations for all n. The goal is to generalize the algebraic models for rational homotopy theory developed by D. Sullivan and D. Quillen, and for p-adic homotopy theory developed by M. Mandell. The natural context to generalize these models is as mapping spaces of (commutative) R-algebras, where R is an E-infinity ring spectrum. We will explore these R-algebra mapping spaces as models for certain localizations of spaces in the cases R = Hk, where k is the algebraic closure of the field F_p, and the case that R is the K(1)-local sphere spectrum.

4:30 pm Tuesday, February 9, 2010 in E203

Algebraic Topology

Homotopic descent and codescent

Kathryn Hess

In this talk I'll introduce a general homotopy-theoretic framework in which to study problems of (co)descent and (co)completion. Since the framework is constructed in the universe of simplicially enriched categories, this approach to homotopic (co)descent and to derived (co)completion can be viewed as ∞-category-theoretic. I'll present general criteria, reminiscent of Mandell's theorem on E_∞-algebra models of p-complete spaces, under which homotopic (co)descent is satisfied. I'll also construct general descent and codescent spectral sequences and explain how to interpret them in terms of derived (co)completion and homotopic (co)descent. To conclude I'll sketch a few applications, such as the close relationship between derived cocompletion and Baum-Connes and Farrell-Jones-type isomorphism conjectures for assembly. I'll also explain how to obtain certain very well-known spectral sequences, such as the unstable and stable Adams spectral sequences, the Adams-Novikov spectral sequence and the descent spectral sequence of a map, as examples of general (co)descent spectral sequences and briefly sketch the relationship between the Lichtenbaum-Quillen conjecture and homotopic descent along the Dwyer-Friedlander map from algebraic K-theory to étale K-theory.

For scheduled Topology Seminars in future quarters or for all of the UC Math Department seminars by date, go to the Math Department Calendar.
For all of the UC Math Department seminars by subject, go to the UC Seminar List.

Past talks: Fall 98, Winter 99, Spring 99, Fall 99, Winter 00, Spring 00, Fall 00, Winter 01, Spring 01, Fall 01, Winter 02, Spring 02, Fall 02, Winter 03, Spring 03, Fall 03, Winter 04, Spring 04, Fall 04, Winter 05, Spring 05, Fall 05, Winter 06, Spring 06, Winter 07, Fall 07, Spring 08, Fall 08, Winter 09. Fall2009.

Other seminars: Northwestern, MIT Topology Seminar, Stanford Topology Seminar.