University of Chicago Algebraic Topology Seminar

2:30 pm   Monday, March 27, 2006

Algebraic Topology: (Colloquium) Computing homotopy groups of spheres using modular forms
by Mark Behrens (MIT) in E 206


The (stable) homotopy groups of spheres are very complicated, and the computations seem at first glance chaotic. This chaos actually breaks up into periodic patterns, however, and understanding these periodic phenomena has been a focal point in the study of the homotopy groups of spheres. The first layer of periodicity is well understood, and is closely linked with Bott periodicity and K-theory. The second layer of periodicity displays much more complicated patterns. I will discuss an emerging picture of how these patterns are actually dictated by the arithmetic of modular forms. The role that K-theory had in the first layer of periodicity is replaced by the cohomology theory of topological modular forms.



4:30 pm   Tuesday, March 28, 2006

Algebraic Topology: Cohomology theories associated to Shimura varieties
by Mark Behrens (MIT) in E 203


Goerss, Hopkins, Miller, and their collaborators constructed a spectrum of topological modular forms, and Goerss, Henn, Mahowald, and Rezk used this theory to give a resolution of the K(2)-local sphere at the prime 3. Half of this resolution may be recovered at all primes using isogenies of elliptic curves and the building for GL_2(Q_l). In this talk, I will describe how recent work of Lurie may be exploited to associate spectra of topological automorphic forms to certain Shimura varieties associated to a form of the unitary group U(1,n-1). A piece of the resolution of the K(n)-local sphere is recovered by considering the building for GL_n(Q_l).


3:00 pm   Tuesday, April 18, 2006

Algebraic Topology: Quandle Algebras
by Alissa Crans (Ohio State University) in E 203


Self-distributive binary operations have appeared extensively in knot theory in recent years, specifically in algebraic structures called `quandles.' A quandle is a set equipped with two binary operations satisfying axioms that capture the essential properties of the operations of conjugation in a group. The self-distributive axioms of a quandle correspond to the third Reidemeister move in knot theory. Thus, quandles give a solution to the Yang-Baxter equation, which is an algebraic distillation of the third Reidemeister move. We formulate analogues of self-distributivity in the categories of coalgebras and Hopf algebras and use these to construct additional solutions to the Yang-Baxter equation. Please note that the seminar meets at 3:00 this week only.


4:30 pm   Tuesday, April 25, 2006

Algebraic Topology: A localization sequence for K(ku)
by Andrew Blumberg (Stanford) in E 203


In this talk, I will briefly outline the emerging program to understand the K-theory of the sphere spectrum via a chromatic filtration in K-theory, due to Rognes (based on an idea of Waldhausen). Part of this program requires the study of the K-theory of certain nonconnective ring spectra, conjecturally via localization sequences. I will discuss the resolution of the first nontrivial case of this conjecture, a particular localization cofiber sequence K(Z) -> K(ku) -> K(KU) which arises from the localization ku -> KU. This is joint work with Mike Mandell.


4:30 pm   Tuesday, May 2, 2006

Algebraic Topology: String topology of classifying spaces
by Kate Gruher (Stanford University) in E 203


Let G be a compact Lie group. One of the first difficulties in studying the string topology of the classifying space BG is that the loop product constructions of Chas-Sullivan and Cohen-Jones do not apply, as BG is not a (finite dimensional) manifold. I will discuss how to generalize their constructions to fiberwise monoids over manifolds, and how to use these constructions to describe the notion of the string topology of BG by associating to BG a pro-ring spectrum related to the loop space LBG. Along the way I will indicate a proof of the homotopy invariance of the loop product for closed manifolds, as well as relationships to recent work of Freed, Hopkins, and Teleman.


4:30 pm   Tuesday, May 16, 2006

Algebraic Topology: The mod 2 homology of the spaces in the connective Omega-spectrum of topological modular forms
by Paul Pearson (Northwestern University) in E 203


The mod 2 homology of topological modular forms (tmf) is a Hopf ring. Every mod 2 Hopf ring is equivalent to a Dieudonne' ring. We calculate the Dieudonne' ring for the nonnegatively graded spaces in the Omega-spectrum for tmf, and show that it splits into two parts that occasionally overlap. One part comes from the homology of the spectrum tmf, and the other part comes from the homology of QS^k, the kth space in the Omega-spectrum of the sphere.


2:00 pm   Thursday, May 18, 2006

Algebraic Topology: Obstructions to Weak A-Cellular Inequalities
by Michele Intermont (Kalamazoo College) in E 203


Let A be a pointed CW complex. The collection of A cellular spaces is the collection of spaces which can be given, up to weak equivalence, in terms of hocolims of A. Weakening this condition by allowing extensions by fibrations, we consider the closed class of spaces this generates. In this talk, we explore this weak cellular inequality and provide a characterization of X being weakly A cellular. This is joint work with Jeff Strom, Western Michigan University. Pretalk at 1:30 in the same room.


4:30 pm   Tuesday, May 23, 2006

Algebraic Topology: Toward an E-infinity structure for Quinn's bordism-type spectra
by Jim McClure (Purdue University) in E 203


This is joint with Gerd Laures. Quinn's formalism of ``bordism-type spectra'' is a basic tool in geometric topology (for example, this is how surgery spectra are constructed). But the construction itself is a purely homotopy-theoretic one and should be of independent interest to algebraic topologists. The goal of my work with Gerd Laures is to give a sufficient condition for bordism-type spectra, and maps between them, to be E-infinity. The lecture will not assume any prior knowledge of Quinn's work (but those who are interested might like to look at his paper Assembly maps in bordism-type theories).


2:00 pm   Thursday, June 1, 2006

Algebraic Topology: On topological lifts of algebraic diagrams
by Jim Turner (Calvin College) in E 203


In a program begun by W. Dwyer, D. Kan, and C. Stover, machinery was developed for determining whether a \Pi-algebra (a kind of twisted product of a graded Lie algebra with a group) can be realized by the homotopy groups of a space. One refinement of this program involves extending this machinery to address lifting diagrams of \Pi-algebras (such as arise from homotopy commutative diagrams of spaces) to diagrams of spaces that commute on the nose. Such a generalization opens the door to interpreting such things as Toda brackets as obstructions to making such extensions. In this talk, we describe such an extension of the Dwyer-Kan-Stover program and describe some aspects of computing the obstruction groups and potential connections to higher homotopy operations. This is joint work with David Blanc and Mark Johnson. Pretalk at 1:30 in same location.