Stanford University Topology Seminar 2006-7

Unless otherwise noted, all seminars are on Tuesdays 4:00 - 5:00 pm in Room 383-N (Third floor of Math Building, Bldg 380).
There is tea at 3:30 on Tuesdays on the 2nd floor in the Common Room

Autumn 2006 Schedule


Sept. 26, 2006

Speaker: No seminar this week.




Oct. 3, 2006

Speaker: Dan Ramras (Stanford)

Title: Deformation K-theory and Group Completion

Abstract: Deformation K-theory provides a homotopy theoretical setting for the study of representations of infinite discrete groups. After introducing deformation K-theory, I'll discuss the McDuff-Segal form of the Group Completion Theorem. This theorem normally provides a homological model for the group completion of a topological monoid, but I'll explain how in certain settings (including deformation K-theory) it actually provides a homotopy theoretical model. Two applications of this result will be discussed. The first uses Yang-Mills theory to relate deformation K-theory of surface groups to topological K-theory of the underlying surface, and the second involves the ``excision problem'' for free products.


Oct. 10, 2006

Speaker: Sverre Lunoe-Nielsen (Oslo and Stanford)

Title: A homological approach to topological cyclic homology

Abstract: I will explain a way of studying the fixed points of Bokstedt's topological Hochschild homology spectrum THH(B) of an S-algebra B, under the action of finite cyclic p-subgroups G of the circle group. While the strict equivariant structure of THH(B) is generally hard to calculate directly, there is always a natural comparison map from the fixed points to the homotopy fixed points. In favorable cases, this map is a p-equivalence or a p-equivalence when restricted to sufficiently high degrees. When B is the sphere spectrum, this statement is equivalent to Segal's Burnside ring conjecture for the group acting. For G = Z/p, Segal's conjecture was proven by W.H. Lin (p=2) and Gunawardena (p>2) via calculations in cohomology and the Steenrod algebra. This homological approach can be generalized to derive good comparison results between the fixed points and the homotopy fixed points when G is cyclic of prime order and B = BP and BP. The motivation for calculating the fixed points of THH comes from their connection with topological cyclic homology and algebraic K-theory via the cyclotomic trace map.


Monday, Oct. 16, 2006 (Note special day!), 4:00 p.m.

Speaker: Andrei Pajitnov (Nantes and Ohio State)

Title: Circle-valued Morse theory for knots and links



Oct. 24, 2006

Speaker: Grace Lyo (Berkeley and Stanford)

Title: Semilinear Representations in K-theory and the Grothendieck Ring of a Semisimple Twisted Group Ring

Abstract: In this talk, we will begin by discussing a conjectural model for the completed K-theory spectrum of a field in terms of the K-theory of the category of continuous semilinear representations of its absolute Galois group G_F. We will outline a strategy for showing that the conjecture holds in a special case. This strategy requires an understanding of the Grothendieck ring K_0 of the twisted group ring k, where k is a field on which G_F acts. In the time remaining, we will discuss semisimple twisted group rings and show how to compute the simple modules and the multiplication structure of the modules.


Oct. 31, 2006

Speaker: Alexandra Pettet (Stanford)

Title: Cohomology of the Torelli subgroup of Aut(F_n)

Abstract: The Torelli subgroup of Aut(F_n) consists of those automorphisms of the free group F_n which act trivially on the homology of F_n. It is often compared with the Torelli group of a surface, the subgroup of the mapping class group which acts trivially on the homology of the surface. Little is known about the finiteness properties of these groups, including whether or not they are finitely presented. I will describe some methods for studying the cohomology of the Torelli subgroup of Aut(F_n).


Nov. 7, 2006


This quarter BATS will be held at Stanford. Both talks will be in Room 383 N on the third floor of the Mathematics Department (Building 380).

 2:40 pm:

Speaker: Eric Babson (UC Davis)

Title: Homotopy theory for graphs


4:15 pm:

Speaker: Ciprian Manolescu (Columbia)

Title: A combinatorial description of knot Floer homology

Abstract: Knot Floer homology is an invariant of knots in the three-sphere, which detects the genus of the knot, and can be used to recover the Heegaard Floer homology of any surgery on that knot. The original definition, due to Ozsvath-Szabo and Rasmussen, involved counts of pseudoholomorphic disks in a symplectic manifold. In joint work with Peter Ozsvath and Sucharit Sarkar, we found a purely combinatorial description of this invariant. Starting with a grid presentation of the knot, we construct a special Heegaard diagram for the knot complement, in which the count of pseudolomorphic disks is elementary.


Note: Special day and room
Nov. 8, 2006 in 381T

Speaker: Andre Henriques (Univ. of Muenster)

Title: Orbispaces are equivalent to a diagram category

Abstract: A classical theorem of Elmendorf says that the homotopy theory of G-spaces is equivalent to that of continuous functors O_G -> Spaces, where O_G is the orbit category of G. We prove an analog of this result for orbispaces. The category that plays the role of O_G is now a topological category whose objects are groups, and whose morphisms for H to G are given by (Mono(H,G) x EG)/G, where G acts by conjugation in the target.


Nov. 14, 2006

Speaker: Boris Botvinnik (Oregon)

Title: Rational homotopy groups of the moduli space of positive scalar curvature metrics

Abstract: The problem of determining when a smooth compact manifold admits a positive scalar curvature (psc) Riemannian metric is comparatively well understood. However, even for the n-sphere, surprisingly little work has been done to date concerning the topological stucture of the space of all psc-metrics. In this talk, I will present some new results concerning the rational homotopy groups of this space for the n-sphere with n>4. My approach uses results on higher analytical/topological torsion due to Hatcher, Igusa, and Goethe.


Nov. 21, 2006

Speaker: There will be no seminar this week due to the Thanksgiving break.




Nov. 28, 2006

Speaker: No seminar this week due to the special algebraic geometry seminars.




Dec. 5, 2006

Speaker: TBA




Dec. 12, 2006