Schedule of events: (All talks are in Swain East 140, Coffee and tea in Rawles Hall 107)
9:30 - 11:00 Coffee, Tea, Bagels, etc.
11:00 - 12:00 Kiyoshi Igusa Axioms for higher torsion (abstract)
Lunch break
1:45 - 2:45 Jacob Rasmussen Khovanov homology and the slice genus (abstract)
3:00 - 4:00 Pisheng Ding Obstructions to certain Lie group actions (abstract)
Tea
4:30 - 5:30 Randy McCarthy On the Witt spectrum of a bimodule (abstract)
After dining at a local restaurant, there will be a dessert party (location to be announced) starting at 7:30.
Axioms for higher torsion:
Three invariants of smooth manifold bundles: higher Franz-Reidemeister (FR) torsion, higher Miller-Morita-Mumford (MMM) classes and Bismut-Lott (BL) higher analytic torsion are related by a formula which has been proved in many cases and is conjectured to hold in all cases. This talk will explain the relationship between these invariants in the nonequivariant case using a simply set of axioms. There are only two axioms. A glueing axiom and a transfer axiom. There are at most two linearly independent invariants satisfying these axioms. Since MMM and FR satisfy the axioms, any ``higher torsion theory'' is a linear combination of MMM and FR. Xiaonan Ma has shown that higher analytic torsion satisfies the transfer axiom. So, the conjecture is basically reduced to showing that it satisfies the glueing axiom. In this talk I will explain the axioms for higher torsion, define the higher MMM classes and show that they satisfy the axioms and show furthermore how any higher torsion theory (anything satisfying the axioms) can be computed up to two parameters. By computing these parameters in the three theories (FR, MMM, BL) we obtain a precise statement of the conjecture. We also obtain the known relationship between higher FR-torsion and the MMM classes on the mapping class group as an easy consequence.
Khovanov homology and the slice genus:
I'll describe a knot invariant s, which is defined using Khovanov's Jones polynomial homology. This invariant is strikingly similar to the invariant \tau which appears in knot Floer homology. In particular, s(K) gives a lower bound for the slice genus of K. As a corollary, we get a Khovanov homology proof of the Milnor conjecture.
Obstructions to certain Lie group actions:
Relating topological data of a G-manifold to that of the fixed-point set has always been of interest. This talk surveys some classical and recent results, and gives generalizations to some. The "flavor" of the results to be discussed can be captured by the slogan: "the nonvanishing of certain cohomological data gives obstruction to certain actions".
On the Witt spectrum of a bimodule:
We will introduce a functor $W(R,M)$ whose input is a (brave new) ring $R$ and a bimodule $M$ which takes values in pro-spectra. A theorem of Ayelet Lindenstrauss and the speaker shows that $W(R,M)$ is the Taylor tower, in the sense of the Goodwillie calculus of functors, for parametrized algebraic K-theory. We will discuss how fairly elementary ideas of calculus can then be used to generalize calculations of Lars Hesselholt and Ib Madsen for Bloch's typical curves ${\rm holim} \tilde K(R[x]/(x^n))$ and a new proof of the calculation by Carlsson, Cohen, Goodwillie and Hsiang of $A(\Sigma X)$ for $X$ a connected space. If time permits we will also discuss how the one-parameter Lefschetz class of Geoghegan and Nicas can be reinterpreted in this framework using work of Iwashita.