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50th Topology Festival

May 4–7, 2012

Abstracts of Talks

Ian Agol, University of California at Berkeley

The Virtual Haken Conjecture

We prove that cubulated hyperbolic groups are virtually special. Work of Haglund and Wise on special cube complexes implies that they are therefore linear groups, and quasi-convex subgroups are separable. A consequence is that closed hyperbolic 3-manifolds have finite-sheeted Haken covers, which resolves the virtual Haken question of Waldhausen and Thurston’s virtual fibering question. The results depend on a recent result of Wise, the malnormal virtually special quotient theorem; the cubulation of closed hyperbolic 3-manifolds by Bergeron-Wise using the existence of nearly geodesic surfaces by Kahn-Markovic; and a generalization of previous work with Groves and Manning to the case of torsion (which is joint with Groves and Manning).

CornellCast video of Agol’s talk

Francis Bonahon, University of Southern California

Hitchin Representations

The talk will be centered around certain group homomorphisms from the fundamental group of a surface S to a Lie group G. Over the past thirty years, much of low-dimensional topology and hyperbolic geometry has been based on the cases where G = SL2(R) or SL2(C). Hitchin, and more recently Fock, Goncharov, Labourie and others have shown how to extend some of the corresponding properties to more general Lie groups such as G = SLn(R). I will discuss additional results in this direction, obtained in collaboration with Guillaume Dreyer.

CornellCast video of Bonahon’s talk

David Gabai, Princeton University

Volumes of Hyperbolic 3-Manifolds

We will survey developments in the field, discuss some ongoing work (with R. Meyerhoff and N. Thurston) and state various open problems.

CornellCast video of Gabai’s talk

Allen Hatcher, Cornell University

A 50-Year View of Diffeomorphism Groups

This talk will be a survey of what is known about the homotopy types of diffeomorphism groups of smooth manifolds, as well as the classifying spaces of these groups.

Jacob Lurie, Harvard University

The Siegel Mass Formula, Tamagawa Numbers, and Nonabelian Poincaré Duality

Let L be a positive definite lattice. There are only finitely many positive definite lattices L′ which are isomorphic to L modulo N for every N > 0: in fact, there is a formula for the number of such lattices, called the Siegel mass formula. In this talk, I’ll review the Siegel mass formula and explain how it was reformulated by Weil as a statement about volumes of adelic groups. I’ll then describe some recent joint work with Dennis Gaitsgory on computing these volumes over function fields using ideas from topology: in particular, a nonabelian version of Poincaré duality.

CornellCast video of Lurie’s talk

Peter May, University of Chicago

What Is Equivariant Cohomology and What Is It Good For?

I’ll give a slanted overview of equivariant cohomology, starting with the use of ordinary (Bredon) cohomology, of which Borel cohomology H*(EG ×X) is a special case, to prove two classical results on the fixed point and orbit spaces of a G-space X, namely P.A. Smith theory and the Connor conjecture. For the second, we must extend our Z-graded cohomology theory to an RO(G)-graded theory, and that is also necessary for equivariant Poincaré duality. The equivariant analogue of an abelian group is a coefficient system, and an ordinary Z-graded theory extends to an RO(G)-graded theory if and only if its coefficient system extends to a “Mackey functor.” That is a standard notion in representation theory when G is finite, but a topological reinterpretation in terms of equivariant stable homotopy groups extends the notion to compact Lie groups. I’ll briefly describe the stable homotopy category of G-spectra, which is the natural home for RO(G)-graded cohomology. This is all ancient history, at least 30 years old, but I’ll give a glimpse of the modern theory. As a matter of sheer good luck, the unreasonable effectiveness of equivariant ideas, the key calculational input to the recent solution of the Kervaire invariant problem by Hill, Hopkins, and Ravenel is an easy calculation of certain RO(G)-graded ordinary cohomology groups of a point.

CornellCast video of May’s talk

Dusa McDuff, Barnard College / Columbia University

Embedding Questions in Symplectic Geometry

Gromov’s work on the nonsqueezing problem showed that embedding questions lie at the heart of symplectic geometry. There has been much recent progress in understanding these questions, especially as far as symplectic ellipsoids are concerned. This talk will discuss a variety of results, in four dimensions and above.

CornellCast video of McDuff’s talk

John Milnor, Stony Brook University

Small Denominators: Adventures Through the Looking Glass

After a brief historical survey of small denominator problems, the talk will focus on cubic rational maps which commute with the antipodal map. Joint work with Araceli Bonifant and Xavier Buff.

CornellCast video of Milnor’s talk

Tom Mrowka, Massachusetts Institute of Technology

Instantons and Knots

In the past few years Kronheimer and I have been reinvestigating a various versions of instanton Floer homology for knots or links in three manifolds. To define these theories we need to label each component of the link by a partial flag manifold for Cn. A couple of years ago we used one version to prove that Khovanov homology detects the unknot. More recently we used this same version to prove that Rasmussen’s s-invariant could not detect homotopy spheres. We’ve now understood how to generalize this theory to knotted trivalent graphs and are beginning to understand its relation to Khovanov-Rozansky homology. This talk will survey some of these developments.

CornellCast video of Mrowka’s talk

Walter Neumann, Barnard College / Columbia University

Local Metric Geometry of Complex Varieties

The local metric geometry of complex varieties seems to have been first addressed by Pham and Teissier in a 1969 preprint (for singular complex curves), and has arisen since in work of Hsiang and Pati, Hardt and Sullivan, etc. It is only relatively recently, starting with a 2008 paper of Birbrair and Fernandes, that its richness in higher dimensions started to be clear. The talk will address mainly the local metric geometry at a point of a complex surface, which is now well understood, in terms of a collection of rational numbers associated to components of a refined JSJ decomposition of the 3-manifold link of the point (recent work of Birbrair, Pichon and the speaker). It is the rigidity of 3-manifold topology which makes the surface case accessible, so less is known in higher dimensions. If time permits, some applications will also be described.

CornellCast video of Neumann’s talk

Hee Oh, Brown University

Circle Packings and Ergodic Theory

We will discuss counting and equidistribution results for circle packings in the plane invariant under a Kleinian group. These questions turn out to be related to the infinite ergodic theory of flows on hyperbolic manifolds where many important problems are wide open. I will discuss some of these problems as well. This talk is based on joint work with Nimish Shah.

CornellCast video of Oh’s talk

John Pardon, Stanford University

Totally Disconnected Groups (Not) Acting on Three-Manifolds

Hilbert’s Fifth Problem asks whether every topological group which is a manifold is in fact a (smooth!) Lie group; this was solved in the affirmative by Gleason and Montgomery-Zippin. A stronger conjecture is that a locally compact topological group which acts faithfully on a manifold must be a Lie group. This is the Hilbert-Smith Conjecture, which in full generality is still wide open. It is known, however (as a corollary to the work of Gleason and Montgomery-Zippin) that it suffices to rule out the case of the additive group of p-adic integers acting faithfully on a manifold. I will present a solution in dimension three. The proof uses tools from low-dimensional topology, for example incompressible surfaces, minimal surfaces, and a property of the mapping class group.

CornellCast video of Pardon’s talk

Ronald Stern, University of California at Irvine

Pinwheels, Smooth Structures, and 4-Manifolds with Euler Characteristic 3

We will develop techniques to construct symplectic 4-manifolds focusing on those with Euler characteristic three. When applied to the case that the first Betti number is zero we construct infinitely many distinct symplectic fake projective planes, i.e. symplectic 4-manifolds with Euler characteristic 3 and first Betti number zero. There are corresponding results for such smooth, but non-symplectic, 4-manifolds. This is joint work with Ron Fintushel.

CornellCast video of Stern’s talk

Peter Teichner, University of California at Berkeley / MPI Bonn

Iterated Disk Constructions in 4-Manifold Topology

We will survey various approaches in the last decades to detecting and resolving the failure of the topological Whitney trick in dimension four.

CornellCast video of Teichner’s talk

William Thurston, Cornell University