Topology Festival

Abstracts of Talks

The 2006 festival included a concentration of talks in algebraic topology and its applications. There were 11 talks in total. Here are the speakers and their abstracts.

Noel Brady, University of Oklahoma

Perron-Frobenius Eigenvalues, Snowflake Groups and Isoperimetric Spectra

The Dehn function of a finitely presented group measures the difficulty of filling loops in the Calyey complex of the finite presentation. These functions offer a geometric measure of the complexity of the word problem in finitely presented groups. They have natural higher order generalizations. The (n – 1)th order Dehn function of a group of type F_n measures the difficulty of filling (n – 1)-spheres with n-balls in suitable complexes associated to the group.

We describe a simple geometric construction (the iteratively suspended snowflake construction) which produces groups of type F_n whose (n – 1)th order Dehn functions are of the type x^a for a large range of exponents a.

Gunnar Carlsson, Stanford University

Algebraic Topology and High Dimensional Data

In recent years techniques have been developed for evaluating Betti numbers of spaces given only a finite but large set of points (a "point cloud") sampled from the space. Such techniques can be applied in statistical settings where the data being considered is high dimensional (and hence cannot be visualized) and non-linear in nature. This talk will describe the techniques required to obtain the Betti numbers, as well as discuss some actual examples where they have been applied.

Thomas Farrell, Binghamton University

Some Applications of Topology to Geometry

I will discuss how smoothing and psuedo-isotopy theories have implications about:

1. harmonic maps between negatively curved Riemanian manifolds,
2. the space of negatively curved Riemanian metrics on a closed high dimensional smooth manifold, and
3. the Ricci flow starting at a negatively curved metric on a high dimensional manifold.

These implications are due to several people working together in various combinations. They include L.E.Jones, P.Ontaneda, M.S.Raghunathan, C.S.Aravinda and myself.

Soren Galatius, Stanford University

Stable Homology of Automorphisms of Free Groups

Let Aut(F_n) denote the automorphism group of a free group on n generators. It is known that group homology H_k(Aut(F_n)) is independent of n as long as n >> k. There is a natural homomorphism from the symmetric group S_n to Aut(F_n), I will sketch a proof that it induces an isomorphism from H_k(S_n) to H_k(Aut(F_n)) for n >> k. An important point of view here is that BAut(F_n) can be thought of as a moduli space of metric graphs, i.e. graphs equipped with metrics, considered up to isometry.

Robert Ghrist, University of Illinois, Urbana-Champaign

Homological Sensor Networks

The ability to engineer and fabricate sensors of increasingly small size and scope has the promise of allowing "smart dust" networks for security applications. However, engineers are finding that it is difficult to extract global information from a network of local sensors. Algebraic topologists have spent the past century perfecting techniques which are directly applicable to these engineering problems. This lecture will outline a homological approach to a variety of sensor networks problems. This talk will be at an elementary level: no background in sensor networks will be assumed, and only basic (e.g. Chapter 2 of Hatcher) homology theory will be needed.

Jesper Grodal, University of Chicago

From Finite Groups to Infinite Groups via Homotopy Theory

I will explain how examining the p-local structure in finite groups naturally lead to certain interesting infinite groups.

Maurice Herlihy, Brown University

Topological Methods in Distributed and Concurrent Computing

Models and techniques borrowed from classical algebraic and combinatorial topology have yielded a variety of new lower bounds and impossibility results for distributed and concurrent computation. This talk explains the basic concepts underlying this approach, surveys some of the results with an emphasis on open problems and research areas.

Tara Holm, Connecticut and Cornell University

Orbifold cohomology of abelian symplectic reductions

I will talk about the topology of symplectic (and other) quotients. Kirwan developed techniques for proving that the restriction map from the equivariant cohomology of the originial space to the ordinary cohomology of the symplectic reduction is a surjection. After a brief review of her results, I will show how they can be used to understand various aspects of the topology of quotients, focusing on the theme of orbifolds and computing the Chen- Ruan orbifold cohomology ring of abelian symplectic reductions. This is based on joint work with Rebecca Goldin (George Mason U) and Allen Knutson (UCSD).

Martin Kassabov, Cornell University

Kazhdan Property T and its Applications

In this talk I will define Kazhdan property T and its variants and discuss several resent results. In particular I will describe how representation theory can be used to study finite covers of 3 dimensional manifolds and outline an ongoing program, which may lead to a proof of the Virtual Hakken Conjecture.

Lee Mosher, Rutgers University

Axes in Outer Space

We develop a notion of axis for the action of a fully irreducible outer automorphism phi of a finite rank free group acting on the Culler-Vogtmann outer space X of the free group. Unlike the situation of a loxodromic isometry acting on hyperbolic space, or a pseudo-Anosov mapping class acting on Teichmüller space, X has no natural metric, and phi seems not to have a single natural axis. Instead our axes for phi, while not unique, fit into an "axis bundle" A with nice topological properties: A is a closed subset of X proper homotopy equivalent to a line, it is invariant under phi, the two ends of A limit on the repeller and attractor of the source--sink action of phi on compactified outer space, and A depends naturally on the repeller and attractor. We propose various definitions for A, each motivated in different ways by train track theory or by properties of axes in Teichmüller space, and we prove their equivalence. (Joint work with Michael Handel.)

Nathalie Wahl, University of Chicago

Mapping Class Groups of Non-orientable Surfaces

We prove that the homology of the mapping class groups of non-orientable surfaces stabilizes with the genus of the surface. Combining our result with recent work of Madsen and Weiss, we obtain that the classifying space of the stable mapping class group of non-orientable surfaces, up to homology isomorphism, is the infinite loop space of a Thom spectrum build from the canonical bundle over the Grassmannians of 2-planes in \RR^{n+2}. In particular, we show that the stable rational cohomology is a polynomial algebra on generators in degrees $4i$---this is the non-oriented analogue of the Mumford conjecture.