Unni Namboodiri Lectures in Geometry and Topology
2011-2012 Speaker: Danny Calegari (CalTech)
Overall theme of talks: Extremal problems in group theory and topology
Abstract: We give an introduction to the theory of stable commutator length, and the dual theory of quasimorphisms, together with applications to rigidity, dynamics, group theory and topology. Some of this work is joint with Alden Walker, Joseph Maher, Joel Louwsma and Koji Fujiwara.
The titles and abstracts for the individual talks are below.
Lecture 1: Topological minimal surface theory and stable commutator length
Monday, April 23, 4 PM - 5 PM, Ryerson 251Abstract: If G is a group and [G,G] is its commutator subgroup, the commutator length of g in [G,G] is the least number of commutators whose product is g, and the stable commutator length is the growth rate of the commutator length of powers of g. Stable commutator length is the algebraic analogue of (relative) 2-dimensional Gromov-Thurston norm. We discuss the Rationality and Rigidity theorems for the stable commutator length norm in free groups, and some applications to the construction of surface subgroups and to symplectic rigidity.
Lecture 2: Dynamics, integer programming, and surgery
Tuesday, April 24, 4:30 PM - 5:30 PM, Eckhart 206Abstract: The phenomenon of Arnold tongues is a well-known example of phase locking of coupled nonlinear oscillators. The frequency spectrum of such nonlinear systems obey power laws; such power laws also turn up in integer programming, for example where one considers the Hermite normal form of a random integer matrix. We discuss how stable commutator length in surgery families is parameterized by families of integer programming problems. On the dynamics side, we show how similar power laws arise in the nonlinear "character varieties" of the group of homeomorphisms of the circle known as ziggurats.
Lecture 3: Statistics, concentration, and compression
Wednesday, April 25, 4 PM - 5 PM, Ryerson 251Abstract: We discuss the statistical distribution of stable commutator length in various classes of groups, and some applications. For certain classes of groups (e.g. central extensions of lattices in Sp(2n,R)) stable commutator length is distributed like distance to the origin for a random walk in a finite dimensional Euclidean space. For other classes of groups (e.g. hyperbolic groups, braid groups) there is a concentration of values, clustered around some fixed scale Cn/log(n) where the constant C should conjecturally be derived in a simple manner from the (growth) entropy. This concentration should be thought of as a "random" analogue of the phenomenon of Mostow rigidity for hyperbolic manifolds. Finally, the growth rate of stable commutator length is an obstruction to the existence of (nonelementary) homomorphisms to hyperbolic groups, or actions on certain hyperbolic spaces.
Previous years
2010-2011 Speaker: Douglas C. Ravenel (Rochester)
The Arf-Kervaire invariant problem in algebraic topology
April 1, 4, 5, 2011In 2009 Mike Hill, Mike Hopkins and I solved the Kervaire invariant problem in stable homotopy theory. In the first talk I will describe the history of the problem beginning with Pontryagin's work on the homotopy groups of spheres in the 1930s and Kervaire-Milnor's work on exotic spheres in the 1960s. In the second and third talks I will outline a portion of our proof.
The Namboodiri Lectures will be held in conjunction with the Midwest Topology Seminar, April 2, 3, 2011. For speakers and details, see here.
2009-2010 Speaker: Farrell
Smooth versus topological rigidity
May 3The best of all possible maps
May 4The space of negatively curved metrics
May 6Abstracts can be found here.
2008-2009 Speaker: Mladen Bestvina
SL2(Z) and Three Generalizations
Tuesday March 31, 4:30 pm, E 206Abstract.
Outer Space and the Topology of Out(Fn)
Wednesday April 1, 4:00 pm, E 206Abstract.
What is the Geometry of Outer Space
Thursday, April 2, 4:30 pm, E 206Abstract.
2007-2008 Speaker: Nigel Higson
Dimensions and C*-algebras
Monday April 7, 4:00 pm, E 206The study of the possible dimensions of projective modules over C*- algebras has been a perennial topic in operator algebras since the earliest work of von Neumann. But the subject took on new life in the 1980's when it emerged that encoded within these dimensions is information of real geometric interest, and when K-theoretic techniques were developed to extract this information. I shall present a survey of these topics, ending with an introductory account of the Baum-Connes conjecture, which is the main organizing principle governing this meeting of geometry, C*-algebras and K-theory.
How to prove the Baum-Connes conjecture
Tuesday April 8, 4:30 pm, E 206As a by-product of his work with Singer on the index theorem, Atiyah discovered a remarkably simple (or at least conceptually simple) proof of the Bott periodicity theorem using elliptic operators. As I shall explain, most of the progress on the Baum-Connes conjecture has been achieved by framing the conjecture as a generalized, equivariant form of Bott periodicity, and then borrowing from Atiyah's argument.
On the positive side, this approach has led to quite a bit of progress on the conjecture and its geometric corollaries. Moreover some of the geometric progress so achieved appears to be very difficult to recast in purely geometric terms. On the negative side, obstacles arise that have more to do with unitary representation theory than geometry.
C*-algebras and Lie group representations
Thursday April 10, 4:30pm, E 206I shall try to give an account of the connection between the Baum- Connes conjecture and the representation theory of semisimple groups. Even though this is, in some sense, one of the oldest aspects of the conjecture, the situation is still very unclear. However there are clear links to the problem of realizing discrete series representations, and to a proposal of Mackey to parametrize the duals of semisimple groups.