Cal Tech

Abstract: Given a group G and an element g \in [G, G], the commutator length of g, denoted cl(g), is the smallest number of commutators in G whose product is g, and the stable commutator length of g is the limit scl(g) := lim_{n \to \infty} cl(g^n)/n. Commutator length in a group extends in a natural way to a pseudo-norm on the real vector space of 1-boundaries (in group homology), and should be thought of as a kind of relative Gromov-Thurston norm. We show that the problem of computing stable commutator length in free products of abelian groups reduces to a (finite dimensional) integer programming problem. Moreover, certain families of elements in such groups (i.e. those obtained by surgery on some element in a bigger group) give rise to families of integer programming problems that are related in explicit ways. In particular one can use this to establish the existence of limit points in the range of scl in such groups, and produce elements whose stable commutator length is congruent to any rational number modulo the integers. This technology relates stable commutator length to the theory of multi-dimensional continued fractions, and Klein polyhedra, and suggests an interesting conjectural picture of scl in free groups. See [1, 2] for background and more details.

References

[1] D. Calegari, scl, MSJ Memoirs, 20. Mathematical Society of Japan, Tokyo, 2009

[2] D. Calegari, Scl, sails and surgery, preprint arXiv:0907.3541

Univ. Texas, Austin

Abstract: If K is a rationally null-homologous knot in a 3-manifold M then the rational genus |K| of K is defined to be (roughly) the infimum of -\chi(S)/2n, over all S and n > 0, where S is an orientable surface in M whose boundary wraps n times homologically around K. If M is a homology sphere then this is essentially the genus of K. One can construct (for example by doing surgery on knots in S^3) knots in 3-manifolds with arbitrarily small rational genus; we show that such knots can be characterized geometrically. More precisely we show that there is a constant C > 0 such that if K in M has 0 < |K| < C then (M,K) belongs to one of a small number of classes: e.g. M is hyperbolic and K is the core of a Margulis tube, M is Seifert fibered and K is a fiber, K lies in a JSJ torus in M, etc. Conversely we show that there are pairs (M,K) in each of these classes with |K| arbitrarily small. This is joint work with Danny Calegari.

USC

Abstract: In this talk I will introduce the contact category of a compact surface \Sigma. The contact category has enjoys many properties of a triangulated category (such as the octahedral axiom), and we can also construct its Grothendieck group. The braid group naturally acts on a version of the contact category, and categorifies several classical representations of the braid group, such as the Burau and Gassner representations. I will also discuss an exact functor from the contact category to a linear category via the Heegaard Floer homology/sutured Floer homology functor.

UCLA

Abstract: Heegaard Floer theory, introduced by Ozsvath and Szabo, is a useful technique in low-dimensional topology: in particular, in four dimensions, it gives rise to invariants that are conjecturally the same as the Seiberg-Witten invariants, and share many of their properties. In this talk, I will describe an algorithm for computing the Heegaard Floer invariants of three- and four-manifolds (modulo 2). The algorithm is based on presenting the manifolds in terms of links in S^3, and then using grid diagrams to represent the links. The talk is based on joint work with P. Ozsvath and D. Thurston.