All talks will be in Skiles 006.

* This will be a colloquium talk.

Pallavi Dani
Title: Divergence in groups
Abstract: Consider a pair of geodesics emanating from a point in a metric space. The ``divergence'' of the pair indicates how quickly they move away from each other, by measuring how far one must travel to get between corresponding points on the geodesics, without backtracking towards the basepoint. Gersten used this idea to de fine a quasi-isometry invariant for groups. I will describe divergence in a variety of groups, and talk about a higher dimensional analog. This will include joint work with A. Abrams, N. Brady, M. Duchin and R. Young on right-angled Artin groups, and with A. Thomas on right-angled Coxeter groups.

Benson Farb
Title: Permutations and polynomiality in algebra and topology
Abstract: Tom Church, Jordan Ellenberg and I recently discovered that the i-th Betti number of the space of configurations of n points on any manifold is given by a polynomial in n. Similarly for the moduli space of n-pointed genus g curves. Similarly for the dimensions of various spaces of homogeneous polynomials arising in algebraic combinatorics. Why? What do these disparate examples have in common? The goal of this talk will be to answer this question by explaining a simple underlying structure shared by these (and many other) examples in algebra and topology.

Josh Greene
Title: Conway mutation and alternating links
Abstract: We will show that a pair of alternating links have homeomorphic branched double-covers if and only if they are mutants. The proof involves some interesting combinatorial invariants coming from Heegaard Floer homology.

Shelly Harvey
Title: Filtering smooth concordance classes of topologically slice knots
Abstract: Recall that two knots are concordant if they cobound a smoothly embedded annulus in S^3 x I. The set of knots modulo concordance forms a group, called the knot concordance group, C. Inside this group lies a very interesting subgroup called the group of topologically slice knots, denoted T. In this talk we will define two monoid filtrations of T that approximate how close a knot is to being (smoothly) slice in #_m CP(2) or #_m -CP(2). Their intersection, denoted T_n, is a filtration of T by subgroups. T /T_0 is large, detected by the tau, s, and and delta-invariants, while T_0/T_1 is detected by certain d-invariants. Going beyond this, our main result is that T_1/T_2 has rank at least one. We also give evidence to believe that Tn/Tn+1 has positive rank for all n.

Lenny Ng
Title: Knot contact homology and unknot detection
Abstract: Knot contact homology is an invariant of knotted submanifolds in R^n obtained by counting holomorphic curves in the cotangent bundle of R^n with certain boundary conditions. For knots in R^3, this produces a fairly strong knot invariant that can be formulated combinatorially. I will describe how this invariant is related to the fundamental group of the knot complement via "string topology", and deduce an elementary proof that knot contact homology detects the unknot. This is partly joint work with Kai Cieliebak, Tobias Ekholm, and Janko Latschev.

Kate Petersen
Title: The Geometry of Character Varieties
Abstract:The character variety, X(M), of a finite volume cusped hyperbolic 3-manifold M is in essence the moduli space of hyperbolic structures on M. I will discuss how the geometry of these algebraic sets is related to the topology of M and the structure of the fundamental group of M. I will focus on curve components of X(M) and the action of symmetries of M on X(M). This is partly joint with Melissa Macasieb and Ronald van Luijk, and Ken Baker.

Bulent Tosun
Title:Legendrian and transverse knots in cabled knot types
Abstract: In this talk we will exhibit many new phenomena in the structure of Legendrian and transverse knots by giving a complete classification of all cables of the positive torus knots. We will also provide two structural theorems to ensure when a cable of an arbitrary Legendrian simple knot type is also Legendrian simple. Part of the results are joint work with John Etnyre and Douglas LaFountain.