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Series: Geometry Topology Seminar

The set of knots up to a four-dimensional equivalence relation can be given the structure of a group, called the (smooth) knot concordance group. We will discuss how to compute concordance invariants using Heegaard Floer homology. We will then introduce the idea of a "reduced" knot Floer complex, see how it can be used to simplify computations, and give examples of how it can be helpful in distinguishing knots which are not concordant.

Series: Geometry Topology Seminar

The Ptolemy coordinates are efficient coordinates for computingboundary-unipotent representations of a 3-manifold group in SL(2,C). Wedefine a slightly modified version which allows you to computerepresentations that are not necessarily boundary-unipotent. This givesrise to a new algorithm for computing the A-polynomial.

Series: Geometry Topology Seminar

We show that each (p,q)-torus knot in the 3-sphere is
determined by
its A-polynomial and its knot Floer homology. This is joint work with Yi
Ni.

Series: Geometry Topology Seminar

Suppose that F is a field with p elements, and let G be the finite-index congruence subgroup of SL(n, F[t]) obtained as the kernel of the homomorphism that reduces entries in SL(n, F[t]) modulo the ideal (t). Then H^(n-1)(G;F) is infinitely generated. I'll explain the ideas behind the proof of the above result, which is a special case of a result that applies to any noncocompact arithmetic group defined over function fields.

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

A non-trivial group G is called left-orderable if there exists a strict total ordering < on its elements such that g

Series: Geometry Topology Seminar

A contact structure on a 3-manifold is called overtwisted ifthere is a certain kind of embedded disk called an overtwisted disk; it istight if no such disk exists. A Legendrian knot in an overtwisted contact3-manifold is loose if its complement is overtwisted and non-loose if itscomplement is tight. We define and compare two geometric invariants, depthand tension, that measure how far from loose is a non-loose knot. This isjoint work with Sinem Onaran.

Series: Geometry Topology Seminar

The question of what conditions guarantee that a symplectic$S^1$ action is Hamiltonian has been studied for many years. Sue Tolmanand Jonathon Weitsman proved that if the action is semifree and has anon-empty set of isolated fixed points then the action is Hamiltonian.Furthermore, Cho, Hwang, and Suh proved in the 6-dimensional case that ifwe have $b_2^+=1$ at a reduced space at a regular level $\lambda$ of thecircle valued moment map, then the action is Hamiltonian. In this paper, wewill use this to prove that certain 6-dimensional symplectic actions whichare not semifree and have a non-empty set of isolated fixed points areHamiltonian. In this case, the reduced spaces are 4-dimensional symplecticorbifolds, and we will resolve the orbifold singularities and useJ-holomorphic curve techniques on the resolutions.

Series: Geometry Topology Seminar

I'll talk about joint work with Sam Taylor. We characterize convex cocompact subgroups of mapping class groups that arise as subgroups of specially embedded right-angled Artin groups. We use this to construct convex cocompact subgroups of Mod(S) whose orbit maps into the curve complex have small Lipschitz constants.

Series: Geometry Topology Seminar

The topic of smooth 4-manifolds is a long established, yetunderdeveloped one. Its mystery lies partly in its wealth of strangeexamples, coupled with a lack of generally applicable tools to putthose examples into a sensible framework, or to effectively study4-manifolds that do not satisfy rather strict criteria. I will outlinerecent work that associates objects from symplectic topology, calledweak Floer A-infinity algebras, to general smooth, closed oriented4-manifolds. As time permits, I will speculate on a "genus-g Fukayacategory of smooth 4-manifolds.