Seminars and Colloquia by Series

Knot invariants and their categorifications via Howe duality

Series
Geometry Topology Seminar
Time
Monday, April 13, 2015 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Aaron Lauda – USC –
It is a well understood story that one can extract linkinvariants associated to simple Lie algebras. These invariants arecalled Reshetikhin-Turaev invariants and the famous Jones polynomialis the simplest example. Kauffman showed that the Jones polynomialcould be described very simply by replacing crossings in a knotdiagram by various smoothings. In this talk we will explainCautis-Kamnitzer-Licata's simple new approach to understanding theseinvariants using basic representation theory and the quantum Weylgroup action. Their approach is based on a version of Howe duality forexterior algebras called skew-Howe duality. Even the graphical (orskein theory) description of these invariants can be recovered in anelementary way from this data. The advantage of this approach isthat it suggests a `categorification' where knot homology theoriesarise in an elementary way from higher representation theory and thestructure of categorified quantum groups. Joint work with David Rose and Hoel Queffelec

Tightness of positive rational surgeries

Series
Geometry Topology Seminar
Time
Monday, April 6, 2015 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Bulent Tosun – University of Virginia
Existence of a tight contact structure on a closed oriented three manifold is still widely open problem. In this talk we will present some work in progress to answer this problem for manifolds that are obtained by Dehn surgery on a knot in three sphere. Our method involves on one side generalizing certain geometric methods due to Baldwin, on the other unfolds certain homological algebra methods due to Ozsvath and Szabo.

Independence of Whitehead Doubles of Torus Knots in the Smooth Concordance Group

Series
Geometry Topology Seminar
Time
Monday, March 30, 2015 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Juanita Pinzon-Caicedo – University of Georgia
In the 1980’s Furuta and Fintushel-Stern applied the theory of instantons and Chern-Simons invariants to develop a criterion for a collection of Seifert fibred homology spheres to be independent in the homology cobordism group of oriented homology 3-spheres. In turn, using the fact that the 2-fold cover of S^3 branched over the Whitehead double of a positive torus knot is negatively cobordant to a Seifert fibred homology sphere, Hedden-Kirk establish conditions under which an infinite family of Whitehead doubles of positive torus knots are independent in the smooth concordance group. In the talk, I will review some of the definitions and constructions involved in the proof by Hedden and Kirk and I will introduce some topological constructions that greatly simplify their argument. Time permiting I will mention some ways in which the result could be generalized to include a larger set of knots.

Lagrangian concordance and contact invariants in sutured Floer theories

Series
Geometry Topology Seminar
Time
Monday, March 23, 2015 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
John Baldwin – Boston College
In 2007, Honda, Kazez, and Matic defined an invariant of contact 3-manifolds with convex boundaries using sutured Heegaard Floer homology (SHF). Last year, Steven Sivek and I defined an analogous contact invariant using sutured Monopole Floer homology (SMF). In this talk, I will describe work with Sivek to prove that these two contact invariants are identified by an isomorphism relating the two sutured theories. This has several interesting consequences. First, it gives a proof of invariance for the contact invariant in SHF which does not rely on the relative Giroux correspondence between contact structures and open books (something whose proof has not yet been written down in full). Second, it gives a proof that the combinatorially computable invariants of Legendrian knots in Heegaard Floer homology can obstruct Lagrangian concordance.

Tight Surgeries on Knots in Overtwisted Contact Manifolds

Series
Geometry Topology Seminar
Time
Monday, March 9, 2015 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jamie Conway – Georgia Tech
Most work on surgeries in contact manifolds has focused upon determining the situations where tightness is preserved. We will discuss an approach to this problem from the reverse angle: when negative surgery on a fibred knot in an overtwisted contact manifold produces a tight one. We will examine the various phenomena that occur, and discuss an approach to characterising them via Heegaard Floer homology.

q-Ehrhart polynomials and knots

Series
Geometry Topology Seminar
Time
Monday, March 2, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Roland van der Veen – University of Amsterdam –
We will start by counting lattice points in a polytope and showhow this produces many familiar objects in mathematics.For example if one scales the polytope, the number of lattice points givesrise to the Ehrhart polynomials, including binomals and other well knownfunctions.Things get more interesting once we take a weighted sum over the latticepoints instead of just counting them. I will explain how toextend Ehrhart's theory in this case and discuss an application to knottheory. We will derive a new state sum for the colored HOMFLYpolynomial using q-Ehrhart polynomials, following my recent preprint Arxiv1501.00123.

Torsion in Homology for Hyperbolic 3-Manifolds

Series
Geometry Topology Seminar
Time
Monday, March 2, 2015 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Pere Menal-Ferrer – Georgia Tech
How is the homological torsion of a hyperbolic 3-manifold related to its geometry? In this talk, I will explain some techniques to address this general question. In particular, I will discuss in detail the case of arithmetic manifolds, where the situation is presumably easier to understand.

Loop Spaces, Operads and the Space of Positive Scalar Curvature Metrics

Series
Geometry Topology Seminar
Time
Monday, February 23, 2015 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Mark Walsh – Wichita State
In this talk we will begin by discussing the problem of understanding the topology of the space of Riemannian metrics of positive scalar curvature on a smooth manifold. Recently much progress has occurred in this topic. We will then look at an application of the theory of operads to this problem in the case when the underlying manifold is an n-sphere. In the case when n>2, this space is a homotopy commutative, homotopy associative H-space. In particular, we show that it admits an action of the little n-disks operad. Via theorems of Stasheff, Boardman, Vogt and May, this allows us to demonstrate that the path component of this space containing the round metric, is weakly homotopy equivalent to an n-fold loop space.

Approximating C^0 foliations with contact structures

Series
Geometry Topology Seminar
Time
Monday, February 9, 2015 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Will Kazez – University of Georgia
I will discuss Eliashberg and Thurston's theorem that C^2 taut foliations can be approximated by tight contact structures. I will try to explain the importance of their work and why it is useful to weaken their smoothness assumption. This work is joint with Rachel Roberts.

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