Date:November 3

Title: Stable Khovanov homology of torus links and volume

Abstract: Let T(n, k) denote the (n, k)-torus braid. It is well known that the Jones polynomial and the Khovanov homology of torus links stabilize as k → ∞ by the work of Champanekar-Kofman and Stosic. In particular, Rozanksy showed that the stable Khovanov homology of torus links exists as the direct limit of the Khovanov homology of T(n, k)-torus links, and the stable Khovanov homology recovers the categorification of the Jones-Wenzl projector. We show that the categorification of the Khovanov homology of a link stabilizes under twisting as a categorial analogue of the result by Champanekar-Kofman, extending the results by Stosic and Rozansky. Since the Jones-Wenzl projector can be used to define the colored Jones polynomial, we will discuss potential relationship between the stable invariant to the hyperbolic volume of a knot in the spirit of the volume conjecture.

 

Name: Ying Hu

Date:November 10 

Title: Stir fry Homeo_+(S^1) representations from pseudo-Anosov flows 

Abstract: A total linear order on a non-trivial group G is a left-order if it’s invariant under group left-multiplication. A result of Boyer, Rolfsen and Wiest shows that a 3-manifold group has a left-order if and only if it admits a non-trivial representation into Homeo_+(S^1) with zero Euler class. Foliations, laminations and flows on 3-manifolds often give rise to natural non-trivial Homeo_+(S^1)-representations of the fundamental groups, which have proven to be extremely useful in studying the left-orderability of 3-manifold groups.  In this talk, we will present a recipe of stir frying these Homeo_+(S^1)-representations. Our operation generalizes a previously known ``flipping'' operation introduced by Calegari and Dunfield. As a consequence, we constructed a surprisingly large number of new Homeo_+(S^1)-representations of the link groups. We then use these newly obtained representations to prove the left-orderablity of cyclic branched covers of links associated with any epimorphisms to Z_n.  This is joint work with Steve Boyer and Cameron Gordon.

 

Name: Francisco Arana Herrera 

Date:November 10

Title: Arithmetic v/s geometric complexity of simple closed geodesics

Abstract: How homologically complicated are long simple closed geodesics on hyperbolic surfaces? We provide an answer to this question which is surprisingly different from the well studied case of general primitive closed geodesics. We explain the relation between these questions and mixing limit theorems for the Kontsevich-Zorich cocycle. Parts of this talk are joint work in progress with Pouya Honaryar and other parts are joint work with Giovanni Forni. Time permitting we discuss several open questions in the field.
 

Name: Antonio Alfieri 

Date:November 17

Title: Involutions and the Chern-Simons filtration in instanton Floer homology

Abstract: In a recent paper in collaboration with Dai, Mallick, and Taniguchi we developed an invariant based on instanton Floer homology aimed at studying strong corks and equivariant bounding. Our construction utilizes the Chern-Simons filtration and is qualitatively different from previous Floer-theoretic methods used to address these questions. I will discuss various topological applications of these methods and outline the main ideas behind this machinery.  

 

Name: Bulent Tosun

Date: December 1

Title: Embedding questions for 3-manifolds

Abstract: In this talk I will discuss 3-manifold embedding questions into 4-dimensional manifolds from the perspectives of smooth topology as well as symplectic and complex geometry. The talk will have two parts: In the first part the focus will be on embedding questions into 4-dimensional Euclidean space (R^4). Given a closed, orientable 3-manifold Y, it is of great interest but often a difficult problem to determine whether Y may be smoothly embedded in R^4. This is the case even for integer homology spheres (where usual obstructions coming from homology disappear), and restricting to special classes such as Seifert manifolds, the problem is open in general. On the other hand, under additional geometric considerations coming from symplectic geometry (such as hypersurfaces of contact type in R^4)and complex geometry (such as the boundaries of pseudo-convex or rationally convex domains in complex Euclidean space C^2), the problems become tractable and in certain cases a uniform answer is possible. For example, recent work shows no correctly oriented Seifert homology sphere admits an embedding as a hypersurface of contact type in R^4. In the second part, I will consider general closed 4-manifolds as target manifolds and mention some recent work in progress. I will provide further context and motivations for these results, and give some details of the proofs.  

 

Name: Diana Hubbard

Date: December 8

Title: Twisting in small and big mapping class groups

Abstract: For a finite type surface with boundary, the fractional Dehn twist coefficient is a rational number that measures, informally, the amount of twisting a mapping class of the surface effects about a chosen boundary component. This quantity has connections to classical knot theory, open book decompositions, and contact geometry. In this talk I will generalize this notion to the infinite type setting by characterizing it as the unique quasimorphism from the mapping class group of the surface to the real numbers with certain properties. Unlike in the finite type setting, for certain infinite type surfaces this quasimorphism has image all of R. To see this I will explain a new construction of maps with irrational rotation behavior. I will discuss how this work may be of use in the search for "Nielsen-Thurston"-type classification results for infinite type surfaces. This work is joint with Peter Feller and Hannah Turner.