# Seminars & Events for Topology Seminar

##### Action-Dimension of Groups

**Please note change of time and location. This is a joint Topology/Algebraic Topology seminar. ** For a group G, we define a notion of dimension in terms of dimension change of the of the top homology between a free G space X and it's quotient X/G. We show that this is well defined and calculate this "Action-dimension" for certain groups, including finitely generated solvable groups, free groups, finite groups, and connected Lie groups. As a consequence we give a positive answer to a conjecture of J Kollar: Let f: R^n ---> T^n be the universal covering and M a closed submanifold in T^n = (S^1)^n such that f^{-1}(M) has the homotopy type of a finite complex, then M = T^n.

##### An introduction to the rational genus of a knot

What is the "simplest" knot in a given three-manifold $Y$? We know that the answer is the unknot when $Y=S^3$, as the unknot happens to be the only knot in the three-sphere with the smallest genus (=0). In this talk, we will discuss the more general notion of the rational genus of knots. In particular, we will show that the simple knots are really the "simplest" knots in the lens spaces in the sense of being a genus minimizer in its homology class. This is a joint work with Yi Ni.

##### TQFT, Hochschild homology and localization

Hochschild homology is a categorification of the trace. We will start by explaining what this means, and how both the trace and Hochschild homology are relevant to topological field theories. We will then explain an algebraic result for Hochschild homology, and how it can sometimes be used to study TQFT invariants of covering spaces. The first half of this talk is classical; the second half is joint work with D. Treumann.

##### A Transcendental Invariant of Pseudo-Anosov Maps

For each pseudo-Anosov map, we will associate it with a $\mathbb{Q}$-submodule of $\mathbb{R}$. This invariant is defined by interaction between Thurston norm and dilatation of pseudo-Anosov map. We will develop a few nice properties of our invariant and give a few examples to show it can be nontrivial. These nontrivial examples give negative answer to a question asked by McMullen.

##### A New Spectral Sequence in Khovanov Homology

Khovanov homology is an invariant of links L in S^3which categorifies the Jones polynomial. In this talk, I will describe a new spectral sequence in Khovanov homology, the link forgetful spectral sequence. The spectral sequence starts at Khovanov homology and converges to the Khovanov homology of the disjoint union of the components of the link -- that is, it forgets the linking between components. The discovery of this construction was partly motivated by looking for an analog in Khovanov homology of the component-forgetting spectral sequence in link Floer homology. As an application, building on results of Kronheimer-Mrowka and Hedden-Ni, I will prove that Khovanov homology detects the unlink. Joint work with Josh Batson.

##### Mom 1.5

This talk will discuss joint work in progress with Robert Haraway and Craig Hodgson. Mom Technology has had considerable success in proving facts about low-volume 1-cusped hyperbolic 3-manifolds. We are attempting to generalize Mom Technology to the case of hyperbolic 3-manifolds with totally geodesic boundary. The generalization of Mom Technology to this setting has a number of interesting aspects.

##### Embeddings of Rational Homology Balls

We will begin with a description of the rational homology balls appearing in Fintushel and Stern's rational blow-down procedure for smooth 4-manifolds, a generalization of the standard blow-down operation. We will then discuss various smooth and symplectic embedding results of these rational homology balls, as well as a description of a symplectic rational blow-up operation. THIS IS A JOINT TOPOLOGY/ALGEBRAIC TOPOLOGY SEMINAR.

##### Fundamental groups of Kähler manifolds and combinatorial group theory

The study of fundamental groups of Kähler manifolds is a fascinating enterprise at the crossroads of various branches of geometry and topology, with strong relations to algebra and analysis as well. I will discuss some results in this area pertaining to groups of interest in low-dimensional topology and in combinatorial and geometric group theory.

##### The Strange World of Quantum Computing

This talk will give an introductory overview of quantum computing in an intuitive and conceptual fashion. No prior knowledge of quantum mechanics will be assumed. This talk is intended to be a preamble to my next talk on quantum knots.

##### Quantum Knots

In this talk, we show how to reconstruct knot theory in such a way that it is intimately related to quantum physics. In particular, we give a blueprint for creating a quantum system that has the dynamic behavior of a closed knotted piece of rope moving in 3-space. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Structure and rigidity of totally periodic pseudo-Anosov flows in graph manifolds

This is joint work with Thierry Barbot. A graph manifold is an irreducible manifold so that all pieces of the torus decomposition are Seifert fibered. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Root systems of torus graphs and automorphism groups of torus manifolds

A torus manifold is a compact oriented 2n-dimensional T^n-manifold with fixed points. We can define a labelled graph from given torus manifold as follows: vertices are fixed points; edges are invariant 2-spheres; edges are labelled by tangential representations around fixed points. THIS IS A JOINT TOPOLOGY / ALGEBRAIC TOPOLOGY SEMINAR.

##### Stable pairs and the HOMFLY polynomial

Given a planar curve singularity, Oblomkov and Shende conjectured a precise relationship between the geometry of its Hilbert scheme of points and the HOMFLY polynomial of the associated link. I will explain a proof of this conjecture, as well as a generalization to colored invariants proposed by Diaconescu, Hua, and Soibelman.

##### Bordered Floer homology and splicing knot complements

We use bordered Floer homology to study 3-manifolds obtained by gluing together two knot complements, in view of several conjectures concerning the classification of L-spaces, manifolds with the simplest possible Heegaard Floer homology. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Normalizing Topologically Minimal surfaces

Topologically minimal surfaces generalize several well-studied classes of surfaces in 3-manifolds, and provide a topological analogue to geometrically minimal surfaces. We will discuss recent progress in obtaining a normal form for any such surface with respect to a fixed triangulation. This provides striking analogues with results of Colding and Minicozzi, and establishes finiteness results which are crucial to understanding how Heegaard splittings are effected by Dehn surgery.

##### Khovanov homotopy type, Steenrod squares, and new s-invariants

I will briefly discuss how we can extend the Khovanov homology link invariant to a stable homotopy type. We use the Pontryagin-Thom construction packaged as a framed flow category. I will then talk about how to compute some induced algebraic structures, like Steenrod squares, on Khovanov homology, and how those can be used to refine Rasmussen's s-invariant to a stronger concordance invariant. This is joint work with Robert Lipshitz.

##### String Topology Operations: A Frog's Eye View

THIS IS A JOINT TOPOLOGY/ALGEBRAIC TOPOLOGY SEMINAR. Chas and Sullivan introduced in 1999 a product on the homology of the free loop space LM of a compact, oriented manifold M. I will discuss some related higher order string topology operations that are the result of joint work with Ralph Cohen and Nathalie Wahl. Some are constructed using Morse's finite dimensional approximation of LM. Others are parameterized by Sullivan diagrams.

##### Stratified String Topology and 3D-Manifolds

PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Knot Floer homology and bordered algebras

##### Finite Energy Instantons on $\Sigma \times \mathbb{C}$

We will outline a proof of a conjecture by Jardim-Biquard on the asymptotic decay of instantons on $\Sigma \times \mathbb{C}$. In the case $\Sigma$ is a torus, coupled with the results of Jardim-Biquard, this establishes a correspondence between finite energy instantons and (stable) holomorphic bundles on $\Sigma \times P^1$. Our techniques are based on the Gromov-Uhlenbeck compactness estimates developed for the Atiyah-Floer Conjecture.