Georgia Institute of Technology

Constructing 3-Manifolds Using Dehn Surgery on Handlebodies

Fri, 11/06/2009 - 3:00pm, Skiles 269

Meredith Casey, Georgia Tech         Organizer: Meredith Casey

The goal of this talk is to describe simple constructions by which we can construct any compact, orientable 3-manifold.  It is well-known that every orientable 3-manifold has a Heegaard splitting.  We will first define Heegaard splittings, see some examples, and go through a very geometric proof of this therem.  We will then focus on the Dehn-Lickorish Theorem, which states that any orientation-preserving homeomorphism of an oriented 2-manifold without boundary can by presented as the composition of Dehn twists and homeomorphisms isotopic to the identity.  We will prove this theorm, and then see some applications and examples.  With both of these resutls together, we will have shown that using only handlebodies and Dehn twists one can construct any compact, oriented 3-manifold.    

Bordered Heegaard-Floer Theory

Fri, 10/30/2009 - 3:00pm, Skiles 269

Shea Vela-Vick, Columbia University        Organizer: John Etnyre

In this talk I will discuss a generalizations and/oo applications of bordered Floer homology.  After reviewing the basic definitions and constructions, I will focus either on an application to sutured Floer homology developed by Rumen Zarev, or on applications of the theory to the knot Floer homology. (While it would be good to have attended the other two talks this week, this talk shoudl be independent of them.)  This is a 2 hour talk.

Introduction to Bordered Heegaard-Floer homology II

Wed, 10/28/2009 - 10:00am, Skiles 255

Shea Vela-Vick, Columbia University        Organizer: John Etnyre

Here we will introduce the basic definitions of bordered Floer homology.  We will discuss bordered Heegaard diagrams as well as the algebraic objects, like A_\infinity algebras and modules, involved in the theory.  We will also discuss the pairing theorem which states that if  Y = Y_1 U_\phi Y_2  is obtained by identifying the (connected) boundaries of  Y_1  and  Y_2, then the closed Heegaard Floer theory of  Y  can be obtained as a suitable tensor product of the bordered theories of  Y_1  and  Y_2.Note the different time and place!This is a 1.5 hour talk.

Introduction to Bordered Heegaard-Floer homology

Mon, 10/26/2009 - 10:00am, Skiles 255

Shea Vela-Vick, Columbia University        Organizer: John Etnyre

We will focus on the "toy model" of bordered Floer homology.  Loosely speaking, this is bordered Floer homology for grid diagrams of knots.  While the toy model unfortunately does not provide us with any knot invariants, it highlights many of the key ideas needed to understand the more general theory. Note the different time and place!  This is a 1.5 hour talk.

Introduction to Heegaard Floer Homology

Fri, 10/23/2009 - 3:00pm, Skiles 269

Amey Kaloti, Georgia Tech         Organizer: Meredith Casey

Abstract: Heegaard floer homology is an invariant of closed 3 manifolds defined by Peter Ozsvath and Zoltan Szabo. It has proven to be a very strong invariant of 3 manifolds with connections to contact topology. In these talks we will try to define the Heegaard Floer homology without assuming much background in low dimensional topology. One more goal is to present the combinatorial description for this theory.

This is a 2 hour talk.

Introduction to Heegaard Floer Homology

Fri, 10/16/2009 - 3:00pm, Skiles 169

Amey Kaloti, Georgia Tech         Organizer: Meredith Casey

Heegaard floer homology is an invariant of closed 3 manifolds defined by Peter Ozsvath and Zoltan Szabo. It has proven to be a very strong invariant of 3 manifolds with connections to contact topology. In these talks we will try to define the Heegaard Floer homology without assuming much background in low dimensional topology. One more goal is to present the combinatorial description for this theory.

This is a 2-hour talk.

On classification of of high-dimensional manifolds-II

Fri, 10/09/2009 - 3:00pm, Skiles 269

Igor Belegradek, Georgia Tech        Organizer: Bulent Tosun

This 2 hour talk is a gentle introduction to simply-connected sugery theory (following classical work by Browder, Novikov, and Wall). The emphasis will be on classification of high-dimensional manifolds and understanding concrete examples.

On classification of of high-dimensional manifolds

Fri, 10/02/2009 - 3:00pm, Skiles 269

Igor Belegradek, Georgia Tech        Organizer: Bulent Tosun

This 2 hour talk is a gentle introduction to simply-connected sugery theory (following classical work by Browder, Novikov, and Wall). The emphasis will be on classification of high-dimensional manifolds and understanding concrete examples.

Introduction to Contact Homology

Fri, 09/25/2009 - 3:00pm, Skiles 269

Anh Tran, Georgia Tech         Organizer: Meredith Casey

In this talk I will give a quick review of classical invariants of Legendrian knots in a 3-dimensional contact manifold (the topological knot type, the Thurston-Bennequin invariant and the rotation number). These classical invariants do not completely determine the Legendrian isotopy type of Legendrian knots, therefore we will consider Contact homology (aka Chekanov-Eliashberg DGA), a new invariant that has been defined in recent years. We also discuss the linearization of Contact homology, a method to extract a more computable invariant out of the DGA associated to a Legendrian knot.

(This is a 2 hour lecture.)

Hyperbolic structures on surfaces and 3-manifolds

Fri, 09/18/2009 - 2:00pm, Skiles 269

John Etnyre, Georgia Tech         Organizer: Meredith Casey

We will discuss how to put a hyperbolic structure on various surface and 3-manifolds. We will being by discussing isometries  of hyperbolic space in dimension 2 and 3. Using our understanding of these isometries we will explicitly construct hyperbolic structures on all close surfaces of genus greater than one and a complete finite volume hyperbolic structure on the punctured torus. We will then consider the three dimensional case where we will concentrate on putting hyperbolic structures on knot complements. (Note: this is a 1.5 hr lecture)

Hyperbolic structures on surfaces and 3-manifolds

Fri, 09/11/2009 - 3:00pm, Skiles 269

John Etnyre, Georgia Tech        Organizer: Meredith Casey

We will discuss how to put a hyperbolic structure on various surface and 3-manifolds. We will being by discussing isometries  of hyperbolic space in dimension 2 and 3. Using our understanding of these isometries we will explicitly construct hyperbolic structures on all close surfaces of genus greater than one and a complete finite volume hyperbolic structure on the punctured torus. We will then consider the three dimensional case where we will concentrate on putting hyperbolic structures on knot complements. (Note: this is a 2 hr seminar)

The Jones polynomial and quantum invariants

Fri, 04/24/2009 - 3:00pm, Skiles 269

Thang Le, School of Mathematics, Georgia Tech        Organizer: John Etnyre

We will develop general theory of quantum invariants based on sl_2 (the simplest Lie algebra): The Jones polynomials, the colored Jones polynomials, quantum sl_2 groups, operator invariants of tangles, and relations with the Alexander polynomial and the A-polynomials. Optional: Finite type invariants and the Kontsevich integral.

These are two hour lectures.

The Jones polynomial and quantum invariants

Fri, 04/17/2009 - 3:00pm, Skiles 269

Thang Le, School of Mathematics, Georgia Tech        Organizer: John Etnyre

We will develop general theory of quantum invariants based on sl_2 (the simplest Lie algebra): The Jones polynomials, the colored Jones polynomials, quantum sl_2 groups, operator invariants of tangles, and relations with the Alexander polynomial and the A-polynomials. Optional: Finite type invariants and the Kontsevich integral.

These are two hour lectures.

The Jones polynomial and quantum invariants

Fri, 04/10/2009 - 3:00pm, Skiles 269

Thang Le, School of Mathematics, Georgia Tech        Organizer: John Etnyre

We will develop general theory of quantum invariants based on sl_2 (the simplest Lie algebra): The Jones polynomials, the colored Jones polynomials, quantum sl_2 groups, operator invariants of tangles, and relations with the Alexander polynomial and the A-polynomials. Optional: Finite type invariants and the Kontsevich integral.

These are two hour talks.

Introduction to metric and comparison geometry

Fri, 02/27/2009 - 3:05pm, Skiles 269

Igor Belegradek, Ga Tech        Organizer: John Etnyre

Comparison geometry studies Riemannian manifolds with a given curvature bound. This minicourse is an introduction to volume comparison (as developed by Bishop and Gromov), which is fundamental in understanding manifolds with a lower bound on Ricci curvature. Prerequisites are very modest: we only need basics of Riemannian geometry, and fluency with fundamental groups and metric spaces. In the third (2 hour) lecture I shall prove volume and Laplacian comparison theorems.