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Wednesday, November 5, 2014 - 14:05 ,
Location: Skiles 006 ,
Shane Scott ,
GaTech ,
Organizer: Shane Scott

The genus of a knot can be thought of as a measure of complexity for a 3 dimensional knot compliment. This notion can be extended to compact 3 manifolds by defining a norm on the second homology group with real coefficients measuring the Euler characteristic of embedded surfaces.

Wednesday, October 29, 2014 - 14:00 ,
Location: Skiles 006 ,
Jonathan Paprocki ,
Georgia Tech ,
Organizer: Jonathan Paprocki

This is a project for Prof. Margalit's course on Low-dimensional Topology and Hyperbolic Geometry.

We will present an introduction to the notion of quantum invariants of knots and links, and in particular the colored Jones polynomial. We will also introduce the Volume Conjecture, which relates a certain limiting behavior of a quantum invariant (the colored Jones polynomial of a link) with a classical invariant (the hyperbolic volume of the hyperbolic part of a link complement in S^3) and has been proven in a number of cases.

Wednesday, October 22, 2014 - 14:00 ,
Location: Skiles 006 ,
Sudipta Kolay ,
Georgia Tech ,
Organizer: Sudipta Kolay

This is a project for Prof. Margalit's course on Low-dimensional Topology and Hyperbolic Geometry.

In this talk we will discuss the Loop Theorem, which is a generalization of Dehn's lemma. We will outline a proof using the "tower construction".

Wednesday, September 10, 2014 - 14:00 ,
Location: Skiles 006 ,
Jonathan Paprocki ,
Georgia Tech ,
Organizer: Jonathan Paprocki

We will present an introduction to gauge theory and classical Chern-Simons theory, a 3-dimensional topological gauge field theory whose quantization yields new insights about knot invariants such as the Jones polynomial. Then we will give a sketch of quantum Chern-Simons theory and how Witten used it as a 3-dimensional method to obtain the Jones polynomial, as well as how it may be used to obtain other powerful knot and 3-manifold invariants. No physics background is necessary.

Wednesday, September 3, 2014 - 14:00 ,
Location: Skiles 006 ,
Jonathan Paprocki ,
Georgia Tech ,
Organizer: Jonathan Paprocki

Canceled due to speaker illness, date will be moved forward.

Thursday, June 26, 2014 - 12:00 ,
Location: Skiles 006. ,
Amey Kaloti ,
Georgia Tech. ,
Organizer:

This is a continuation of the previous talk.

Tuesday, June 24, 2014 - 12:05 ,
Location: Skiles 006. ,
Amey Kaloti ,
Georgia Tech. ,
Organizer:

We start studying open book foliations in this series of seminars. We will go through the theory and see how it is used in applications to contact topology.

Friday, May 2, 2014 - 14:00 ,
Location: Skiles 006 ,
John Dever ,
Georgia Tech ,
Organizer: John Dever

This is a final project for Dr. Etnyre's Differential Geometry class.

After briefly considering embeddings of the category of smooth manifolds into so called smooth toposes and arguing that we may ignore the details of the embedding and work from axioms if we agree to use intuitionistic logic, we consider axiomatic synthetic differential geometry. Key players are a space R playing the role of the "real line" and a space D consisting of null-square infinitesimals such that every function from D to R is "microlinear". We then define microlinear spaces and translate many definitions from differential geometry to this setting. As an illustration of the ideas, we prove Stokes' theorem. Time permitting, we show how synthetic differential geometry may be considered as an extension of differential geometry in that theorems proven in the synthetic setting may be "pulled back" to theorems about smooth manifolds.

Wednesday, April 30, 2014 - 14:05 ,
Location: Skiles 006. ,
Amey Kaloti ,
Georgia Tech. ,
ameyk@math.gatech.edu ,
Organizer:

Given a Riemannian manifold $(M,g)$, does there exist a metric $g'$ on $M$ conformal to $g$ such that $g'$ has constant scalar curvature? This question is known as the Yamabe problem. Aim of this talk is to give an overview of the problem and discuss and develop methods that go into solving a few of intermediate results in the solution to the problem in full generality.