Seminars and Colloquia by Series

Sphere eversion: From Smale to Gromov I

Series
Geometry Topology Student Seminar
Time
Wednesday, September 12, 2018 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Hyunki Min – Georgia Tech
In 1957, Smale proved a striking result: we can turn a sphere inside out without any singularity. Gromov in his thesis, proved a generalized version of this theorem, which had been the starting point of the h-principle. In this talk, we will prove Gromov's theorem and see applications of it.

Introduction to jet bundle and Whitney embedding theorem

Series
Geometry Topology Student Seminar
Time
Wednesday, September 5, 2018 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Anubhav Mukherjee – GaTech
This is the second lecture of the series on h-principle. We will introduce jet bundle and it's various properties. This played a big role in the devloping modern geometry and topology. And using this we will prove Whitney embedding theorem. Only basic knowledge of calculus is required.

Whitney–Graustein theorem

Series
Geometry Topology Student Seminar
Time
Wednesday, August 22, 2018 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Sudipta Kolay – Georgia Tech

Please Note: This theorem is one of earliest instance of the h-principle, and there will be a series of talks on it this semester.

The Whitney-Graustein theorem classifies immersions of the circle in the plane by their turning number. In this talk, I will describe a proof of this theorem, as well as a related result due to Hopf.

The Dehn-Nielsen-Baer Theorem

Series
Geometry Topology Student Seminar
Time
Wednesday, April 18, 2018 - 14:10 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Sarah Davis – GaTech
The theorem of Dehn-Nielsen-Baer says the extended mapping class group is isomorphic to the outer automorphism group of the fundamental group of a surface. This theorem is a beautiful example of the interconnection between purely topological and purely algebraic concepts. This talk will discuss the background of the theorem and give a sketch of the proof.

Exotic 7-sphere

Series
Geometry Topology Student Seminar
Time
Wednesday, April 4, 2018 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Hongyi Zhou (Hugo) – GaTech
Exotic sphere is a smooth manifold that is homeomorphic to, but not diffeomorphic to standard sphere. The simplest known example occurs in 7-dimension. I will recapitulate Milnor’s construction of exotic 7-sphere, by first constructing a candidate bundle M_{h,l}, then show that this manifold is a topological sphere with h+l=-1. There is an 8-dimensional bundle with M_{h,l} its boundary and if we glue an 8-disc to it to obtain a manifold without boundary, it should possess a natural differential structure. Failure to do so indicates that M_{h,l} cannot be mapped diffeomorphically to 7-sphere. Main tools used are Morse theory and characteristic classes.

Period three implies chaos

Series
Geometry Topology Student Seminar
Time
Wednesday, March 28, 2018 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Justin Lanier – GaTech
We will discuss a celebrated theorem of Sharkovsky: whenever a continuous self-map of the interval contains a point of period 3, it also contains a point of period n , for every natural number n.

Convexity and Contact Sphere Theorem

Series
Geometry Topology Student Seminar
Time
Wednesday, March 14, 2018 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Surena Hozoori – GaTech
Assuming some "compatibility" conditions between a Riemannian metric and a contact structure on a 3-manifold, it is natural to ask whether we can use methods in global geometry to get results in contact topology. There is a notion of compatibility in this context which relates convexity concepts in those geometries and is well studied concerning geometry questions, but is not exploited for topological questions. I will talk about "contact sphere theorem" due to Etnyre-Massot-Komendarczyk, which might be the most interesting result for contact topologists.

A discussion on 3 dim Lens spaces.

Series
Geometry Topology Student Seminar
Time
Wednesday, March 7, 2018 - 14:00 for 1 hour (actually 50 minutes)
Location
Atlanta
Speaker
Agniva Roy – GaTech
Three dimensional lens spaces L(p,q) are well known as the first examples of 3-manifolds that were not known by their homology or fundamental group alone. The complete classification of L(p,q), upto homeomorphism, was an important result, the first proof of which was given by Reidemeister in the 1930s. In the 1980s, a more topological proof was given by Bonahon and Hodgson. This talk will discuss two equivalent definitions of Lens spaces, some of their well known properties, and then sketch the idea of Bonahon and Hodgson's proof. Time permitting, we shall also see Bonahon's result about the mapping class group of L(p,q).

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