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Wednesday, August 22, 2018 - 14:00 ,
Location: Skiles 005 ,
Sudipta Kolay ,
Georgia Tech ,
Organizer: Sudipta Kolay

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.

Wednesday, April 18, 2018 - 14:10 ,
Location: Skiles 006 ,
Sarah Davis ,
GaTech ,
Organizer: Anubhav Mukherjee

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.

Wednesday, April 4, 2018 - 14:00 ,
Location: Skiles 006 ,
Hongyi Zhou (Hugo) ,
GaTech ,
Organizer: Anubhav Mukherjee

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.

Friday, March 30, 2018 - 14:00 ,
Location: Skiles 006 ,
Sudipta Kolay ,
Georgia Tech ,
Organizer: Sudipta Kolay

We will give eight different descriptions of the Poincaré homology sphere, and outline the proof of equivalence of the definitions.

Wednesday, March 28, 2018 - 14:00 ,
Location: Atlanta ,
Justin Lanier ,
GaTech ,
Organizer: Anubhav Mukherjee

Wednesday, March 28, 2018 - 14:00 ,
Location: Skiles 006 ,
Justin Lanier ,
GaTech ,
Organizer: Anubhav Mukherjee

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.

Wednesday, March 14, 2018 - 14:00 ,
Location: Skiles 006 ,
Surena Hozoori ,
GaTech ,
Organizer: Anubhav Mukherjee

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.

Wednesday, March 7, 2018 - 14:00 ,
Location: Atlanta ,
Agniva Roy ,
GaTech ,
Organizer: Anubhav Mukherjee

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).

Wednesday, February 28, 2018 - 14:00 ,
Location: Skiles 006 ,
Hyun Ki Min ,
GaTech ,
Organizer: Anubhav Mukherjee

I will introduce the notion of satellite knots and show that a knot in a 3-sphere is either a torus knot, a satellite knot or a hyperbolic knot.

Wednesday, February 21, 2018 - 14:00 ,
Location: Skiles 006 ,
Kevin Kodrek ,
GaTech ,
Organizer: Anubhav Mukherjee

There are a number of ways to define the braid group. The traditional definition involves equivalence classes of braids, but it can also be defined in terms of mapping class groups, in terms of configuration spaces, or purely algebraically with an explicit presentation. My goal is to give an informal overview of this group and some of its subgroups, comparing and contrasting the various incarnations along the way.