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Wednesday, September 19, 2018 - 14:00 ,
Location: Skiles 006 ,
Hyunki Min ,
Georgia Tech ,
Organizer: Hyun Ki Min

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.

Wednesday, September 12, 2018 - 14:00 ,
Location: Skiles 006 ,
Hyunki Min ,
Georgia Tech ,
Organizer: Hyun Ki Min
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.

Wednesday, September 5, 2018 - 14:00 ,
Location: Skiles 005 ,
Anubhav Mukherjee ,
GaTech ,
Organizer: Anubhav Mukherjee

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.

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.