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Wednesday, October 10, 2018 - 14:00 ,
Location: Skiles 006 ,
Sudipta Kolay ,
Georgia Tech ,
Organizer: Sudipta Kolay

This talk will be an introduction to the homotopy principle
(h-principle). We will discuss several examples. No prior knowledge
about h-principle will be assumed.

Wednesday, October 3, 2018 - 14:00 ,
Location: Skiles 006 ,
Stephen Mckean ,
GaTech ,
Organizer: Anubhav Mukherjee

Many problems in algebraic geometry involve counting solutions to
geometric problems. The number of intersection points of two projective
planar curves and the number of lines on a cubic surface are two
classical problems in this enumerative
geometry. Using A1-homotopy theory, one can gain new insights to old
enumerative problems. We will outline some results in A1-enumerative
geometry, including the speaker’s current work on Bézout’s Theorem.

Wednesday, September 26, 2018 - 14:00 ,
Location: Skiles 006 ,
Agniva Roy ,
Georgia Tech ,
Organizer: Sudipta Kolay

The Schoenflies' conjecture proposes the following: An embedding of the n-sphere in the (n+1)-sphere bounds a standard (n+1)-ball. For n=1, this is the well known Jordan curve theorem. Depending on the type of embeddings, one has
smooth and topological versions of the conjecture. The topological version was settled in 1960 by Brown.
In the smooth setting, the answer is known to be yes for all
dimensions other than 4, where apart from one special case, nothing is
known. The talk will review the question and attempt to describe some of
the techniques that have been used in low dimensions,
especially in the special case, that was worked out by Scharlemann in
the 1980s. There are interesting connections to the smooth 4-dimensional
Poincare conjecture that will be mentioned, time permitting. The talk
is aimed to be expository and not technical.

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.