Seminars and Colloquia by Series

A1-enumerative geometry

Series
Geometry Topology Student Seminar
Time
Wednesday, October 3, 2018 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Stephen MckeanGaTech
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.

A discussion about the smooth Schoenflies' conjecture

Series
Geometry Topology Student Seminar
Time
Wednesday, September 26, 2018 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Agniva RoyGeorgia Tech
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.

Sphere eversion: From Smale to Gromov II

Series
Geometry Topology Student Seminar
Time
Wednesday, September 19, 2018 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Hyunki MinGeorgia 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.

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 MinGeorgia 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 MukherjeeGaTech
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 KolayGeorgia 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 DavisGaTech
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 LanierGaTech
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.

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