Seminars and Colloquia by Series

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).

Classification of knots in 3-sphere

Series
Geometry Topology Student Seminar
Time
Wednesday, February 28, 2018 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Hyun Ki Min – GaTech
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.

An introduction to the braid group.

Series
Geometry Topology Student Seminar
Time
Wednesday, February 21, 2018 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Kevin Kodrek – GaTech
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.

Dehn surgery on the figure 8 knot

Series
Geometry Topology Student Seminar
Time
Wednesday, February 7, 2018 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Hyun Ki Min – GaTech
The figure 8 knot is the simplest hyperbolic knot. In the late 1970s, Thurston studied how to construct hyperbolic manifolds from ideal tetrahedra. In this talk, I present the Thurston’s theory and apply this to the figure 8 knot. It turns out that every Dehn surgery on the figure 8 knot results in a hyperbolic manifold except for 10 exceptional surgery coefficients. If time permits, I will also introduce the classification of tight contact structures on these manifolds. This is a joint work with James Conway.

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