Seminars and Colloquia by Series

An introduction to 4-manifolds

Series
Geometry Topology Student Seminar
Time
Wednesday, August 30, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Sierra Knavel – Georgia Tech –

In the early 80's, Freedman discovered that the Whitney trick could be performed in 4-dimensions which quickly led to a complete classification of closed, simply connected topological 4-manifolds. With gauge theory, Donaldson showed that 4-manifolds differ greatly from their higher dimensional counterparts which uncovered the stark differences between topological and smooth results in dimension 4. In this introductory talk, we will give a brief overview this classification and why dimension 4 is so unique. Then, we will describe handlebody decompositions of 4-manifolds and draw several Kirby pictures representing some basic 4-mfds.

An Introduction to Teichmüller Theory

Series
Geometry Topology Student Seminar
Time
Wednesday, August 23, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Alex Nolte – Rice University –

Say you’ve got an (orientable) surface S and you want to do geometry with it. Well, the complex plane C has dimension 2, so you might as well try to model S on C and see what happens. The objects you get from following this thought are called complex structures. It turns out that most surfaces have a rich but manageable amount of genuinely different complex structures. I’ll focus in this talk on how to think about comparing and deforming complex structures on S. I’ll explain the remarkable result that there are highly structured “best” maps between (marked) complex structures, and how this can be used to show the right space of complex structures on S is a finite-dimensional ball. This is known as Teichmüller’s theorem, and I’ll be following Bers’ proof.

A Visual Journey via Unicorn Paths

Series
Geometry Topology Student Seminar
Time
Wednesday, April 12, 2023 - 02:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Katherine Williams Booth – Georgia Tech

Are you tired of having to read a bunch of words during a seminar talk? Well, you’re in luck! This talk will be a (nearly) word-free exploration of a construction called unicorn paths. These paths are incredibly useful and can be used to show that both the curve graph and the arc graph of a surface are hyperbolic. 

Benoist’s Limit Cone Theorem

Series
Geometry Topology Student Seminar
Time
Wednesday, April 5, 2023 - 14:00 for
Location
Skiles 006
Speaker
Alex Nolte – Rice

I'll talk about the structure of the collection of all n-ples of eigenvalues of elements of Zariski-dense subgroups D of SL(n,R). Subgroups like this appear, for instance, as the images of holonomy representations of geometric structures. Our focus is a deep and useful result of Benoist, which states that the natural cone one is led to consider here has strong convexity and non-degeneracy properties, and a succinct, qualitative characterization of the cones that so arise from Zariski-dense subgroups. The theorem comes out of a study of the dynamics of the actions of D on spaces of flags such as RP^n and the collection of open subsemigroups of SL(n,R). Everything in this talk is from Benoist’s paper Propriétés Asymptotiques des Groupes Linéaires (GAFA, 2002), and holds in far more generality than I'll state.

The belt trick and spin groups

Series
Geometry Topology Student Seminar
Time
Wednesday, March 29, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Sean Eli – Georgia Tech –

This talk includes an interactive prop demonstration. There exist non-trivial loops in SO(3) (the familiar group of real life rotations) which can be visualized with Dirac's belt trick. Although the belt trick offers a vivid picture of a loop in SO(3), a belt is not a proof, so we will prove SO(n) is not simply connected (n>2), and find its universal covering group Spin(n) (n >2). Along the way we'll introduce the Clifford algebra and study its basic properties. 

The pants complex and More-s

Series
Geometry Topology Student Seminar
Time
Wednesday, March 8, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Roberta Shapiro – Georgia Tech –

The pants complex of a surface has as its 0-cells the pants decompositions of a surface and as its 1-cells some elementary moves relating two pants decompositions; the 2-cells are disks glued along certain cycles in the 1-skeleton of the complex. In "Pants Decompositions of Surfaces," Hatcher proves that this complex is contractible.

 

 During this interactive talk, we will aim to understand the structure of the pants complex and some of the important tools that Hatcher uses in his proof of contractibility.

Common fixed points of commuting homeomorphisms of S^2.

Series
Geometry Topology Student Seminar
Time
Wednesday, March 1, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Cindy Tan – University of Chicago –

When do commuting homeomorphisms of S^2 have a common fixed point? Christian Bonatti gave the first sufficient condition: Commuting diffeomorphisms sufficiently close to the identity in Diff^+(S^2) always admit a common fixed point. In this talk we present a result of Michael Handel that extends Bonatti's condition to a much larger class of commuting homeomorphisms. If time permits, we survey results for higher genus surfaces due to Michael Handel and Morris Hirsch, and connections to certain compact foliated 4-manifolds.

Uniform perfection: a DIFF-icult situation

Series
Geometry Topology Student Seminar
Time
Wednesday, February 15, 2023 - 14:00 for
Location
Skiles 006
Speaker
Roberta Shapiro – Georgia Tech

Have you ever wanted to marry topology, hyperbolic geometry, and geometric group theory, all at once?* Bowden-Hensel-Webb do this and more when they embark on their study of Diff0(S). In this talk, we will discuss the main theorems of Bowden-Hensel-Webb's paper, the most notable of which is (arguably) the lack of uniform perfection of Diff0(S). We will then summarize the main tools they use to prove these results. (Note: the perspectives on Diff0(S) in this talk will DIFFer greatly from those used in the diffeomorphism groups class.) 

 

*If you answered "yes" for your personal life as opposed to your academic life: that's ok, I won't judge (if you don't tell me).

The Braid Group and the Burau Representation

Series
Geometry Topology Student Seminar
Time
Wednesday, February 8, 2023 - 14:00 for
Location
Speaker
Jacob Guynee – Georgia Tech –

The braid group has many applications throughout the world of math due to its simple yet rich structure. In this talk we will focus on the Burau representation of the braid group, which has important implications in knot theory. Most notably, the open problem of faithfulness of the Burau representation of the braid group on 4 strands is equivalent to whether or not the Jones polynomial can detect the unknot. The Burau representation has a topological interpretation that uses the mapping class definition of the braid group. We'll introduce the braid group first and then discuss the Burau representation. We will go through examples for small n and discuss the proof of nonfaithfulness for n > 4. Time permitting, we may introduce the Gassner representation.

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