Seminars and Colloquia by Series

An introduction to Morse theory and Morse homology

Series
Geometry Topology Student Seminar
Time
Wednesday, September 20, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Akash NarayananGeorgia Institute of Technology

Morse theory analyzes the topology of a smooth manifold by studying the behavior of its real-valued functions. From this, one obtains a well-behaved homology theory which provides further information about the manifold and places constraints on the smooth functions it admits. This idea has proven to be useful in approaching topological problems, playing an essential role in Smale's solution to the generalized Poincare conjecture in dimensions greater than 4. Morse theory has been adapted to study complex manifolds, and even algebraic varieties over more general fields, but the underlying principles remain the same. In this talk, we will define the basic notions of Morse theory and describe some of the fundamental results. Then we will define Morse homology and discuss some important corollaries and applications. 

An Interactive Introduction to Surface Bundles

Series
Geometry Topology Student Seminar
Time
Wednesday, September 13, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jaden WangGeorgia Tech

Surface bundles lie in the intersection of many areas of math: algebraic topology, 2–4 dimensional topology, geometric group theory, algebraic geometry, and even number theory! However, we still know relatively little about surface bundles, especially compared to vector bundles. In this interactive talk, I will present the general (and beautiful) fiber bundle theory, including characteristic classes, as a starting point, and you the audience will get to specialize the general theory to surface bundles, with rewards! The talk aims to be accessible to anyone who had exposure to algebraic topology. This is also part one of three talks about surface bundles I will give this semester.

Spheres can knot in 4 dimensions

Series
Geometry Topology Student Seminar
Time
Wednesday, September 6, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Sean EliGeorgia Tech

You are probably familiar with the concept of a knot in 3 space: a tangled string that can't be pushed and pulled into an untangled one. We briefly show how to prove mathematical knots are in fact knotted, and discuss some conditions which guarantee unknotting. We then give explicit examples of knotted 2-spheres in 4 space, and discuss 2-sphere version of the familiar theorems. A large part of the talk is practice with visualizing objects in 4 dimensional space. We will also prove some elementary facts to give a sense of what working with these objects feels like. Time permitting we will describe know knotted 2-spheres were used to give evidence for the smooth 4D Poincare conjecture, one of the guiding problems in the field.

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 KnavelGeorgia 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 NolteRice 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 BoothGeorgia 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 NolteRice

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 EliGeorgia 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 ShapiroGeorgia 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.

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