Seminars and Colloquia by Series

Smooth infinitesimal analysis

Series
Geometry Topology Student Seminar
Time
Friday, May 2, 2014 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
John Dever – Georgia Tech

Please Note: This is a final project for Dr. Etnyre's Differential Geometry class.

After briefly considering embeddings of the category of smooth manifolds into so called smooth toposes and arguing that we may ignore the details of the embedding and work from axioms if we agree to use intuitionistic logic, we consider axiomatic synthetic differential geometry. Key players are a space R playing the role of the "real line" and a space D consisting of null-square infinitesimals such that every function from D to R is "microlinear". We then define microlinear spaces and translate many definitions from differential geometry to this setting. As an illustration of the ideas, we prove Stokes' theorem. Time permitting, we show how synthetic differential geometry may be considered as an extension of differential geometry in that theorems proven in the synthetic setting may be "pulled back" to theorems about smooth manifolds.

Yamabe Problem.

Series
Geometry Topology Student Seminar
Time
Wednesday, April 30, 2014 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 006.
Speaker
Amey Kaloti – Georgia Tech. –
Given a Riemannian manifold $(M,g)$, does there exist a metric $g'$ on $M$ conformal to $g$ such that $g'$ has constant scalar curvature? This question is known as the Yamabe problem. Aim of this talk is to give an overview of the problem and discuss and develop methods that go into solving a few of intermediate results in the solution to the problem in full generality.

Legendrian Torus Knots

Series
Geometry Topology Student Seminar
Time
Wednesday, April 23, 2014 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ece Gülşah Çolak – Bülent Ecevit University and Georgia Tech –
We will discuss Etnyre and Honda's proof of the classification of Legendrian positive torus knots in the tight contact 3-sphere up to Legendrian isotopy by using the tools from convex surface theory.

Taut Foliations on 3-manifolds.

Series
Geometry Topology Student Seminar
Time
Wednesday, March 12, 2014 - 13:59 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Dheeraj Kulkarni – Georgia Tech. –
In this talk, we will discuss a result due to Gabai which states that a minimal genus Seifert surface for a knot in 3-sphere can be realized as a leaf of a taut foliation of the knot complement. We will give a fairly detailed outline of the proof. In the process, we will learn how to construct taut foliations on knot complements.

Overview of Yamabe problem

Series
Geometry Topology Student Seminar
Time
Wednesday, February 12, 2014 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 006.
Speaker
Amey Kaloti – Georgia Tech.
We will give an overview of ideas that go into solution of Yamabe problem: Given a compact Riemannian manifold (M,g) of dimension n > 2, find a metric conformal to g with constant scalar curvature.

Knot Contact Homology

Series
Geometry Topology Student Seminar
Time
Wednesday, February 5, 2014 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jamie Conway – GeorgiaTech
Knot Contact Homology is a powerful invariant assigning to each smooth knot in three-space a differential graded algebra. The homology of this algebra is in general difficult to calculate. We will discuss the cord algebra of a knot, which allows us to calculate the grading 0 knot contact homology. We will also see a method of extracting information from augmentations of the algebra.

Non-lifting of a subgroup of the mapping class group

Series
Geometry Topology Student Seminar
Time
Wednesday, December 4, 2013 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Robert Krone – Georgia Tech
The mapping class group of a surface is a quotient of the group of orientation preserving diffeomorphisms. However the mapping class group generally can't be lifted to the group of diffeomorphisms, and even many subgroups can't be lifted. Given a surface S of genus at least 2 and a marked point z, the fundamental group of S naturally injects to a subgroup of MCG(S,z). I will present a result of Bestvina-Church-Souto that this subgroup can't be lifted to the diffeomorphisms fixing z.

Hirzebruch's signature theorem in dimension 4

Series
Geometry Topology Student Seminar
Time
Wednesday, November 20, 2013 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Alan Diaz – School of Math, Georgia Tech –
We'll prove the simplest case of Hirzebruch's signature theorem, which relates the first Pontryagin number of a smooth 4-manifold to the signature of its intersection form. If time permits, we'll discuss the more general case of 4k-manifolds. The result is relevant to Prof. Margalit's ongoing course on characteristic classes of surface bundles.

Chern-Weil theory for vector bundles

Series
Geometry Topology Student Seminar
Time
Friday, November 15, 2013 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 006.
Speaker
Amey Kaloti – Georgia Tech.
Given a vector bundle over a smooth manifold, one can give an alternate definition of characteristic classes in terms of geometric data, namely connection and curvature. We will see how to define Chern classes and Euler class for the a vector bundle using this theory developed in mid 20th century.

Pages