Seminars and Colloquia by Series

Smooth concordance, homology cobordism, and the figure-8 knot

Series
Geometry Topology Student Seminar
Time
Wednesday, October 20, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006 (also in BlueJeans)
Speaker
Sally CollinsGeorgia Tech

Please Note: BlueJeans link: https://bluejeans.com/936509442/0487

Given two knots K_1 and K_2, their 0-surgery manifolds S_0^3(K_1) and S_0^3(K_2) are homology cobordant rel meridian if they are homology cobordant preserving the homology class of the positively oriented meridian. It is known that if K_1 ∼ K_2, then S_0^3(K_1) and S_0^3(K_2) are homology cobordant rel meridian. The converse of this statement was first disproved by Cochran-Franklin-Hedden-Horn.  In this talk we will provide a new counterexample, the pair of knots 4_1 and M(4_1) where M is the Mazur satellite operator. S_0^3(4_1) and S_0^3(M(4_1)) are homology cobordant rel meridian, but 4_1 and M(4_1) are non-concordant and have concordance orders 2 and infinity, respectively. 

SLn skein algebra and quantum matrices

Series
Geometry Topology Student Seminar
Time
Wednesday, October 13, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006 (also in BlueJeans)
Speaker
Tao YuGeorgia Tech

Please Note: BlueJeans link: https://bluejeans.com/248767326/2767

Since Jones introduced his knot polynomial using representation theory, there has been a wide variety of invariants defined this way, e.g., HOMFLY-PT and Reshetikhin-Turaev. Recently, through the work of Bonahon-Wong and Constantino-Le, some of these invariants are reinterpreted as quantum matrices. In this talk, we will review the history of these representation theoretical knot invariants. Then we will discuss one particular connection to the quantum special linear group.

On Anosovity, divergence and bi-contact surgery

Series
Geometry Topology Student Seminar
Time
Wednesday, October 6, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006 (also on BlueJeans)
Speaker
Surena HozooriGeorgia Tech

Please Note: BlueJeans link: https://bluejeans.com/844708532/5458

I will revisit the relation between Anosov 3-flows and invariant volume forms, from a contact geometric point of view. Consequently, I will give a contact geometric characterization of when a flow with dominated splitting is Anosov based on its divergence, as well as a Reeb dynamical interpretation of when such flows are volume preserving. Moreover, I will discuss the implications of this study on the surgery theory of Anosov 3-flows. In particular, I will conclude that the Goodman-Fried surgery of Anosov flows can be reconstructed, using a bi-contact surgery of Salmoiraghi.

Legendrians, Contact Structures, and Time Travel

Series
Geometry Topology Student Seminar
Time
Wednesday, September 29, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Agniva RoyGeorgia Tech

A general theme in studying manifolds is understanding lower dimensional submanifolds that encode information. For contact manifolds, these are Legendrians. I will discuss some low and high dimensional examples of Legendrians, their invariants, and how they are applied to understand manifolds. I will also talk about the Legendrian Low Conjecture, which says that understanding linking of certain Legendrians is the key to understanding causal relations between events in a globally hyperbolic spacetime.

Mapping Class Group of 4-Manifolds

Series
Geometry Topology Student Seminar
Time
Wednesday, September 22, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Anubhav MukherjeeGeorgia Tech

One interesting question in low-dimensional topology is to understand the structure of mapping class group of a given manifold. In dimension 2, this topic is very well studied. The structure of this group is known for various 3-manifolds as well (ref- Hatcher's famous work on Smale's conjecture). But virtually nothing is known in dimension 4. In this talk I will try to motivate why this problem in dimension 4 is interesting and how it is different from dimension 2 and 3. I will demonstrate some "exotic" phenomena and if time permits, I will talk a few words on my upcoming work with Jianfeng Lin. 

Enumerating Knots and Links

Series
Geometry Topology Student Seminar
Time
Wednesday, September 15, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Hugo ZhouGeorgia Tech

How do we build a knot table? We will discuss Conway’s paper “an enumeration of knots and links” and Hoste, Thistlethwaite and Weeks’ paper “the first 1701936 knots”.

An Alexander method for infinite-type surfaces

Series
Geometry Topology Student Seminar
Time
Wednesday, April 21, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Roberta Shapiro

Given a surface S, the Alexander method is a combinatorial tool used to determine whether two homeomorphisms are isotopic. This statement was formalized in A Primer on Mapping Class Groups in the case that S is of finite type. We extend the Alexander method to include infinite-type surfaces, which are surfaces with infinitely generated fundamental groups.

In this talk, we will introduce a technique useful in proofs dealing with infinite-type surfaces. Then, we provide a "proof by example" of an infinite-type analogue of the Alexander method.

Exotic smooth structures and H-slice knots

Series
Geometry Topology Student Seminar
Time
Wednesday, April 7, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
Speaker
Hyunki MinGeorgia Tech

One of interesting topic in low-dimensional topology is to study exotic smooth structures on closed 4-manifolds. In this talk, we will see an example to distinguish exotic smooth structure using H-slice knots.

Symplectic rigidity, flexibility, and embedding problems

Series
Geometry Topology Student Seminar
Time
Wednesday, March 31, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
Online
Speaker
Agniva RoyGeorgia Tech

Embedding problems, of an n-manifold into an m-manifold, can be heuristically thought to belong to a spectrum, from rigid, to flexible. Euclidean embeddings define the rigid end of the spectrum, meaning you can only translate or rotate an object into the target. Symplectic embeddings, depending on the object, and target, can show up anywhere on the spectrum, and it is this flexible vs rigid philosophy, and techniques developed to study them, that has lead to a lot of interesting mathematics. In this talk I will make this heuristic clearer, and show some examples and applications of these embedding problems.

Pages