Seminars and Colloquia by Series

Every surface is a leaf

Series
Geometry Topology Student Seminar
Time
Wednesday, May 6, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
Online
Speaker
Justin LanierGeorgia Tech

Every closed 3-manifold admits foliations, where the leaves are surfaces. For a given 3-manifold, which surfaces can appear as leaves? Kerékjártó and Richards gave a classification up to homeomorphism of noncompact surfaces, which includes surfaces with infinite genus and infinitely many punctures. In their 1985 paper "Every surface is a leaf", Cantwell--Conlon prove that for every orientable noncompact surface L and every closed 3-manifold M, M has a foliation where L appears as a leaf. We will discuss their paper and construction and the surrounding context.

Bordered Floer Homology via Immersed Curves

Series
Geometry Topology Student Seminar
Time
Wednesday, April 22, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
Online
Speaker
Sally CollinsGeorgia Tech

In the setting of manifolds with connected torus boundary, we can reinterpret bordered invariants as immersed curves in the once punctured torus. This machinery, due to Hanselman, Rasmussen, and Watson, is particularly useful in the context of knot complements. We will show how a type D structure can be viewed as a multicurve in the boundary of a manifold, and we will consider how the operation of cabling acts on this new invariant. If time permits, we will discuss how to extract concordance invariants from the curves.

Bordered Floer Homology

Series
Geometry Topology Student Seminar
Time
Wednesday, April 15, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Sally CollinsGeorgia Tech

Bordered Floer homology, due to Lipshitz, Ozsváth, and Thurston, is a Heegaard Floer homology theory for 3-manifolds with connected boundary. This theory associates to the boundary surface (with suitable parameterization) a differential graded algebra A(Z). Our invariant comes in two versions: a left differential (type D) module over A(Z), or its dual, a right A-infinity (type A) module over A(Z). In this talk, we will focus on the case of 3-manifolds with torus boundary, and will explicitly describe how to compute type D structures of knot complements.

The Jones polynomial via quantum group representations

Series
Geometry Topology Student Seminar
Time
Wednesday, April 8, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
Online
Speaker
Tao YuGeorgia Tech

Continuing the theme of Hopf algebras, we will discuss a recipe by Reshetikhin and Turaev for link invariants using representations of quantum groups, which are non-commutative, non-cocommutative Hopf algebras. In the simplest case with the spin 1/2 representation of quantum sl2, we recover the Kauffman bracket and the Jones polynomial when combined with writhe. Time permitting, we will also talk about colored Jones polynomials and connections to 3-manifold invariants.

Hopf Algebras and Cohomology of Lie Groups

Series
Geometry Topology Student Seminar
Time
Wednesday, April 1, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
Online
Speaker
Tao YuGeorgia Tech

In 1941, Hopf gave a proof of the fact that the rational cohomology of a compact connected Lie group is isomorphic to the cohomology of a product of odd dimensional spheres. The proof is natural in the sense that instead of using the classification of Lie groups, it utilizes the extra algebraic structures, now known as Hopf algebras. In this talk, we will discuss the algebraic background and the proof of the theorem.

Noncollapsed Ricci limit spaces and the codimension 4 conjecture

Series
Geometry Topology Student Seminar
Time
Wednesday, March 11, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Xingyu ZhuGeorgia Tech

In this talk we will survey some of the developments of Cheeger and Colding’s conjecture on a sequence of n dimensional manifolds with uniform two sides Ricci Curvature bound, investigated by Anderson, Tian, Cheeger, Colding and Naber among others. The conjecture states that every Gromov-Hausdorff limit of the above-mentioned sequence, which is a metric space with singularities,  has the singular set with Hausdorff codimension at least 4. This conjecture was proved by Colding-Naber in 2014, where the ideas and techniques like \epsilon-regularity theory, almost splitting and quantitative stratification were extensively used. I will give an introduction of the background of the conjecture and talk about the idea of the part of the proof that deals with codimension 2 singularities.

Cosmetic surgeries on knots in S^3

Series
Geometry Topology Student Seminar
Time
Wednesday, February 26, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Hugo ZhouGeorgia Tech

Cosmetic surgeries (purely cosmetic surgeries) are two distinct surgeries on a knot that produce homeomorphic 3-manifolds (as oriented manifolds). It is one of the ways Dehn surgeries on knots could fail to be unique. Gordon conjectured that there are no nontrivial purely cosmetic surgeries on nontrivial knots in S^3. We will recap the progress of the problem over time, and mainly discuss Ni and Wu's results in their paper "Cosmetic surgeries on knots in S^3".

Lifting Covers to Braided Embeddings

Series
Geometry Topology Student Seminar
Time
Wednesday, February 19, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Sudipta KolayGeorgia Tech

An embedding of a manifold into a trivial disc bundle over another manifold is called braided if projection onto the first factor gives a branched cover. This notion generalizes closed braids in the solid torus, and gives an explicit way to construct many embeddings in higher dimensions. In this talk, we will discuss when a covering map of surfaces lift to a braided embedding.

TBA by Stavros Garoufalidis

Series
Geometry Topology Student Seminar
Time
Wednesday, February 12, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker

Please Note: This is an ordinary research Geometry/Topology seminar: https://math.gatech.edu/seminars-colloquia/series/geometry-topology-seminar/s-garoufalidis-20200212

Fillings of Contact 3 Manifolds and Relations in Mapping Class Groups of Surfaces

Series
Geometry Topology Student Seminar
Time
Wednesday, February 5, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Agniva RoyGeorgia Tech

A useful way of studying contact 3 manifolds is by looking at their open book decompositions. A result of Akbulut-Ozbagci, Ghiggini, and Loi-Piergallini showed that the manifold is filled by a Stein manifold if and only if the monodromy of an open book can be factorised as the product of positive Dehn twists. Then, the problem of classifying minimal fillings of contact 3 manifolds, or answering questions about which manifolds can be realised by Legendrian surgery, becomes questions about finding factorisations for a given mapping class. This talk will be expository and expand upon how these mapping classes come up, and also discuss known results, techniques, and future directions for research.

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