Seminars and Colloquia by Series

A Study of Knots & Links derived from Doubly Periodic Knitted Fabric Patterns

Series
Geometry Topology Student Seminar
Time
Wednesday, November 6, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Shashank MarkandeGeorgia Tech

The emergent shape of a knitted fabric is highly sensitive to the underlying stitch pattern. Here, by a stitch pattern we mean a periodic array of symbols encoding a set of rules or instructions performed to produce a swatch or a piece of fabric. So, it is crucial to understand what exactly these instructions mean in terms of mechanical moves performed using a yarn (a smooth piece of string) and a set of knitting needles (oriented sticks). Motivated by the fact that locally every knitting move results in a slip knot, we use tools from topology to model the set of all doubly periodic stitch patterns, knittable & non-knittable, as knots & links in a three manifold. Specifically, we define a map from the set of doubly-periodic stitch patterns to the set of links in S^3 and use link invariants such as the linking number, multivariable Alexander polynomial etc. to characterize them. We focus on such links derived from knitted stitch patterns in an attempt to tackle the question: whether or not a given stitch pattern can be realized through knitting.

Heegaard Floer obstruction to knot surgery

Series
Geometry Topology Student Seminar
Time
Wednesday, October 23, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Hongyi ZhouGeorgia Tech

Which manifold can be obtained from surgery on a knot? Many obstructions to this have been studied. We will discuss some of them, and use Heegaard Floer homology to give an infinite family of seifert fibered integer spheres that cannot be obtained by surgery on a knot in S^3. We will also discuss a recipe to compute HF+ of surgery on a knot (Short review on Heegaard Floer homology included).

Partial Torelli Groups

Series
Geometry Topology Student Seminar
Time
Wednesday, October 16, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Daniel MinahanGeorgia Tech

The Torelli group is the subgroup of the mapping class group acting trivially on homology.  We will discuss some basic properties of the Torelli group and explain how to define it for surfaces with boundary.  We will also give some Torelli analogues of the Birman exact sequence.

Obstructions to nice branch sets for branched coverings

Series
Geometry Topology Student Seminar
Time
Wednesday, October 9, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Sudipta KolayGeorgia Tech

It is a classical theorem of Alexander that every closed oriented manifold is a piecewise linear branched covering of the sphere. In this talk, we will discuss some obstructions to realizing a manifold as a branched covering of the sphere if we require additional properties (like being a submanifold) on the branch set.

 

H-cobordisms and corks

Series
Geometry Topology Student Seminar
Time
Wednesday, October 2, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Agniva RoyGeorgia Tech

Stephen Smale’s h-cobordism Theorem was a landmark result in the classification of smooth manifolds. It paved the way towards solutions for the topological Poincaré and Schoenflies conjectures in dimensions greater than 5. Later, building on this, Freedman’s work applied these techniques to 4 manifolds. I shall discuss the ideas relating to h-cobordisms and the proof, which is a wonderful application of handlebody theory and the Whitney trick. Time permitting, we shall explore the world of smooth 4 manifolds further, and talk about cork twists.

Graph Theory and Heegaard Floer Homology

Series
Geometry Topology Student Seminar
Time
Wednesday, September 25, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Hyun Ki MinGeorgia Tech

I will talk about a connection between graph theory and sutured Floer homology. In fact, there is a one to one correspondence between hypergraphs of a planar bipartite graph and the dimension of sutured Floer homology of a complement of a neighborhood of special alternating link In a three sphere. This is based on the work of Juhas, Kalman and Rasmussen.

Surface bundles in topology, algebraic geometry, and group theory

Series
Geometry Topology Student Seminar
Time
Wednesday, September 18, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Justin LanierGeorgia Tech

I will give an introduction to surface bundles and will discuss several places where they arise naturally. A surface bundle is a fiber bundle where the fiber is a surface. A first example is the mapping torus construction for 3-manifolds, which is a surface bundle over the circle. Topics will include a construction of 4-manifolds as well as section problems related to surface bundles. The talk will be based on a forthcoming Notices survey article by Salter and Tshishiku.

Unfoldings of 3D Polyhedra

Series
Geometry Topology Student Seminar
Time
Wednesday, September 11, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Nicholas Barvinok

Cutting a polyhedron along some spanning tree of its edges will yield an isometric immersion of the polyhedron into the plane. If this immersion is also injective, we call it an unfolding. In this talk I will give some general results about unfoldings of polyhedra. There is also a notion of pseudo-edge unfolding, which involves cutting on a pseudo edge graph, as opposed to an edge graph. A pseudo edge graph is a 3-connected graph on the surface of the polyhedron, whose vertices coincide with the vertices of the polyhedron, and whose edges are geodesics. I will explain part of the paper "Pseudo-Edge Unfoldings of Convex Polyhedra," a joint work of mine with Professor Ghomi, which proves the existence of a convex polyhedron with a pseudo edge graph along which it is not unfoldable. Finally, I will discuss some connections between pseudo edge graphs and edge graphs. 

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