Graduate Student Math Conference

Brown University

Tuesday, April 5th 2016

Titles & Abstracts

10:30 - 11:20

Floer homology and its topological applications

Floer originally introduced his instanton homology groups of three manifolds to study the gluing properties of the Donaldson invariants of four manifolds. On the other hand, this invariant and its relatives turned out to be extremely useful when studying problems which have in principle nothing to do with four dimensional topology. In this talk we introduce the main ideas behind the construction of these Floer theoretic invariants and discuss some of their ground-breaking applications, including Kronheimer-Mrowka's proof of Property P for knots and Manolescu's disproof of the Triangulation conjecture in high dimensions.

11:30 - 12:20

Khovanov homology and transverse knots

Transverse links have two classical invariants: smooth isotopy type, and self-linking number. These are not complete invariants: there are families of smoothly isotopic links with the same self-linking number but different transverse isotopy classes. Plamenevskaya defined an invariant of transverse links using Khovanov homology in the hope of obtaining a more powerful invariant. Using the annular refinement of Khovanov homology and a correspondence between transverse links and braids, we define a conjugacy class invariant in the braid group which may be more powerful than Plamenevskaya's invariant. The invariant has applications to the word problem in the braid group, the lengths of certain spectral sequences, and some classical questions about braids. I hope to also wave my hands at some connections with Floer homology and contact topology. This is joint work with Diana Hubbard.

12:30 - 1:30


Lunch will be provided for those who attend.

1:30- 2:20

Ren Yi


Introduction to Tetrahedral Twist Maps

We introduce a family of piecewise isometries f_s parametrized by s in [0,1] on the surface of a regular tetrahedron, which we call the tetrahedral twists. This family of maps is similar to the polygon exchange transformations (PETs) constructed by Hooper. We study the dynamics of the tetrahedral twists through the notion of renormalization. With computer assistance, we conjecture that the renormalization scheme exists on the entire interval [0,1]. We will show that this system is renormalizable when s lies in the subintervals I_1=[53/128,29/70] or I_2=[1/2,1].

2:30 - 3:20

Hyperbolic 3-manifolds with low cusp volume

The past fifteen years have seen a great deal of progress towards a complete picture of hyperbolic manifolds of low volume. The volume of a hyperbolic manifold is a topological invariant and can be viewed as a measure of complexity. In fact, this invariant is finite-to-one. For manifolds with cusps, one can also consider the volume of the maximal horoball neighborhood of a cusp. In this talk, we will present preliminary results for classifying the infinite families of hyperbolic 3-manifolds of cusp volume < 2.62 and the implications of this classification. These families are of particular interest as they exhibit the largest number of exceptional Dehn fillings. As in some other results on hyperbolic 3-manifolds of low volume, our technique utilizes a rigorous computer assisted search. The talk will focus on providing sufficient background to explain our approach and describe our conclusions. This work is joint with David Gabai, Robert Meyerhoff, Nathaniel Thurston, and Robert Haraway.

3:30 - 4:00


4:00 - 4:50

Ryan Mickler


Chern-Simons invariants of Seifert manifolds via loop spaces

Over the past 30 years the Chern-Simons functional for connections on G-bundles over three-manfolds has lead to a deep understanding of the geometry of three-manfiolds, as well as knot invariants such as the Jones polynomial. Here we study this functional for three-manfolds that are topologically given as the total space of a principal circle bundle over a compact Riemann surface base, which are known as Seifert manifolds. We show that on such manifolds the Chern-Simons functional reduces to a particular gauge-theoretic functional on the 2d base, that describes a gauge theory of connections on an infinite dimensional bundle over this base with structure group given by the level-k affine central extension of the loop group LG. We show that this formulation gives a new understanding of results of Beasley-Witten on the computability of quantum Chern-Simons invariants of these manifolds as well as knot invariants for knots that wrap a single fiber of the circle bundle.

5:00 - 5:50

Matthew Cordes


Morse boundaries of proper geodesic metric spaces

I will introduce a new type of boundary for proper geodesic spaces, called the Morse boundary, that is constructed with equivalence classes of geodesic rays that identify the "hyperbolic directions" in that space. (A ray is Morse if quasi-geodesics with endpoints on the ray stay bounded distance from the ray.) This boundary is a quasi-isometry invariant and a visibility space. In the case of a proper CAT(0) space the Morse boundary generalizes the contracting boundary of Charney and Sultan and in the case of a proper Gromov hyperbolic space this boundary is the Gromov boundary. Time permitting I will also discuss some results on the Morse boundary of the mapping class group.

Organized by Ken Ascher, Dori Bejleri, Mamikon Gulian, and Laura Walton at the Brown Math Department

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