Seminars and Colloquia by Series

Joint GT-UGA Seminar at GT - Unoriented skein relations for link and tangle invariants in Heegaard Floer theory

Series
Geometry Topology Seminar
Time
Monday, February 19, 2018 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Mike WongLSU
Although the Alexander polynomial does not satisfy an unoriented skein relation, Manolescu (2007) showed that there exists an unoriented skein exact triangle for knot Floer homology. In this talk, we will describe some developments in this direction since then, including a combinatorial proof using grid homology and extensions to the Petkova-Vertesi tangle Floer homology (joint work with Ina Petkova) and Zarev's bordered sutured Floer homology (joint work with Shea Vela-Vick).

Non-Orientable Lagrangian Fillings of Legendrian Knots

Series
Geometry Topology Seminar
Time
Monday, February 12, 2018 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Josh SabloffHaverford College
Lagrangian fillings of Legendrian knots are interesting objects that are related, on one hand, to the 4-genus of the underlying smooth knot and, on the other hand, to Floer-type invariants of Legendrian knots. Most work on Lagrangian fillings to date has concentrated on orientable fillings. I will present some first steps in constructions of and obstructions to the existence of (decomposable exact) non-orientable Lagrangian fillings. In addition, I will discuss links between the 4-dimensional crosscap number of a knot and the non-orientable Lagrangian fillings of its Legendrian representatives. This is joint work in progress with Linyi Chen, Grant Crider-Philips, Braeden Reinoso, and Natalie Yao.

Geometry and dynamics of free group automorphisms

Series
Geometry Topology Seminar
Time
Monday, January 29, 2018 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Caglar UyanikVanderbilt University
I will talk about the long standing analogy between the mapping class group of a hyperbolic surface and the outer automorphism group of a free group. Particular emphasis will be on the dynamics of individual elements and applications of these results to structure theorems for subgroups of these groups.

Symplectic K-theory of the integers and Galois groups.

Series
Geometry Topology Seminar
Time
Monday, December 4, 2017 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Soren GalatiusStanford University
The general linear groups GL_n(A) can be defined for any ring A, and Quillen's definition of K-theory of A takes these groups as its starting point. If A is commutative, one may define symplectic K-theory in a very similar fashion, but starting with the symplectic groups Sp_{2n}(A), the subgroup of GL_{2n}(A) preserving a non-degenerate skew-symmetric bilinear form. The result is a sequence of groups denoted KSp_i(A) for i = 0, 1, .... For the ring of integers, there is an interesting action of the absolute Galois group of Q on the groups KSp_i(Z), arising from the moduli space of polarized abelian varieties. In joint work with T. Feng and A. Venkatesh we study this action, which turns out to be an interesting extension between a trivial representation and a cyclotomic representation.

A large abelian quotient of the level 4 braid group

Series
Geometry Topology Seminar
Time
Monday, November 20, 2017 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Kevin KordekGeorgia Institute of Technology
It is generally a difficult problem to compute the Betti numbers of a given finite-index subgroup of an infinite group, even if the Betti numbers of the ambient group are known. In this talk, I will describe a procedure for obtaining new lower bounds on the first Betti numbers of certain finite-index subgroups of the braid group. The focus will be on the level 4 braid group, which is the kernel of the mod 4 reduction of the integral Burau representation. This is joint work with Dan Margalit.

Growth of torsion homology in finite coverings and hyperbolic volume

Series
Geometry Topology Seminar
Time
Monday, November 13, 2017 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Thang LeGeorgia Tech
We discuss the growth of homonoly in finite coverings, and show that the growth of the torsion part of the first homology of finite coverings of 3-manifolds is bounded from above by the hyperbolic volume of the manifold. The proof is based on the theory of L^2 torsion.

Joint GT-UGA Seminar at UGA - Conway mutation and knot Floer homology by Peter Lambert-Cole and A non-standard bridge trisection of the unknot by Alex Zupan

Series
Geometry Topology Seminar
Time
Monday, November 6, 2017 - 14:30 for 2.5 hours
Location
Boyd 304
Speaker
Peter Lambert-Cole and Alex ZupanGeorgia Tech and Univ. Nebraska Lincoln
Peter Lambert-Cole: Mutant knots are notoriously hard to distinguish. Many, but not all, knot invariants take the same value on mutant pairs. Khovanov homology with coefficients in Z/2Z is known to be mutation-invariant, while the bigraded knot Floer homology groups can distinguish mutants such as the famous Kinoshita-Terasaka and Conway pair. However, Baldwin and Levine conjectured that delta-graded knot Floer homology, a singly-graded reduction of the full invariant, is preserved by mutation. In this talk, I will give a new proof that Khovanov homology mod 2 is mutation-invariant. The same strategy can be applied to delta-graded knot Floer homology and proves the Baldwin-Levine conjecture for mutations on a large class of tangles. -----------------------------------------------------------------------------------------------------------------------------------------------Alex Zupan: Generally speaking, given a type of manifold decomposition, a natural problem is to determine the structure of all decompositions for a fixed manifold. In particular, it is interesting to understand the space of decompositions for the simplest objects. For example, Waldhausen's Theorem asserts that up to isotopy, the 3-sphere has a unique Heegaard splitting in every genus, and Otal proved an analogous result for classical bridge splittings of the unknot. In both cases, we say that these decompositions are "standard," since they can be viewed as generic modifications of a minimal splitting. In this talk, we examine a similar question in dimension four, proving that -- unlike the situation in dimension three -- the unknotted 2-sphere in the 4-sphere admits a non-standard bridge trisection. This is joint work with Jeffrey Meier.

Transverse invariants, knot Floer homology and branched covers

Series
Geometry Topology Seminar
Time
Monday, October 30, 2017 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Shea Vela-VickLSU
Heegaard Floer theory provides a powerful suite of tools for studying 3-manifolds and their subspaces. In 2006, Ozsvath, Szabo and Thurston defined an invariant of transverse knots which takes values in a combinatorial version of this theory for knots in the 3—sphere. In this talk, we discuss a refinement of their combinatorial invariant via branched covers and discuss some of its properties. This is joint work with Mike Wong.

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