Seminars and Colloquia by Series

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.

Thurston equivalence is decidable

Series
Geometry Topology Seminar
Time
Thursday, October 26, 2017 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Nikita SelingerUniversity of Alabama-Birmingham
In a joint work with M. Yampolsky, we gave a classification of Thurston maps with parabolic orbifolds based on our previous results on characterization of canonical Thurston obstructions. The obtained results yield a solution to the problem of algorithmically checking combinatorial equivalence of two Thurston maps.

The immersed cross-cap number of a knot

Series
Geometry Topology Seminar
Time
Monday, October 23, 2017 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Mark HughesBYU
The immersed Seifert genus of a knot $K$ in $S^3$ can be defined as the minimal genus of an orientable immersed surface $F$ with $\partial F = K$. By a result of Gabai, this value is always equal to the (embedded) Seifert genus of $K$. In this talk I will discuss the embedded and immersed cross-cap numbers of a knot, which are the non-orientable versions of these invariants. Unlike their orientable counterparts these values do not always coincide, and can in fact differ by an arbitrarily large amount. In further contrast to the orientable case, there are families of knots with arbitrarily high embedded 4-ball cross-cap numbers, but which are easily seen to have immersed cross-cap number 1. After describing these examples I will discuss a classification of knots with immersed cross-cap number 1. This is joint work with Seungwon Kim.

Complex curves through a contact lens

Series
Geometry Topology Seminar
Time
Monday, October 16, 2017 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Kyle HaydenBoston College
Every four-dimensional Stein domain has a Morse function whoseregular level sets are contact three-manifolds. This allows us to studycomplex curves in the Stein domain via their intersection with thesecontact level sets, where we can comfortably apply three-dimensional tools.We use this perspective to understand links in Stein-fillable contactmanifolds that bound complex curves in their Stein fillings.

Joint GT-UGA Seminar at GT - Fibered, homotopy-ribbon disk-knots by Jeff Meier

Series
Geometry Topology Seminar
Time
Monday, October 2, 2017 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Jeff MeierUGA
I'll introduce you to one of my favorite knotted objects: fibered, homotopy-ribbon disk-knots. After giving a thorough overview of these objects, I'll discuss joint work with Kyle Larson that brings some new techniques to bear on their study. Then, I'll present new work with Alex Zupan that introduces connections with Dehn surgery and trisections. I'll finish by presenting a classification result for fibered, homotopy-ribbon disk-knots bounded by square knots.

Joint GT-UGA Seminar at GT - Homology Cobordism of Seifert Spaces

Series
Geometry Topology Seminar
Time
Monday, October 2, 2017 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Matt StoffregenMIT
We use Manolescu's Pin(2)-equivariant Floer homology to study homology cobordisms among Seifert spaces. In particular, we will show that the subgroup of the homology cobordism group generated by Seifert spaces admits a \mathbb{Z}^\infty summand. This is joint work with Irving Dai.

On boundaries of relatively hyperbolic right-angled Coxeter groups

Series
Geometry Topology Seminar
Time
Monday, September 25, 2017 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Hung TranGeorgia
We give "visual descriptions" of cut points and non-parabolic cut pairs in the Bowditch boundary of a relatively hyperbolic right-angled Coxeter group. We also prove necessary and sufficient conditions for a relatively hyperbolic right-angled Coxeter group whose defining graph has a planar flag complex with minimal peripheral structure to have the Sierpinski carpet or the 2-sphere as its Bowditch boundary. We apply these results to the problem of quasi-isometry classification of right-angled Coxeter groups. Additionally, we study right-angled Coxeter groups with isolated flats whose $\CAT(0)$ boundaries are Menger curve. This is a joint work with Matthew Haulmark and Hoang Thanh Nguyen.

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