Seminars and Colloquia by Series

The Frohman-Kania-Bartoszynska invariant is the 3D index

Series
Geometry Topology Seminar
Time
Wednesday, February 12, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
S. GaroufalidisSUSTECH and MPI Bonn
We prove that a power series invariant of suitable ideal triangulations, defined by Frohman-Kania-Bartoszynska coincides with the power series invariant of Dimofte-Gaiotto-Gukov known as the 3D index. In partucular, we deduce that the FKB invariant is topological, and that the tetrahedron weight of the 3D index is a limit of quantum 6j symbols. Joint work with Roland van der Veen.

Joint UGA-GT Topology Seminar at GT: Brieskorn spheres bounding rational balls

Series
Geometry Topology Seminar
Time
Monday, February 10, 2020 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Kyle LarsonUGA

Fintushel and Stern showed that the Brieskorn sphere Σ(2, 3, 7) bounds a rational homology ball, while its non-trivial Rokhlin invariant obstructs it from bounding an integral homology ball. It is known that their argument can be modified to show that the figure-eight knot is rationally slice, and we use this fact to provide the first additional examples of Brieskorn spheres that bound rational homology balls but not integral homology balls, including two infinite families. This is joint work with Selman Akbulut.

Joint UGA-GT Topology Seminar at GT: Homotopy invariants of homology cobordism and knot concordance

Series
Geometry Topology Seminar
Time
Monday, February 10, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Kent OrrIndiana University
Modern homotopy invariants of links derive from Gauss’ work on linking numbers.  Many modern examples have arisen following Milnor’s early work.  I will define and investigate a `universal' homotopy invariant of homology cobordism classes of orientable 3-manifolds.  Time permitting (unlikely,) the resulting equivalence classes yield further invariants using filtrations, and classical and von Neumann signatures.  Primary focus will be given to defining these
invariants, and the tools essential to their definition.

Annular Rasmussen invariants: Properties and 3-braid classification

Series
Geometry Topology Seminar
Time
Monday, February 3, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Gage MartinBoston College

Annular Rasmussen invariants are invariants of braid closures which generalize the Rasmussen s invariant and come from an integer bifiltration on Khovanov-Lee homology. In this talk we will explain some connections between the annular Rasmussen invariants and other topological information. Additionally we will state theorems about restrictions on the possible values of annular Rasmussen invariants and a computation of the invariants for all 3-braid closures, or conjugacy classes of 3-braids. Time permitting, we will sketch some proofs.

The coalgebra of singular chains and the fundamental group

Series
Geometry Topology Seminar
Time
Monday, January 27, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Manuel RiveraPurdue University

The goal of this talk is to explain the sense in which the natural algebraic structure of the singular chains on a path-connected space determines its fundamental group functorially. This new basic piece about the algebraic topology of spaces, which tells us that the fundamental group may be determined from homological data, has several interesting and deep implications. An example of a corollary of our statement is the following extension of a classical theorem of Whitehead: a continuous map between path-connected pointed topological spaces is a weak homotopy equivalence if and only if the induced map between the differential graded coalgebras of singular chains is a Koszul weak equivalence (i.e. a quasi-isomorphism after applying the cobar functor). A deeper implication, which is work in progress, is that this allows us to give a complete description of infinity groupoids in terms of homological algebra.

There are three main ingredients that come into play in order to give a precise formulation and proof of our main statement: 1) we extend a classical result of F. Adams from 1956 regarding the “cobar construction” as an algebraic model for the based loop space of a simply connected space, 2) we make use of the homotopical symmetry of the chain approximations to the diagonal map on a space, and 3) we apply a duality theory for algebraic structures known as Koszul duality. This is joint work with Mahmoud Zeinalian and Felix Wierstra.

Joint UGA-GT Topology Seminar at GT: Branched covers bounding rational homology balls

Series
Geometry Topology Seminar
Time
Monday, January 13, 2020 - 16:00 for 1 hour (actually 50 minutes)
Location
Boyd
Speaker
JungHwan ParkGeorgia Tech

Prime-power-fold cyclic branched covers along smoothly slice knots all bound rational homology balls. This phenomenon, however, does not characterize slice knots: In this talk, we give examples of non-slice knots that have the above property. This is joint work with Aceto, Meier, A. Miller, M. Miller, and Stipsicz.

Joint UGA-GT Topology Seminar at UGA: Knot Floer homology and cosmetic surgeries

Series
Geometry Topology Seminar
Time
Monday, January 13, 2020 - 14:30 for 1 hour (actually 50 minutes)
Location
Boyd
Speaker
Jonathan HanselmanPrinceton University

The cosmetic surgery conjecture states that no two different Dehn surgeries on a given knot produce the same oriented 3-manifold (such a pair of surgeries is called purely cosmetic). For knots in S^3, I will describe how knot Floer homology provides a strong obstruction to the existence of purely cosmetic surgeries. For many knots, including all alternating knots with genus not equal to two as well as all but 337 of the first 1.7 million knots, this is enough to confirm the conjecture. For the remaining knots, all but finitely many surgery slopes are obstructed, so checking the conjecture for a given knot reduces to distinguishing finitely many pairs of manifolds. Using a computer search, the conjecture has been verified for all prime knots with up to 16 crossings, as well as for arbitrary connected sums of such knots. These results significantly improve on earlier work of Ni and Wu, who also used Heegaard Floer homology to obstruct purely cosmetic surgeries. The improvement comes from using the full graded Heegaard Floer invariant, which is facilitated by a recent recasting of knot Floer homology as a collection of immersed curves in the punctured torus.

Localization in Khovanov homology

Series
Geometry Topology Seminar
Time
Monday, January 6, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
Skile 006
Speaker
Melissa ZhangUGA

When a topological object admits a group action, we expect that our invariants reflect this symmetry in their structure. This talk will explore how link symmetries are reflected in three generations of related invariants: the Jones polynomial; its categorification, Khovanov homology; and the youngest invariant in the family, the Khovanov stable homotopy type, introduced by Lipshitz and Sarkar. In joint work with Matthew Stoffregen, we use Lawson-Lipshitz-Sarkar's construction of the Lipshitz-Sarkar Khovanov homotopy type to produce localization theorems and Smith-type inequalities for the Khovanov homology of periodic links.

Classifying contact structures on hyperbolic 3-manifolds

Series
Geometry Topology Seminar
Time
Monday, December 9, 2019 - 14:30 for 1 hour (actually 50 minutes)
Location
Skiles 202
Speaker
James ConwayUC, Berkeley

Please Note: Note time and place of seminar

Two of the most basic questions in contact topology are which manifolds admit tight contact structures, and on those that do, can we classify such structures. In dimension 3, these questions have been answered for large classes of manifolds, but with a notable absence of hyperbolic manifolds. In this talk, we will see a new classification of contact structures on an family of hyperbolic 3-manifolds arising from Dehn surgery on the figure-eight knot, and see how it suggests some structural results about tight contact structures. This is joint work with Hyunki Min.

Residual Torsion-Free Nilpotence and Two-Bridge Knot Groups

Series
Geometry Topology Seminar
Time
Monday, December 2, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Speaker
Jonathan JohnsonThe University of Texas at Austin

I will discuss how a graph theoretic construction used by Hirasawa and Murasugi can be used to show that the commutator subgroup of the knot group of a two-bridge knot is a union of an ascending chain of parafree groups. Using a theorem of Baumslag, this implies that the commutator subgroup of a two-bridge knot group is residually torsion-free nilpotent which has applications to the anti-symmetry of ribbon concordance and the bi-orderability of two-bridge knots. In 1973, E. J. Mayland gave a conference talk in which he announced this result. Notes on this talk can be found online. However, this result has never been published, and there is evidence, in later papers, that a proper proof might have eluded Mayland.

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