Seminars and Colloquia by Series

Ribbon disks for the square knot

Series
Geometry Topology Seminar
Time
Monday, April 1, 2024 - 16:30 for 1 hour (actually 50 minutes)
Location
Georgia Tech
Speaker
Alex ZupanUniversity of Nebraska - Lincoln

A knot K in S^3 is (smoothly) slice if K is the boundary of a properly embedded disk D in B^4, and K is ribbon if this disk can be realized without any local maxima with respect to the radial Morse function on B^4. In dimension three, a knot K with nice topology – that is, a fibered knot – bounds a unique fiber surface up to isotopy. Thus, it is natural to wonder whether this sort of simplicity could extend to the set of ribbon disks for K, arguably the simplest class of surfaces bounded by a knot in B^4. Surprisingly, we demonstrate that the square knot, one of the two non-trivial ribbon knots with the lowest crossing number, bounds infinitely many distinct ribbon disks up to isotopy. This is joint work with Jeffrey Meier.

A Staircase Proof for Contact Non-Squeezing

Series
Geometry Topology Seminar
Time
Monday, April 1, 2024 - 15:00 for 1 hour (actually 50 minutes)
Location
Georgia Tech
Speaker
Lisa TraynorBryn Mawr College

Gromov's non-squeezing theorem established symplectic rigidity and is widely regarded as one of the most important theorems in symplectic geometry. In contrast, in the contact setting, a standard ball of any radius can be contact embedded into an arbitrarily small neighborhood of a point. Despite this flexibility, Eliashberg, Kim, and Polterovich discovered instances of contact rigidity for pre-quantized balls in $\mathbb R^{2n} \times S^1$ under a more restrictive notion of contact squeezing. In particular, in 2006 they applied holomorphic techniques to show that for any {\it integer} $R \geq 1$, there does not exist a contact squeezing of the pre-quantized ball of capacity $R$ into itself; this result was reproved by Sandon in 2011 as an application of the contact homology groups she defined using the generating family technique. Around 2016, Chiu applied the theory of microlocal sheaves to obtain the stronger result that squeezing is impossible for all $R \geq 1$. Very recently, Fraser, Sandon, and Zhang, gave an alternate proof of Chiu’s nonsqueezing result by developing an equivariant version of Sandon’s generating family contact homology groups. I will explain another proof of Chiu’s nonsqueezing, one that uses a persistence module viewpoint to extract new obstructions from the contact homology groups as defined by Sandon in 2011. This is joint work in progress with Maia Fraser.

Monopole Floer spectra of Seifert spaces

Series
Geometry Topology Seminar
Time
Monday, March 4, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Matt StoffregenMSU

We'll give a short description of what exactly monopole Floer spectra are, and then explain how to calculate them for AR plumbings, a class of 3-manifolds including Seifert spaces.  This is joint work with Irving Dai and Hirofumi Sasahira.

Contact surgeries and symplectic fillability

Series
Geometry Topology Seminar
Time
Friday, February 23, 2024 - 10:30 for 1.5 hours (actually 80 minutes)
Location
Skiles 249
Speaker
Bülent TosunIAS and U. Alabama

Please Note: Note unusual date and length for the seminar!

It is well known that all contact 3-manifolds can be obtained from the standard contact structure on the 3-sphere by contact surgery on a Legendrian link. Hence, an interesting and much studied question asks what properties (e.g. tightness, fillability, vanishing or non-vanishing of various Floer or symplectic homology classes) of contact structures are preserved under various types of contact surgeries. The case for the negative contact surgeries is fairly well understood. The case of positive contact surgeries much more subtle. In this talk, extending an earlier work of the speaker with Conway and Etnyre, I will discuss some new results about symplectic fillability of positive contact surgeries, and in particular we will provide a necessary and sufficient condition for contact (positive) integer surgery along a Legendrian knot to yield a fillable contact manifold. When specialized to knots in the three sphere with its standard tight structure, this result can be rather efficient to find many examples of fillable surgeries along with various obstructions and surprising topological applications. This will report on joint work with T. Mark.

Corks for exotic diffeomorphisms

Series
Geometry Topology Seminar
Time
Monday, February 19, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Terrin WarrenUGA

In dimension 4, there exist simply connected manifolds which are homeomorphic but not diffeomorphic; the difference between the distinct smooth structures can be localized using corks. Similarly, there exist diffeomorphisms of simply connected 4-manifolds which are topologically but not smoothly isotopic to the identity. In this talk, I will discuss some preliminary results towards an analogous localization of this phenomena using corks for diffeomorphisms. This project is joint work with Slava Krushkal, Anubhav Mukherjee, and Mark Powell.

New algebraic invariants of Legendrian links

Series
Geometry Topology Seminar
Time
Monday, February 12, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Lenhard NgDuke

For the past 25 years, a key player in contact topology has been the Floer-theoretic invariant called Legendrian contact homology. I'll discuss a package of new invariants for Legendrian knots and links that builds on Legendrian contact homology and is derived from rational symplectic field theory. This includes a Poisson bracket on Legendrian contact homology and a symplectic structure on augmentation varieties. Time permitting, I'll also describe an unexpected connection to cluster theory for a family of Legendrian links associated to positive braids. Parts of this are joint work in progress with Roger Casals, Honghao Gao, Linhui Shen, and Daping Weng.

Projective Rigidity of Circle Packings

Series
Geometry Topology Seminar
Time
Monday, February 5, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Mike WolfGeorgia Tech

We prove that the space of circle packings consistent with a given triangulation on a surface of genus at least two is projectively rigid, so that a packing on a complex projective surface is not deformable within that complex projective structure.  More broadly, we show that the space of circle packings is a (smooth)  submanifold within the space of complex projective structures on that surface.

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