Seminars and Colloquia by Series

Joint GT-UGA Seminar at UGA - Homological knot invariants and the unknotting number

Series
Geometry Topology Seminar
Time
Monday, August 27, 2018 - 14:30 for 1 hour (actually 50 minutes)
Location
Boyd 328
Speaker
Akram Alishahi – Columbia University
Unknotting number is one of the simplest, yet mysterious, knot invariants. For example, it is not known whether it is additive under connected sum or not. In this talk, we will construct lower bounds for the unknotting number using two homological knot invariants: knot Floer homology, and (variants of) Khovanov homology. Unlike most lower bounds for the unknotting number, these invariants are not lower bound for the slice genus and they only vanish for the unknot. Parallely, we will discuss connections between knot Floer homology and (variants of) Khovanov homology. One main conjecture relating knot Floer homology and Khovanov homology is that there is a spectral sequence from Khovanov homology to knot Floer homology. If time permits, we will sketch an algebraically defined knot invariant, for which there is a spectral sequence from Khovanov homology converging to it. The construction is inspired by counting holomorphic discs, so we expect it to recover the knot Floer homology. This talk is based on joint works with Eaman Eftekhary and Nathan Dowlin.

Geometric realizations of cyclic actions on surfaces

Series
Geometry Topology Seminar
Time
Friday, July 20, 2018 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Kashyap Rajeevsarathy – IISER Bhopal
Let Mod(Sg) denote the mapping class group of the closed orientable surface Sg of genus g ≥ 2. Given a finite subgroup H < Mod(Sg), let Fix(H) denote the set of fixed points induced by the action of H on the Teichmuller space Teich(Sg). In this talk, we give an explicit description of Fix(H), when H is cyclic, thereby providing a complete solution to the Modular Nielsen Realization Problem for this case. Among other applications of these realizations, we derive an intriguing correlation between finite order maps and the filling systems of surfaces. Finally, we will briefly discuss some examples of realizations of two-generator finite abelian actions.

Fillability of positive contact surgeries and Lagrangian disks

Series
Geometry Topology Seminar
Time
Wednesday, May 23, 2018 - 14:00 for 1.5 hours (actually 80 minutes)
Location
Skiles 006
Speaker
Bulent Tosun – University of Alabama

Please Note: This will be a 90 minute 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 are preserved under various types of contact surgeries. The case for the negative contact surgeries is fairly well understood. In this talk, we will discuss some new results about positive contact surgeries and in particular completely characterize when contact r surgery is symplectically/Stein fillable when r is in (0,1]. This is joint work with James Conway and John Etnyre.

The h-principle and totally convex immersions

Series
Geometry Topology Seminar
Time
Monday, April 30, 2018 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Michael Harrison – Lehigh University –
The h-principle is a powerful tool in differential topology which is used to study spaces of functionswith certain distinguished properties (immersions, submersions, k-mersions, embeddings, free maps, etc.). Iwill discuss some examples of the h-principle and give a neat proof of a special case of the Smale-HirschTheorem, using the "removal of singularities" h-principle technique due to Eliashberg and Gromov. Finally, I willdefine and discuss totally convex immersions and discuss some h-principle statements in this context.

Novikov Fundamental Group

Series
Geometry Topology Seminar
Time
Monday, April 23, 2018 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Hong Van Le – Institute of Mathematics CAS, Praha, Czech Republic –
Novikov homology was introduced by Novikov in the early 1980s motivated by problems in hydrodynamics. The Novikov inequalities in the Novikov homology theory give lower bounds for the number of critical points of a Morse closed 1-form on a compact differentiable manifold M. In the first part of my talk I shall survey the Novikov homology theory in finite dimensional setting and its further developments in infinite dimensional setting with applications in the theory of symplectic fixed points and Lagrangian intersection/embedding problems. In the second part of my talk I shall report on my recent joint work with Jean-Francois Barraud and Agnes Gadbled on construction of the Novikov fundamental group associated to a cohomology class of a closed 1-form on M and its application to obtaining new lower bounds for the number of critical points of a Morse 1-form.

Joint GT-UGA Seminar at GT - Augmentations and immersed exact Lagrangian fillings by Yu Pan

Series
Geometry Topology Seminar
Time
Monday, April 16, 2018 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Yu Pan – MIT
Augmentations and exact Lagrangian fillings are closely related. However, not all the augmentations of a Legendrian knot come from embedded exact Lagrangian fillings. In this talk, we show that all the augmentations come from possibly immersed exact Lagrangian fillings. In particular, let ∑ be an immersed exact Lagrangian filling of a Legendrian knot in $J^1(M)$ and suppose it can be lifted to an embedded Legendrian L in J^1(R \times M). For any augmentation of L, we associate an induced augmentation of the Legendrian knot, whose homotopy class only depends on the compactly supported Legendrian isotopy type of L and the homotopy class of its augmentation of L. This is a joint work with Dan Rutherford.

Joint GT-UGA Seminar at GT - Asymmetric L-space Knots by Ken Baker

Series
Geometry Topology Seminar
Time
Monday, April 16, 2018 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ken Baker – University of Miami
Based on the known examples, it had been conjectured that all L-space knots in S3 are strongly invertible. We show this conjecture is false by constructing large families of asymmetric hyperbolic knots in S3 that admit a non-trivial surgery to the double branched cover of an alternating link. The construction easily adapts to produce such knots in any lens space, including S1xS2. This is joint work with John Luecke.

Iterated planar contact manifolds

Series
Geometry Topology Seminar
Time
Monday, April 9, 2018 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Bahar Acu – Northwestern University

Planar contact manifolds have been intensively studied to understand several aspects of 3-dimensional contact geometry. In this talk, we define "iterated planar contact manifolds", a higher-dimensional analog of planar contact manifolds, by using topological tools such as "open book decompositions" and "Lefschetz fibrations”. We provide some history on existing low-dimensional results regarding Reeb dynamics, symplectic fillings/caps of contact manifolds and explain some generalization of those results to higher dimensions via iterated planar structure. This is partly based on joint work in progress with J. Etnyre and B. Ozbagci.

Truncated Heegaard Floer homology and concordance invariants

Series
Geometry Topology Seminar
Time
Monday, April 2, 2018 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Linh Truong – Columbia University
Heegaard Floer homology has proven to be a useful tool in the study of knot concordance. Ozsvath and Szabo first constructed the tau invariant using the hat version of Heegaard Floer homology and showed it provides a lower bound on the slice genus. Later, Hom and Wu constructed a concordance invariant using the plus version of Heegaard Floer homology; this provides an even better lower-bound on the slice genus. In this talk, I discuss a sequence of concordance invariants that are derived from the truncated version of Heegaard Floer homology. These truncated Floer concordance invariants generalize the Ozsvath-Szabo and Hom-Wu invariants.

Joint GT-UGA Seminar at UGA - On Uniqueness of End Sums and TBA

Series
Geometry Topology Seminar
Time
Monday, March 26, 2018 - 14:30 for 2.5 hours
Location
Room 304
Speaker
Bob Gompf and Sergei Gukov – UT Austin and Cal Tech
For oriented manifolds of dimension at least 4 that are simply connected at infinity, it is known that end summing (the noncompact analogue of boundary summing) is a uniquely defined operation. Calcut and Haggerty showed that more complicated fundamental group behavior at infinity can lead to nonuniqueness. We will examine how and when uniqueness fails. There are examples in various categories (homotopy, TOP, PL and DIFF) of nonuniqueness that cannot be detected in a weaker category. In contrast, we will present a group-theoretic condition that guarantees uniqueness. As an application, the monoid of smooth manifolds homeomorphic to R^4 acts on the set of smoothings of any noncompact 4-manifold. (This work is joint with Jack Calcut.)

Pages