Seminars and Colloquia by Series

Joint GT-UGA Seminar at GT - A contact Fukaya category

Series
Geometry Topology Seminar
Time
Monday, October 1, 2018 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Lenny Ng – Duke University
I'll describe a way to construct an A-infinity category associated to a contact manifold, analogous to a Fukaya category for a symplectic manifold. The objects of this category are Legendrian submanifolds equipped with augmentations. Currently we're focusing on standard contact R^3 but we're hopeful that we can extend this to other contact manifolds. I'll discuss some properties of this contact Fukaya category, including generation by unknots and a potential application to proving that ``augmentations = sheaves''. This is joint work in progress with Tobias Ekholm and Vivek Shende.

Link Concordance and Groups

Series
Geometry Topology Seminar
Time
Monday, September 24, 2018 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Miriam Kuzbary – Rice University
Since its introduction in 1966 by Fox and Milnor the knot concordance group has been an invaluable algebraic tool for examining the relationships between 3- and 4- dimensional spaces. Though knots generalize naturally to links, this group does not generalize in a natural way to a link concordance group. In this talk, I will present joint work with Matthew Hedden where we define a link concordance group based on the “knotification” construction of Peter Ozsvath and Zoltan Szabo. This group is compatible with Heegaard Floer theory and, in fact, much of the work on Heegaard Floer theory for links has implied a study of these objects. Moreover, we have constructed a generalization of Milnor’s group-theoretic higher order linking numbers in a novel context with implications for our link concordance group.

Non-isotopic embeddings of contact manifolds.

Series
Geometry Topology Seminar
Time
Monday, September 17, 2018 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
John Etnyre – Georgia Tech
The study of transverse knots in dimension 3 has been instrumental in the development of 3 dimensional contact ge- ometry. One natural generalization of transverse knots to higher dimensions is contact submanifolds. Embeddings of one contact manifold into another satisfies an h-principle for codimension greater than 2, so we will discuss the case of codimension 2 contact embeddings. We will give the first pair of non-isotopic contact embeddings in all dimensions (that are formally isotopic).

Rational cobordisms and integral homology (JungHwan Park)

Series
Geometry Topology Seminar
Time
Monday, September 10, 2018 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Rational cobordisms and integral homology – School of Mathematics Georgia Institute of Technology –
We show that for any connected sum of lens spaces L there exists a connected sum of lens spaces X such that X is rational homology cobordant to L and if Y is rational homology cobordant to X, then there is an injection from H_1(X; Z) to H_1(Y; Z). Moreover, as a connected sum of lens spaces, X is uniquely determined up to orientation preserving diffeomorphism. As an application, we show that the natural map from the Z/pZ homology cobordism group to the rational homology cobordism group has large cokernel, for each prime p. This is joint work with Paolo Aceto and Daniele Celoria.

Joint GT-UGA Seminar at UGA - Khovanov homology via immersed curves in the 4-punctured sphere

Series
Geometry Topology Seminar
Time
Monday, August 27, 2018 - 16:00 for 1 hour (actually 50 minutes)
Location
Boyd 328
Speaker
Artem Kotelskiy – Indiana University
We will describe a geometric interpretation of Khovanov homology as Lagrangian Floer homology of two immersed curves in the 4-punctured 2-dimensional sphere. The main ingredient is a construction which associates an immersed curve to a 4-ended tangle. This curve is a geometric way to represent Khovanov (or Bar-Natan) invariant for a tangle. We will show that for a rational tangle the curve coincides with the representation variety of the tangle complement. The construction is inspired by a result of [Hedden, Herald, Hogancamp, Kirk], which embeds 4-ended reduced Khovanov arc algebra (or, equivalently, Bar-Natan dotted cobordism algebra) into the Fukaya category of the 4-punctured sphere. The main tool we will use is a category of peculiar modules, introduced by Zibrowius, which is a model for the Fukaya category of a 2-sphere with 4 discs removed. This is joint work with Claudius Zibrowius and Liam Watson.

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.

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