Speakers

Titles and Abstracts (main program)

Shaoyun Bai

Arnold conjecture and integral counts of pseudo-holomorphic curves (2 talks)

Abstract: 

Since the early 80s, all fundamental progress towards the Arnold conjecture witnesses important technical advances which have wide applicability to various problems in symplectic topology. In the recent resolution of the Arnold conjecture over the integers with Xu, we systematically developed the so-called normally complex polynomial perturbation method, which allows us to extract integer-valued counts of pseudo-holomorphic curves in a broad setting. This contrasts with the belief that the orbifold nature of moduli spaces would force us to introduce denominators. In my two lectures, I will talk about bits of such a perturbation scheme and indicate how to regularize moduli spaces from Hamiltonian Floer theory so that we can obtain a "smooth" flow category, to which this method can be applied to define Floer theory over the integers.

Erkao Bao

Contact homology through semi-global Kuranishi structures

Abstract: 

The semi-global Kuranishi structure is a simpler form of Kuranishi structure. It uses one interior chart and several boundary charts. When the moduli space is compact, only one interior chart is needed. In this talk, we construct a semi-global Kuranishi structure for the moduli spaces of J-holomorphic curves used in defining contact homology. This part is based on joint work with Ko Honda. If time allows, we will discuss a further simplification of Kuranishi structures when moduli spaces are cleanly cut out, which generalizes the Hutchings-Taubes obstruction bundle gluing techniques. This second part is based on joint work with Ke Zhu.

Fabio Gironella

Tight non-fillable contact structures on spheres

Abstract: 

One of the foundational results in 3-dimensional contact topology is the fact that the 3-sphere admits a unique tight contact structure, which is moreover symplectically fillable. The higher dimensional situation is characterized by much more diversity, with plenty of tight contact structures on the sphere which are exotic, i.e. not contactomorphic to the standard one (even in the same almost contact class). The fact that these exotic contact structures are tight is in fact a consequence of the fact that they are symplectically fillable. In this talk I will present a joint work with Jonathan Bowden, Agustin Moreno and Zhengyi Zhou, in which we contribute to this zoo of exotic creatures by giving the first examples of tight and non-fillable contact structures on spheres of odd dimension at least 5.

Amanda Hirschi

Global Kuranishi charts for moduli spaces of stable maps

Abstract:

I will describe the construction of a global Kuranishi chart for moduli spaces of stable pseudoholomorphic maps of arbitrary genus and discuss the proof of a product formula for symplectic GW invariants. If time permits, I will sketch a comparison between our invariants and the pseudocycle invariants of Ruan and Tian. The first part is based on joint work with Mohan Swaminathan.

Michael Hutchings

An example of obstruction bundle gluing

Abstract:

Suppose that one wants to compute curve counting invariants by choosing a generic almost complex structure, in a situation in which full transversality cannot be obtained, but without abstract perturbations. In this case one needs to determine how to count certain non-transverse objects. We explain how to do this in a specific situation that arises in defining equivariant contact homology for dynamically convex contact forms in three dimensions, in joint work with Jo Nelson. The proof that this count has the desired properties uses obstruction bundle gluing techniques based on earlier joint work with Taubes.

Eleny Ionel

Counting real curves in 3-folds

Abstract:

There are several ways of counting holomorphic curves in Calabi-Yau 3-folds. Counting them as maps gives rise to Gromov-Witten invariants, which tend to overcount multiple covers. One can instead consider images of such maps (possibly with multiplicity), regarded as integral currents. This allowed us to prove a structure theorem for the GW invariants of symplectic 6-manifolds and the Gopakumar-Vafa conjecture. The latter states that the GW invariants of CY 3-folds are obtained from another set of invariants called ``BPS states" which have better properties: integrality and finiteness. The integrality statement was proved in joint work with Thomas Parker and the finiteness in joint work with Aleksander Doan and Thomas Walpuski. 

In this talk I will discuss joint work with Penka Georgieva proving a related structure theorem for Real Gromov-Witten invariants of real symplectic 3-folds (with an anti-symplectic involution). As a consequence, we prove that the "real BPS states" satisfy integrality, finiteness and a parity property. 

Dusa McDuff

From polyfolds to fundamental classes (3 talks)

Abstract:

The objective in these lectures will be using the Hofer-Wysocki-Zehnder definition of polyfolds to construct the virtual fundamental class in Gromov-Witten settings. Scale calculus and polyfolds were introduced by Hofer-Wysocki-Zehnder to give a firm analytic footing to the construction of the virtual fundamental class (VFC) for Gromov-Witten moduli spaces. The lectures will describe a joint project with Katrin Wehrheim that aims to give an explicit  construction for the VFC which should make it easier to use.

Lecture 1: Scale calculus and M-polyfolds

This will be an elementary introduction to sc-calculus.

Lecture 2: Outline of the construction

Lecture 3: More details

I will also explain how to use these ideas to construct a Kuranishi atlas from a polyfold Fredholm section.

Agustin Moreno

A landscape of contact manifolds via rational SFT

Abstract:

In this talk, I will introduce the formalism of BL-infinity-algebras, as a means to formulate the genus zero specialization of SFT. Time permitting, I will discuss examples and applications to contact topology. 

Joint work with Zhengyi Zhou, whose talk will be a continuation of this one.

Jo Nelson

Nonequivariant, S^1-equivariant, and cylindrical contact homology

Abstract:

While waiting in line for the Pergamon at the Lange Nacht der Museen,  I realized it would make more sense to give additional context and motivation for the obstruction bundle gluing example as described by Hutchings in his talk.  The gluing configuration involving a branched cover of a trivial cylinder arises in our joint work on providing foundations for nonequivariant and equivariant flavors of cylindrical contact homology for dynamically convex contact forms in three dimensions, which does not rely on abstract perturbations.  I'll describe our construction of a nonequivariant contact homology chain complex and sketch how we use it to recover cylindrical contact homology and establish invariance of this theory. 

John Pardon

Derived moduli spaces of pseudo-holomorphic curves (2 talks)

Abstract:

My lectures will sketch the construction of a canonical derived (log) smooth manifold structure on moduli spaces of pseudo-holomorphic curves.  Derived smooth manifolds form an infinity-category, which may be obtained from the (ordinary) category of smooth manifolds by freely adjoining finite infty-limits, modulo transverse limits.  The derived smooth structure on moduli spaces of pseudo-holomorphic curves comes from a (quite tautological) moduli functor on derived smooth manifolds (the main result is thus that this functor is representable).  Log smooth manifolds (essentially defined by Melrose, and recently developed further by Parker and Joyce) are used to capture in precisely what sense moduli spaces of pseudo-holomorphic curves are "smooth" near nodal curves.

Mohan Swaminathan

Counting embedded curves in Calabi-Yau 3-folds using bifurcation analysis

I will report on joint work with Shaoyun Bai where we study bifurcations of embedded holomorphic curves in Calabi-Yau 3-folds in order to define an integer-valued invariant analogous to Taubes' Gromov invariant. We are able to understand bifurcations completely in some cases and this leads to the definition of invariants which are conjecturally equivalent to Gopakumar-Vafa's BPS invariants. Rather than relying on global virtual techniques, our approach instead builds on a recent generalization (due to Wendl) of generic transversality results in holomorphic curve theory.

Thomas Walpuski

The Gopakumar-Vafa finiteness conjecture (a glimpse of geometric measure theory for symplectic geometers)

Abstract:

The purpose of this talk is to illustrate an application of the powerful machinery of geometric measure theory to a conjecture in Gromov-Witten theory arising from physics.  Gromov-Witten invariants tend to be quite complicated (or "have a rich internal structure" if one wants to be more generous). This is true especially if X is a symplectic Calabi-Yau 3-fold (that is: 6-dimensional with vanishing first Chern class). In 1998, using arguments from M-theory, Gopakumar and Vafa argued that there are integer BPS invariants of symplectic Calabi-Yau 3-folds. Unfortunately, they did not give a direct mathematical definition of their BPS invariants, but they predicted that they are related to the Gromov-Witten invariants by a transformation of the generating series. The Gopakumar-Vafa conjecture asserts that if one defines the BPS invariants indirectly through this procedure, then they satisfy an integrality and a (genus) finiteness condition.

The integrality conjecture has been resolved by Ionel and Parker. A key innovation of their proof is the introduction of the cluster formalism: an ingenious device to side-step questions regarding multiple covers and super-rigidity. Their argument could not resolve the finiteness conjecture, however. The reason for this is that it relies on Gromov’s compactness theorem for pseudo-holomorphic maps, which requires an a priori genus bound. It turns out, however, that Gromov’s compactness theorem can (and should!) be replaced with the work of Federer-Flemming, Allard, and De Lellis-Spadaro-Spolaor. This upgrade of Ionel and Parker’s cluster formalism proves both the integrality and finiteness conjecture.

This talk is based on joint work with Eleny Ionel and Aleksander Doan.

Yuan Yao

Cascades and Morse-Bott ECH

Embedded contact homology (ECH) is a Floer theory defined for contact 3-manifolds (whose Reeb flows are nondegenerate) that counts J-holomorphic curves, I will explain how to compute embedded contact homology in certain (degenerate) Morse-Bott settings by counting cascades. The key to this construction is an ECH index inequality for cascades and a gluing theorem that glues cascades into J-holomorphic curves. I will explain in some detail how to use the obstruction bundle gluing setup of Hutchings-Taubes, along with some further analytic estimates, to prove this gluing theorem, which may be of independent interest beyond ECH.

Zhengyi Zhou

RSFT functors for strong cobordisms and finite algebraic torsion

I will explain the functoriality of the hierarchy functors from rational SFT in strong cobordisms. In particular, I will show that the concave boundary of a strong cobordism has finite algebraic torsion if the convex boundary does. Then I will explain another SFT functoriality for exact cobordisms motived from the connecting map in the tautological long exact sequence of symplectic cohomology, which will produce many algebraically overtwisted contact manifolds (including all overtwisted manifolds) using cobordisms from contact +1 surgeries.  Finally, with the above two ideas combined, contact manifolds with algebraic torsion k and algebraic planar torsion k for any k in all dimensions greater than 3 can be constructed using spinal open books,  which confirms a conjecture of Latschev and Wendl. This is partially based on joint work with Agustin Moreno.