Research Seminar: Symplectic Geometry (Winter Semester 2021/22)

For a holomorphic curve in symplectic manifold with Lagrangian boundary conditions, it is not always guaranteed that the curve factors through a branched cover of a simple curve. This is much unlike the case for closed holomorphic curves. Lazzarini, and later Zehmisch, showed that in the case of holomorphic discs in dimension at least 4, one can perturb J so that such a disc does factor through a branched cover of a simple disk. We will talk about these results, and make some observations about the case for higher genus holomorphic curves with Lagrangian boundary conditions.

Every Legendrian knot leaves a traces in the 4-dimensional symplectic world. In this talk we will investigate whether a 4-dimensional tracker (with the necessary mathematical education) can determine the 3-dimensional creature that left the trace. This is based on joint work with Roger Casals and John Etnyre.

The topic of the talk will be Floer theories on exact symplectic orbifolds with smooth contact boundary. More precisely, I will first describe the construction, which only uses classical transversality techniques, of a symplectic cohomology group on such symplectic orbifolds. Then, I will give a geometrical application to the smooth setting, namely the existence, in any odd dimension at least 5, of a pair of contact manifolds with no exact symplectic (smooth) cobordisms in either direction. This is joint work with Zhengyi Zhou.

In the mid-1990's Taubes discovered a deep relationship between the Seiberg-Witten invariants on symplectic 4-manifolds and a counting invariant for embedded holomorphic curves, called the Gromov invariant. It is related to the Gromov-Witten invariants, but it packages its curve counts differently so that they are always integers, and its construction does not require any virtual techniques, but relies instead on bifurcation analysis. In this talk, I will say nothing about Seiberg-Witten theory, but will give a basic sketch of how the Gromov invariant is defined and how the underlying bifurcation theory works. The tricky part is to understand a pitchfork-type bifurcation that can emerge from doubly covered holomorphic tori, which forces the invariant to include terms counting these double covers.

It has long been conjectured that in symplectic Calabi-Yau 6-manifolds, there should exist an analogue of the Gromov invariant that gives integer-valued counts of embedded holomorphic curves, together with contributions from their multiple covers that are determined by bifurcation analysis. It is still an open question how to define such an invariant in general, but a recent preprint of Bai and Swaminathan explains how to do it in situations where the multiple covers making contributions have degree at most 2. This talk will give a general overview of the construction.

Continuing on the topic from the previous week, this talk will focus on one of the main technical tools that Bai and Swaminathan use in their construction: it is a new result that gives a necessary condition for a sequence of embedded holomorphic curves (with a converging sequence of almost complex structures) to converge to a nodal multiple cover. This can be used to rule out bifurcations involving nodal curves with ghost bubbles, so that the proof of invariance of curve counts does not require obstruction bundle gluing.

For a symplectic manifold we will construct a Floer homology for a symplectomorphism with coefficients in a Novikov ring. The resulting homology might be trivial, so Y.-J. Lee used the Floer complex to construct a more refined invariant. We will sketch the construction of this invariant and have an outlook at the bifurcation analysis that is needed to prove the invariance under Hamiltonian isotopies.

Computing Floer type invariants is typically very difficult due to the use of generic data. In my talk I will present how to use Morse-Bott techniques, Action estimates and Symmetry to make explicit computations possible in certain situations.

As symplectic geometers, we see that we naturally gravitate towards (or sometimes it is necessary to consider) situations where we want to retain both symmetry and transversality (and even multiple covering relation of holomorphic curves), two generally conflicting requirements. Understanding the example below could be one such situation. In counting holomorphic curves in Calabi-Yau 3-folds, these Gromov-Witten invariants can be expressed in terms of mysterious integral invariants through a beautiful relation by physicists Gopakumar and Vafa. Although this relation has been recently proved mathematically, it might still be curious and illuminating to see how physicists arrive at such a relation through physical principles and considerations. This talk is an attempt to explain this in a reasonable way, while some suspension of mathematical disbelief might be required at places.

In 2011, Richard Siefring defined a homotopy invariant intersection number for punctured pseudoholomorphic curves. This definition involved an asymptotic contribution called “the asymptotic intersection index” to the usual intersection number. The study of this index leads to a stratification of a moduli space pseudoholomorphic curves which helps in characterizing subsets with positive intersection index. The focus of this talk will be to discuss these in detail.

About 20 years ago, Dominic Joyce proposed an invariant for Calabi-Yau 3-folds based on a weighted count of special Lagrangian homology spheres. In the first of two talks we will explain what is special about these Lagrangians, give some examples, and indicate why a (weighted) count of such homology 3-spheres might be a reasonable invariant to consider. Furthermore, we will discuss reasons for the total number of special Lagrangians to vary along generic 1-parameter families of (almost) Calabi-Yau structures.

This is a continuation of the previous talk. The invariant of counting special Lagrangian rational homology 3-spheres in an almost Calabi-Yau 3-fold will be interesting if it is stable under deformations of almost Calabi-Yau structures or at least changes in a predictable way in these deformations. Joyce proposed a conjectural weighted counting invariant which might have these properties based on two particular bifurcation phenomenas, one is special Lagrangians with transverse intersections and another is special Lagrangian with conical singularity modeled on a cone over Clifford tori. We will first discuss results about conically singular special Lagrangians and their desingularizations. Later we will talk about the above two bifurcations in detail to justify Joyce’s proposal.

Take a closed semi-positive symplectic manifold, remove a small ellipsoid, and consider its symplectic completion. We explain that the singed count of punctured simple pseudo-holomorphic spheres with negative end on a suitable cover of the short Reeb orbit is well-defined for generic almost complex structure in the index zero case. The count does not depend on the choice of the generic almost complex structure. Using a neck-stretching argument and a higher-dimensional analog of Siefring’s intersection theory we argue that if the ellipsoid we remove is skinny enough then the count does not depend on the ellipsoid as well.
(Reference: §3 in arXiv:1906.02394)