Northern California Symplectic Geometry
Seminar, 2023-2024
Berkeley - Davis - Santa Cruz - Stanford
The Northern California Symplectic Geometry Seminar usually meets on the first Monday of each month. Established in 1992,
the Andreas Floer Memorial Lecture usually takes place in the Fall, during the first meeting of the seminar.
2:30pm-3pm, room 736 Evans Kai Cieliebak (Augsburg University),
Symplectic geometry of Anosov flows Abstract: Every Anosov flow on a closed oriented three-manifold gives rise to a four-dimensional
Liouville domain, whose Liouville homotopy class depends only on the homotopy class of the
Anosov flow. The goal of this talk is to explain this construction and discuss geometric properties
of the resulting Liouville domains. This is joint work with Oleg Lazarev, Thomas Massoni, and
Agustin Moreno.
4pm-5pm, room 736 Evans Mohammed Abouzaid (Stanford)
Commutative rings from Fukaya categories
Abstract:
Subotic was the first to implement the construction of a Floer-theoretic mirror to the
tensor product of coherent sheaves on algebraic varieties, via Lagrangian correspondences. His
thesis was concerned with the case of the 2-torus, but it is clear that it can be implemented more
generally for manifolds admitting smooth torus fibrations. In this lecture, I will discuss ongoing
joint work with Nate Bottman as well as with Yunpeng Niu on extending this construction to
the first cases in which non-trivial singularities can appear, including that of K3 surfaces. The
construction has intriguing potential consequences for mirror symmetry with coefficients in the
sphere spectrum.
2:30pm-3:30pm, room 736 Evans Andreas Floer Memorial Lecture Roger Casals (UC Davis),
A program to classify Lagrangian fillings of Legendrian links Abstract: This talk will present recent advances on the study of embedded exact Lagrangians in the standard Darboux 4-ball. We will discuss a three step strategy to classify Hamiltonian isotopy classes of Lagrangian fillings of Legendrian links. The main results for the two first steps, existence and surjectivity, have now been established for a wide class of Legendrians. I will discuss the statements and the geometric insights of their proofs in detail. The techniques combine a range of new ideas, including weaves, understanding topological polygons in surfaces, and the study of infinitesimal deformations of quivers with potential. We will motivate these arguments with examples and build the ideas from the ground up. At the end, there will be comments on the general case and the third step, injectivity, which closely relates to the study of Lagrangian skeleta for Weinstein 4-manifolds.
4pm-5pm, room 748 Evans Thomas Massoni (Princeton),
Taut foliations through a contact lens Abstract: In the late '90s, Eliashberg and Thurston established a remarkable
connection between foliations and contact structures in dimension
three: any co-oriented, aspherical foliation on a closed, oriented
3-manifold can be approximated by positive and negative contact
structures. Additionally, when the foliation is taut, its contact
approximations are (universally) tight.
In this talk, I will present a converse result concerning the
construction of taut foliations from suitable pairs of contact
structures. I will also describe a comprehensive dictionary between
the languages of foliations and of (pairs of) contact structures.
Although taut foliations are usually considered rigid objects, this
contact viewpoint reveals some degree of flexibility. As an
application, I will show that taut foliations survive after performing
large slope surgeries along transverse knots.
2:30pm-3:30pm, room 380D Pazit Haim-Kislev (Tel-Aviv University)
On the existence of symplectic barriers Abstract: In his seminal 2001 paper, Biran introduced the concept of Lagrangian Barriers, a symplectic rigidity phenomenon coming from obligatory intersections with Lagrangian submanifolds which don't come from mere topology.
In this joint work with Richard Hind and Yaron Ostrover, we present what appears to be the first illustration of Symplectic Barriers, a form of symplectic rigidity stemming from necessary intersections of symplectic embeddings with symplectic submanifolds (and in particular not Lagrangian). In our work, we also tackle a question by Sackel–Song–Varolgunes–Zhu and provide bounds on the capacity of the ball after removing a codimension 2 hyperplane with a prescribed Kähler angle.
4pm-5pm, room 383N Richard Hind (University of Notre Dame)
Isotopies and squeezing of monotone Lagrangian tori Abstract: Distinct Hamiltonian isotopy classes of Lagrangian tori in \(\mathbb{CP}^2\) can be associated to Markov triples. With two exceptions, each of these tori are symplectomorphic to exactly three Hamiltonian isotopy classes of tori in the ball (the affine part of \(\mathbb{CP}^2\)). We investigate quantitative invariants, which can distinguish the tori corresponding to at least one sequence of Markov triples. A similar analysis for \(S^2 \times S^2\) produces symplectomorphic tori which are not Hamiltonian diffeomorphic. This is joint work with Grigory Mikhalkin and Felix Schlenk.
2:30pm-3:30pm, room 736 Evans Rohil Prasad (UC Berkeley),
On the dense existence of compact invariant sets Abstract: This is joint work in progress with Dan Cristofaro-Gardiner. We explore the topological
dynamics of Reeb flows beyond periodic orbits and find the following rather general
phenomenon. For any Reeb flow for a torsion contact structure on a closed 3-manifold, any
point is arbitrarily close to a proper compact invariant subset of the flow. Such a statement
is false if the invariant subset is required to be a periodic orbit. Stronger results can also be
proved that parallel theorems of Le Calvez-Yoccoz, Franks, and Salazar for homeomorphisms of
the 2-sphere. In fact, we can also extend their results to Hamiltonian diffeomorphisms of closed
surfaces of any genus.
4pm-5pm, room 736 Evans Mohan Swaminathan (Stanford)
Constructing smoothings of stable maps Abstract: The moduli space of closed holomorphic curves in a closed symplectic manifold
can be compactified using stable maps. However, even in the nicest of situations (e.g., degree
d curves of genus g in a complex projective space, with d ¿¿ g), counting dimensions shows
that most stable maps which have ghost components are not “smoothable”, i.e., they can never
appear as the limit of a sequence of non-singular holomorphic curves. It is therefore natural to
ask which stable maps are smoothable (with the aim of obtaining a compact moduli space which
is smaller than the full space of stable maps). In this talk, I will describe recent work (joint
with Fatemeh Rezaee) which provides a partial answer to this question, in all genera, when the
target is a smooth projective variety. We do this via a gluing construction, with the key new
input being a class of explicit model solutions which dictate how to smooth a stable map near
its ghost components.
2:30pm-3:30pm, room 380W Sheel Ganatra (University of Southern California)
Arclike Lagrangians in Liouville sectors Abstract: Sectorial descent, established in earlier work with Pardon-Shende, gives a local-to-global formula computing the wrapped Fukaya category of a Weinstein manifold from a sectorial cover. If one has a specific fixed global Lagrangian in mind that isn't contained in a single subsector, the resulting formula is only implicit, as it begins by appealing to the generation of this object by "local" Lagrangians. In this talk I will introduce and study the class of (global) "arclike" Lagrangian submanifolds with respect to a sectorial covering, which are allowed to run through subsector boundaries but in a controlled fashion. For arclike Lagrangians, a more explicit local-to-global analysis is possible. Based on works in progress with Hanlon-Hicks-Pomerleano-Sheridan and Hanlon-Hicks-Ward.
4pm-5pm, room 383N Daniel Pomerleano (UMass Boston)
The quantum connection on a monotone symplectic manifold Abstract: The small quantum connection on a monotone symplectic manifold M is one of the simplest objects in enumerative geometry. Nevertheless, the poles of the connection have a very rich structure. After reviewing this background, I will outline a proof that, under suitable assumptions, the quantum connection of M is of unramified exponential type. This is joint work (partially in progress) with Paul Seidel.
2pm-3pm, room 736 Evans(Note special time) Roman Krutovski (UCLA),
Heegaard Floer symplectic cohomology and generalized Viterbo’s isomorphism
theorem Abstract: In recent years several groups of authors introduced various invariants that are based
on Lagrangian Floer homology of a symmetric product of a symplectic manifold. In this talk, I
will introduce Heegaard Floer symplectic cohomology (HFSH), an invariant of a Liouville domain
M which mimics symplectic cohomology of the k-th symmetric product of M. This invariant can
also be regarded as a deformation of the k-th symmetric version of symplectic cohomology,
obtained by counting curves of higher genus. I will also introduce a multiloop Morse complex
and show that for cotangent bundles this complex computes HFSH. This is a joint work with
Tianyu Yuan.
4pm-5pm, room 736 Evans Shaoyun Bai (Columbia),
Gauged linear sigma model and infinitude of Hamiltonian periodic orbits Abstract: Take an irrational rotation of the two-sphere; it only has the north and south poles
as its periodic points. However, Franks proved that for any area-preserving diffeomorphism of
the two-sphere, if it has more than two fixed points, then it must have infinitely many periodic
points. I will discuss a generalization with Guangbo Xu of this result to all compact toric
manifolds in the form of a “Betti number or infinity” dichotomy. The Floer theory package
and mirror symmetry considerations from gauged linear sigma models, also known as symplectic
vortices, play a surprising role.
2:30pm-3:30pm, room 380D Yoel Groman (Hebrew University),
Relative symplectic cohomology and quantitative deformation theory Abstract:
Consider a Liouville domain D embedded in a closed symplectic manifold M. To D one can associate two types of Floer theoretic invariants: intrinsic ones like the wrapped Fukaya category which depend on D only, and relative ones which involve both D and M. It is often the case that the intrinsic invariant is amenable to computation. On the other hand, the relative invariants are important, at least in SYZ mirror symmetry, as one can reconstruct the global Floer theory by a local to global principle. Thus it is a fundamental question if the relative invariants can be understood as a deformation of the intrinsic invariant. It turns out the question need to be approached quantitatively. By shrinking the Liouville domain, the answer is often positive. This circle of ideas is at the heart of a program joint with Mohammed Abouzaid and Umut Varolgunes for a general approach to homological mirror symmetry. I will discuss a work in progress on the application of this circle of ideas to the reconstruction problem in mirror symmetry via relative symplectic cohomology.
4pm-5pm, room 383N Felix Schenk (Université de Neuchâtel),
Symplectic almost squeezings Abstract:
Around 2000, Biran introduced the notion of polarization of a symplectic manifold,
and showed that the associated Lagrangian skeleta exhibit remarkable rigidity properties.
He proved in particular that their complements may have small Gromov width.
In this work, we introduce a version of polarization on affine symplectic manifolds.
These polarizations are more flexible than those of closed symplectic manifolds,
which provides a wider range of applications.
For instance, given an affine symplectic manifold V and any closed symplectic 4-manifold M
of larger volume, there exists an isotropic CW complex in V such that its complement symplectically embeds into M.
Specifically, after removing from a 4-ball of any radius finitely Lagrangian planes,
one finds an embedding into the standard cylinder,
extending a result by Sackel-Song-Varolgunes-Zhu and Brendel. This is work joint with Emmanuel Opshtein.
Organizers: Mohammed Abouzaid (Stanford), Roger Casals (Davis), Yasha Eliashberg (Stanford), Dmitry Fuchs (Davis), Viktor Ginzburg (Santa Cruz), Michael Hutchings (Berkeley), Eleny Ionel (Stanford), Richard Montgomery (Santa Cruz), Vivek Shende (Berkeley), Laura Starkston (Davis), Katrin Wehrheim (Berkeley), Alan Weinstein (Berkeley/Stanford).
For further information, please contact
Eleny Ionel