Previous talks

September 2, 1 pm ET: Ipsita Datta (Institute for Advanced Study) - Lagrangian cobordisms, enriched knot diagrams, and algebraic invariants

Abstract: We introduce new invariants to the existence of Lagrangian cobordisms in R^4. These are obtained by studying holomorphic disks with corners on Lagrangian tangles, which are Lagrangian cobordisms with flat, immersed boundaries.

We develop appropriate sign conventions and results to characterize boundary points of 1-dimensional moduli spaces with boundaries on Lagrangian tangles. We then use these to define (SFT-like) algebraic structures that recover the previously described obstructions.

This talk is based on my thesis work under the supervision of Y. Eliashberg and on work in progress joint with J. Sabloff.

September 23, 1 pm ET: Bingyu Zhang (University of Southern Denmark) - Circle Action of Microlocal Kernels

Abstract: The sheaf theoretical proof of the contact non-squeezing theorem relies on a $\mathbb{Z}/\ell$-equivariant invariant constructed using microlocal kernels of balls. We call it the ($\mathbb{Z}/\ell$- equivariant-)Chiu-Tamarkin complex. Chiu uses a numerical way to approximate $S^1$ in his proof. Later, his idea is packed as some symplectic/contact capacities by me. However, it is interesting to know if we can construct an $S^1$ action directly and algebraically. In this talk, I will explain how we build an $S^1$ action using the simplest property of the microlocal kernel. Then we can define the $S^1$-equivariant Chiu-Tamarkin complex. If time permits, I will also explain some computational results.

September 30, 1 pm ET: Rohil Prasad (Princeton) - A strong closing lemma for ellipsoids

Abstract: In a recent preprint, Irie conjectured a "strong closing property" for the Reeb flow on the boundary of any ellipsoid. This conjecture asserts that a Reeb orbit can be created in any open set by a C^\infty-small compactly supported perturbation of the contact form. In this talk, I will explain a proof of this conjecture, which uses spectral invariants from contact homology and higher-dimensional holomorphic intersection theory. This is joint work with J. Chaidez, I. Datta, and S. Tanny.

October 7, 1 pm ET: Luya Wang (UC Berkeley) - A connected sum formula of embedded contact homology

Abstract: The contact connected sum is a well-understood operation for contact manifolds. I will focus on the 3-dimensional case where the contact connected sum can be seen as a Weinstein 1-handle attachment. I will discuss how pseudo-holomorphic curves in the symplectization behave under this operation, and as a result a connected sum formula of embedded contact homology.

October 14, 1 pm ET: Maxim Jeffs (Harvard) - Functoriality for Fukaya categories of very affine hypersurfaces

Abstract: A very affine hypersurface is the vanishing locus of a Laurent polynomial inside a complex torus; its complement is also a very affine hypersurface, in one of two subtly-different ways. The (partially) wrapped Fukaya categories of the hypersurface and its complement are closely related: Auroux sketched the definitions of several new acceleration and restriction functors between them. I'll explain how we can define these functors in terms of gluings of Liouville sectors and how this implies conjectures of Auroux about their mirror counterparts, building on work of Gammage-Shende. On the way, I'll explain how the different realizations of the complement lead to very different Fukaya categories, related by a non-geometric equivalence mediated by derived Knorrer periodicity. This is joint work with Benjamin Gammage.

October 21, 1 pm ET: Kai Hugtenburg (Edinburgh) - The cyclic open-closed map, u-connections and R-matrices

Abstract: This talk will review recent progress on obtaining Gromov-Witten invariants from the Fukaya category. A crucial ingredient is showing that the cyclic open-closed map, which maps the cyclic homology of the Fukaya category of X to its S1-equivariant quantum cohomology, respects connections. Along the way we will encounter R-matrices, which were used in the Givental-Teleman classification of semisimple cohomological field theories, and allow one to determine higher genus Gromov-Witten invariants from genus 0 invariants. I will then present some evidence that this approach might extend beyond the semisimple case. Time permitting, I will also explain work in progress on obtaining open Gromov-Witten invariants from the Fukaya category.

October 28, 1 pm ET: Yash Deshmukh (Columbia) - A homotopical description of Deligne-Mumford compactifications

Abstract: I will describe how the Deligne-Mumford compactifications of moduli spaces of curves (of all genera) arise from the moduli spaces of framed curves by homotopically trivializing certain circle actions in an appropriate sense. I will sketch how such a description is relevant to the problem of relating GW invariants (in all genera) with Fukaya categories. Finally, I will indicate how our result relates to other statements available in the literature. Time permitting, I will talk about a variation on this result which gives rise to a partial compactification of moduli spaces of curves which is relevant to the study of symplectic cohomology.

November 11, 1 pm ET: Yuan Yao (UC Berkeley) - Computing Embedded Contact Homology in the Morse-Bott Setting using Cascades

Abstract: I will first give an overview of ECH. Then I will describe how to compute ECH in the Morse-Bott setting a la Bourgeois. I will discuss some classes of examples where this approach works. Finally I will sketch the gluing results that allow us to compute ECH using cascades.

December 2, 1 pm ET: Lea Kenigsberg (Columbia) - Coproduct structures, a tale of two outputs

Abstract: I will tell the elusive story of coproduct structures in Floer theory and string topology, and explain why we care about them. I will then define a new coproduct structure on the symplectic cohomology of Liouville manifolds and compute it in an example to show that this structure is not trivial. This is based on my thesis work, in progress.

December 16, 1 pm ET: Debtanu Sen (University of Southern California) - The Z/p Gysin sequence in symplectic cohomology

Abstract: Symplectic cohomology is used as an invariant of a class of exact symplectic manifolds with boundary. There’s also an S^1-equivariant refinement of the theory that has been defined in literature. In our talk, we define Z/p-equivariant symplectic cohomology for any prime p and prove the existence of a Gysin type long exact sequence relating the S^1 and Z/p equivariant versions. We also use it to answer affirmatively a conjecture made by Seidel regarding the structure of localized S^1-equivariant symplectic cohomology.

January 28, 12 pm ET: Basak Gurel (University of Central Florida) - Topological entropy, barcodes and Floer theory

Abstract: Topological entropy is one of the fundamental invariants of a dynamical system, measuring its complexity. In this talk, we discuss a connection between the topological entropy of compactly supported Hamiltonian diffeomorphisms and Floer theory. We introduce a new invariant associated with the Floer complexes of the iterates of such a diffeomorphism, which we call barcode entropy. We show that barcode entropy is closely related to topological entropy and that these invariants are equal in dimension two. The talk is based on joint work with Erman Cineli and Viktor Ginzburg.

February 4, 12 pm ET: Sarah Blackwell (University of Georgia) - Triple Knot Grid Diagrams

Abstract: In this talk I will introduce a project I have been working on which uses trisections of 4-manfiolds to represent ``Lagrangian-like’’ surfaces in $\mathbb{CP}^2$ by ``triple knot grid diagrams.’’ Gay and Kirby defined a decomposition of (smooth, closed, connected, oriented) 4-manifolds called a trisection, and proved that every such 4-manifold admits this decomposition. Meier and Zupan showed that surfaces embedded in 4-manifolds inherit a trisection from the trisection of the 4-manifold. Their work includes a description of how to represent these surfaces with ``shadow diagrams.’’ In this project I consider specific shadow diagrams of surfaces in $\mathbb{CP}^2$ that naturally arise as grid diagrams on the central surface of the standard (genus one) trisection of $\mathbb{CP}^2$. The result is a process for encoding Lagrangian-like surfaces which appear to be combinatorial representations of Lagrangian surfaces in $\mathbb{CP}^2$. Surprisingly, triple knot grid diagrams representing Lagrangian-like surfaces are sparse; adding the extra necessary condition makes such diagrams hard to find. 

February 11, 12 pm ET: Dahye Cho (Stony Brook) - Symplectic Criteria on Stratified Uniruledness of Affine Varieties and Applications to the Minimal Model Program

Abstract: We develop criteria for affine varieties to admit uniruled subvarieties of certain dimensions. A projective variety defined over complex numbers is uniruled if for a generic point, there exists a rational curve of genus 0 passing through that point. An affine variety is uniruled if for a generic point of it, there exists a once-punctured rational curve of genus 0, in other words a complex line passing through that point. The measurements are from long exact sequences of versions of symplectic cohomology, which is a Hamiltonian Floer theory for some open symplectic manifolds including affine varieties. Symplectic cohomology is hard to compute, in general. However, certain vanishing and invariance properties of symplectic cohomology can be used to prove that our criteria for finding uniruled subvarieties hold in some cases. We provide applications of the criteria in birational geometry of log pairs in the direction of the Minimal Model Program.

February 18, 12 pm ET: Urs Frauenfelder (Augsburg) - The little sibling of Rabinowitz action functional

Abstract: Rabinowitz action functional is the Lagrange multiplier functional of the area functional to the constraint given by the mean value of a Hamiltonian. Its little sibling is the restriction of the area to the constraint. The two action functionals have the same critical points but in general different gradient flow lines. The motivation for studying the little sibling is that it not only has the same symmetry behaviour as the Rabinowitz action functional but additionally is Chas-Sullivan additive under concatenation of loops. In the talk I will explain that on a symplectization there is a one-to-one correspondence of gradient flow lines of the two functionals.

February 25, 12 pm ET: Aliakbar Daemi (WUSTL) - 3-manifold representations and the Atiyah--Floer conjecture (part 1)

Abstract: The space of all representations of the fundamental group of a Riemann surface S into a Lie group (e.g. SO(3) and SU(2)) determines a (possibly singular) symplectic manifold. One obtains (possibly singular and immersed) Lagrangians of this symplectic manifold by considering representations of the fundamental group of a 3-manifold with boundary S. The Atiyah--Floer conjecture states that Lagrangian Floer homology groups of such Lagrangians can be defined, and the resulting group can be expressed in terms of a 3-manifold invariant defined in the context of Yang--Mills gauge theory, called instanton Floer homology. In a series of two talks, we will review some of the main elements of the proof of a version of the Atiyah--Floer conjecture, where the 3-manifold Lagrangians are smooth and embedded. In this first part, we will focus on 3-manifold Lagrangians and their properties. We will also discuss the mixed equation, which is the key geometrical tool in relating the moduli of homomorphic curves to gauge theory.

March 4, 12 pm ET: Kenji Fukaya (SCGP) - 3-manifold representations and the Atiyah--Floer conjecture (part 2)

Abstract: This is a continuation of the talk by A. Daemi last week. We will discuss:

1) A formulation of Atiyah-Floer type conjecture as a homotopy equivalence of A infinity categories.

2) How bounding cochain (in Lagrangian Floer theory) and immersed Floer theory (Akaho-Joyce) is used.

3) Sketch of the proof.

March 11, 12 pm ET: Mandy Cheung (Harvard University) - Family Floer mirror and mirror symmetry for rank 2 cluster varieties

Abstract: The Gross-Hacking-Keel mirror is constructed in terms of scattering diagrams and theta functions. The ground of the construction is that scattering diagrams inherit the algebro-geometric analogue of the holomorphic disks counting. With Yu-shen Lin, we made use this idea and gave first non-trivial examples of family Floer mirror. Then with Sam Bardwell-Evans, Hansol Hong, and Yu-shen LIn, we construct a special Lagrangian fibration on the non-toric blowups of toric surfaces that contains nodal fibres, and prove that the fibres bounding Maslov 0 discs reproduce the scattering diagrams. As a consequence, we can then illustrate the mirror duality between the A and X cluster varieties.

March 25, 12 pm ET: James Pascaleff (UIUC) - Categories obtained from pants decompositions

Abstract:  The Fukaya categories of Riemann surfaces, and the singularity categories of their Hori-Vafa mirrors, may be obtained by gluing together several copies of the Fukaya category of the pair of pants. A natural generalization is to ask for a classification of all categories that can be so obtained. We will present a solution to this problem, and describe how categories in this class may be realized on the B-side in terms of normal crossings surfaces, and (at least conjecturally) on the A-side in terms of certain 4-manifolds built from Riemann surfaces. We may also characterize how the Fukaya categories of surfaces are distinguished within this class of categories. This is joint work with Nicolo Sibilla.

April 22, 12 pm ET: Surena Hozoori (Georgia Tech) - Contact and symplectic geometry of Anosov flows and their invariant volume forms

Abstract: Since their introduction in the early 1960s, Anosov flows have defined an important class of dynamics, thanks to their many interesting chaotic features and rigidity properties. Moreover, their topological aspects have been deeply explored, in particular in low dimensions, thanks to the use of foliation theory in their study. Although the connection of Anosov flows to contact and symplectic geometry was noted in the mid 1990s by Mitsumatsu and Eliashberg-Thurston, such interplay has been left mostly unexplored. I will present some recent results on the contact and symplectic geometric aspects of Anosov flows in dimension 3, including in the presence of an invariant volume form, which is known to have grave consequences for the dynamics of these flows. Time permitting, the interplay of Anosov flows with Reeb dynamics, Liouville geometry and surgery theory will be briefly discussed as well.

April 29, 12 pm ET: Yaniv Ganor (Technion - Israel Institute of Technology) - Big Fiber Theorems and Ideal-Valued Measures in Symplectic Topology

Abstract: In various areas of mathematics there exist "big fiber theorems", these are theorems of the following type: "For any map in a certain class, there exists a 'big' fiber", where the class of maps and the notion of size changes from case to case.

We will discuss three examples of such theorems, coming from combinatorics, topology and symplectic topology from a unified viewpoint provided by Gromov's notion of ideal-valued measures.

We adapt the latter notion to the realm of symplectic topology, using an enhancement of Varolgunes’ relative symplectic cohomology to include cohomology of pairs. This allows us to prove symplectic analogues for the first two theorems, yielding new symplectic rigidity results.

Necessary preliminaries will be explained.

The talk is based on a joint work with Adi Dickstein, Leonid Polterovich and Frol Zapolsky.

September 17, 12 pm ET: Caitlin Leverson (Bard College) - Lagrangian Realizations of Ribbon Cobordisms

Abstract: Similarly to how every smooth knot has a Legendrian representative (in fact, infinitely many different representatives), in this talk we will discuss why every ribbon cobordism has a Legendrian representative. Meaning, if $C$ is a ribbon cobordism in $[0,1]\times S^3$ from the link $K_0$ to $K_1$, then there are Legendrian realizations $\Lambda_0$ and $\Lambda_1$ of $K_0$ and $K_1$, respectively, such that $C$ may be isotoped to a decomposable Lagrangian cobordism from $\Lambda_0$ to $\Lambda_1$. We will also give examples of some interesting constructions of such decomposable Lagrangian cobordisms. This is joint work with John Etnyre.

September 24, 12 pm ET: Shaoyun Bai (Princeton University) - Rouquier dimension, quantitative intersection of skeleta and Orlov’s conjecture

Abstract: For a given triangulated category, its Rouquier dimension is defined to be the minimal generation time of all of its split generators shifted down by 1. The main objects of this talk are the Rouquier dimensions of derived wrapped Fukaya categories of Weinstein manifolds/sectors. In particular, I will explain two results among some others:

1. Given a Weinstein manifold of dimension 2n, if its wrapped Fukaya category admits a homological section, then any generic push-off of its skeleton by a compactly supported Hamiltonian diffeomorphism intersects its skeleton at least at n points.

2. Orlov’s conjecture which relates the Krull dimension of a variety to the Rouquier dimension of its derived category holds even for certain singular complex surface.

This is a joint work with Laurent Cote.

October 1, 12 pm ET: Shamuel Auyeung (Stony Brook University) - Local Lagrangian Floer Homology of Quasi-Minimally Degenerate Intersections

Abstract: Given two Lagrangian submanifolds, if they intersect cleanly along a submanifold, then, by the work of Pozniak, their local Lagrangian Floer homology is isomorphic to the singular homology of the submanifold. Moreover, as Seidel observed, if the Lagrangian intersection is decomposable into disjoint clean intersections, there is a local-to-global spectral sequence whose E_1 page depends only on the topology, rather than symplectic geometry, of the intersections. In this talk, we introduce a generalization of clean intersections that is inspired by the notion of minimal degeneracy defined by Kirwan. We will then extract analogues of the above results for such intersections.

October 8, 12 pm ET: Chi Hong Chow (CUHK) - Peterson/Lam-Shimozono's theorem via Seidel representations

Abstract: I will present a new proof of a theorem of Peterson and Lam-Shimozono which gives an explicit ring homomorphism from the based loop homology of a compact semi-simple Lie group to the quantum cohomology of any of its coadjoint orbits. This map is in general surjective after localization with explicit kernel and bijective when the coadjoint orbit has maximal dimension. The idea is to compute Savelyev's parametrized version of Seidel representations. Surprisingly, it turns out to be equal to the map of Peterson and Lam-Shimozono. I will also discuss some applications to the Hamiltonian groups of coadjoint orbits.

October 15, 12 pm ET: Alex Pieloch (Columbia) - Sections and Unirulings of families over the projective line

Abstract: We will discuss the existence of rational (multi)sections and unirulings for projective families f: X -> P^1 over the field of complex numbers with at most two singular fibres.  In particular, we will discuss two ingredients that are used to construct the above rational curves.  The first is local symplectic cohomology groups associated to compact subsets of convex symplectic domains.  The second is a degeneration to the normal cone argument that allows one to produce closed curves in X from open curves (which are produced using local symplectic cohomology) in the complement of X by a singular fibre.

October 22, 12 pm ET: Yuan Gao (University of Georgia) - The symplectic formal neighborhood

Abstract: For a stopped Liouville manifold coming from a Liouville sector, we construct a symplectic analogue of the formal neighborhood of the stop from the viewpoint of Fukaya categories. This construction can be performed in two ways: first, via certain variants of Floer theory for a Liouville sector; second, via a purely algebraic construction from the partially wrapped Fukaya category. Our main result shows that the two approaches are equivalent under a geometric properness condition, thereby providing a formula for understanding the new Floer-theoretic structures. Under mirror symmetry for pairs, this is mirror to the formal neighborhood of a divisor in an ambient projective variety. 

October 29, 12 pm ET: Mohamed El Alami (Stony Brook University) - SYZ for index 1 Fano hypersurfaces in projective space

Abstract: I will explain how to construct a monotone Lagrangian torus in the smooth index one Fano hypersurface in projective space, how to compute its associated super-potential, and I will discuss some of its implications on homological mirror symmetry.

November 5, 12 pm ET: Oliver Edtmair (UC Berkeley) - PFH spectral gaps and quantitative C^\infty closing lemmas

Abstract: Asaoka-Irie proved the C^\infty closing lemma for Hamiltonian diffeomorphisms of closed surfaces. In this talk, I will explain how to extend this result to area-preserving diffeomorphisms which are not necessarily Hamiltonian. Moreover, I will explain how the closing lemma can be made quantitative, i.e. how to obtain an upper bound on the periods of the new orbits in terms of the “size” of the perturbation. The main tool to prove these results are spectral numbers defined via periodic Floer homology. This is joint work with Michael Hutchings.

November 12, 12 pm ET: Mark McLean (Stony Brook University) - Complex cobordism, Hamiltonian loops and global Kuranishi charts

Abstract: Consider a smooth submersion from a symplectic manifold P to the complex line with symplectic fibers. Then we prove that the cohomology of P over the integers is additively isomorphic to the cohomology of the fiber times the base. More generally, we prove such an isomorphism holds with respect to any complex oriented cohomology theory, such as complex cobordism. These results are new even in the special case of smooth projective morphisms to the complex line. To prove our result we use Morava K theories, which are generalized cohomology theories approximating cohomology over the integers modulo each prime power and which admit virtual fundamental classes. Our proof also contains a new construction of a global Kuranishi chart for the moduli space of curves. This is joint work with Abouzaid and Smith.

November 19, 12 pm ET: Ralph Cohen (Stanford University) - Floer homotopy theory, new and old

Abstract: In 1995 the speaker, Jones, and Segal introduced the notion of "Floer homotopy theory". The proposal was to attach a (stable) homotopy type to the geometric data given in a version of Floer homology.  More to the point, the question was asked, "When is the Floer homology isomorphic to the (singular) homology of a naturally occurring spectrum  defined from the properties of the moduli spaces inherent in the Floer theory?". Years passed before this notion found some genuine applications to  symplectic geometry and low dimensional topology.  However in recent years several striking applications have been found, and the theory has been developed on a much deeper level. In this expository talk I will sketch both the interesting algebraic topology and symplectic topology involved in the theory and talk about  recent applications by Lipshitz-Sarkar, Manolescu, Abouzaid-Blumberg, and Hopkins-Lin-Shi-Xu.

Note from WHVSS: for more on this topic, you can apply for the MSRI Floer homotopy theory program in Fall 2022 (applications due December 1, 2021).

November 26: No WHVSS seminar today, but Symplectic Zoominar is being held

Abstract at the Symplectic Zoominar webpage: http://www.math.tau.ac.il/~sarabt/zoominar/

December 3, 12 pm ET: Floer Homotopy Panel (not recorded)

Abstract: As background for this panel discussion, we refer the audience to 

• the WHVSS November 12 talk by Mark McLean (Stony Brook University), 

• the WHVSS November 19 talk by Ralph Cohen (Stanford University), and 

• the Symplectic Zoominar November 26 talk by Mohammed Abouzaid (Columbia University).

Panelists: Andrew Blumberg (Columbia), Kristen Hendricks (Rutgers), Xin Jin (Boston College), Robert Lipshitz (University of Oregon), Paul Seidel (MIT), Hiro Lee Tanaka (Texas State University), Nathalie Wahl (University of Copenhagen)

Please submit your questions for the panelists here: https://forms.gle/BGEdZkLZWbdYsxZCA. For more on this topic, you can apply for the MSRI Floer homotopy theory program in Fall 2022 (applications due December 1, 2021).

December 10, 12 pm ET: Olguta Buse (IUPUI) - Isotopy and Homotopic Stability Chambers in Blow-up Ruled Symplectic Surfaces

Abstract: Spaces of holomorphic curves in symplectic ruled surfaces are rich, and well structured, and have been exploited, starting with Gromov, to study the corresponding spaces of symplectomorphism groups. In this talk, we extent inflation and J-holomorphic techniques (and other homotopic tools) to spaces of non-rational, non-minimal symplectic ruled surfaces. We will explain new results regarding $\pi_*$ of the symplectomorphism groups and discuss future results. The results are in collaboration with Jun Li.

December 17, 12 pm ET: Gage Martin (Boston College) - Annular links, double branched covers, and annular Khovanov homology

Abstract: Given a link in the thickened annulus, you can construct an associated link in a closed 3-manifold through a double branched cover construction. In this talk we will see that perspective on annular links can be applied to show annular Khovanov homology detects certain braid closures. Unfortunately, this perspective does not capture all information about annular links. We will see a shortcoming of this perspective inspired by the wrapping conjecture of Hoste-Przytycki. This is partially joint work with Fraser Binns.

June 4, 12 pm ET: Andy Manion (University of Southern California), Higher representations and cornered Heegaard Floer homology

Abstract: I will survey recent work with Raphael Rouquier from the perspective of partially wrapped Fukaya categories of symmetric products of surfaces and related algebras known as bordered strands algebras. I will try to motivate how higher representation theory enters the story when studying how to recover Fukaya categories of symmetric powers of a glued-together surface from the pieces as in cornered Heegaard Floer homology, and to show how formulas describing such gluings are related to a new type of tensor product operation in higher representation theory.

June 11, 11 am ET (note unusual time!): Yoel Groman (Hebrew University), Torsion of non-exact embeddings of Liouville domains

Abstract: Consider a symplectic embedding of a Liouville domain D inside a closed or geometrically bounded symplectic manifold M. We introduce a numerical invariant \tau(D,M) called embedding torsion based on a version  over the Novikv ring of the symplectic cohomology of D relative to M. In the limit as D is shrunk to the  skeleton, \tau(D,M) serves as a quantitative measure of how much the Floer theory of D relative to M is deformed from the Floer theory of D relative to its completion. We show that \tau(D,M) depends only on the skeleton. Thus $\tau(D,M)$ can be seen as a closed string shadow of the notion of obstructedness in Lagrangian intersection Floer theory, but which applies to singular objects. We also show that for isotopies of symplectic embeddings of D the torsion \tau is concave as a function of the flux parameters. Based on work in progress. 

June 18, 12 pm ET: Eleny Ionel (Stanford University), Counting embedded 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 the Gromov-Witten invariants. In general, these are not integer counts due to the presence of multiple covers with symmetries. But one can consider instead images of such maps (possibly with multiplicity), regarded either as subsets or as integral currents. Generically these images are smoothly embedded curves.

In earlier joint work with Thomas Parker we constructed an integer count of embedded pseudo-holomorphic curves in symplectic Calabi-Yau 3-folds, and related it to the Gromov-Witten invariants. In recent work with Aleksander Doan and Thomas Walpuski we extended these arguments to also prove that the former invariants satisfy a finiteness property. The new ingredients are compactness (and regularity) results for pseudo-holomorphic cycles/currents without an a priori genus bound, instead of the Gromov compactness for pseudo-holomorphic maps. In this talk I will outline some of the key ideas involved in these constructions.

June 25, 12 pm ET: Marco Castronovo (Rutgers University), Fukaya category of Grassmannians: bootstrap and mutation

Abstract: I will report on the first steps of a program whose aim is to prove that (a suitable version of) the Fukaya category of complex Grassmannians is split-generated by Lagrangian tori.

July 2, 12 pm ET: Aleksander Doan (Columbia), Equivariant transversality and curve counting

Abstract: This talk is motivated by the problem of counting embedded pseudo-holomorphic curves in symplectic manifolds. Typically, naive counting embedded curves does not lead to a symplectic invariant as they can degenerate to singular or multiply covered curves. I will discuss a result, obtained in collaboration with Thomas Walpuski, which excludes such degenerations in certain situations. The proof of this result combines Wendl's recent theorem on equivariant transversality for multiply covered curves with methods of geometric measure theory. Time permitting, I will talk about an idea of defining invariants of symplectic six-manifolds by counting embedded pseudo-holomorphic curves and solutions to gauge-theoretic equations.

July 9, 12 pm ET: Kai Cieliebak (Universität Augsburg), Sullivan's relation in Rabinowitz Floer homology and loop space homology

Abstract: This talk is about ongoing joint work with Nancy Hingston and Alexandru Oancea. I will explain that Rabinowitz Floer homology is an infinitesimal bialgebra in the sense of Joni-Rota and Aguiar. In particular, it satisfies Sullivan’s relation. Then I will discuss why Sullivan’s relation fails in general for loop space homology.

July 16, 12 pm ET: Mina Aganagic (UC Berkeley), Knot homologies from mirror symmetry

Abstract: Khovanov showed, more than 20 years ago, that the Jones polynomial emerges as an Euler characteristic of a homology theory. The knot categorification problem is to find a uniform description of this theory, for all gauge groups, which originates from physics, or from geometry. I recently discovered two solutions to the problem, related by a version of two dimensional homological mirror symmetry. The result is a new geometric formulation of Khovanov homology, which generalizes to all groups. The focus of this talk is the second, symplectic geometry based approach. 

July 23, 12 pm ET: Sobhan Seyfaddini (Jussieu), The algebraic structure of groups of area-preserving homeomorphisms

Abstract: I will review recent joint work with Dan Cristofaro-Gardiner, Vincent Humilière, Cheuk Yu Mak and Ivan Smith constructing a new family of spectral invariants associated to certain Lagrangian links in compact and connected surfaces of any genus. We show that our invariants recover the Calabi invariant of Hamiltonians in their limit. As applications, we resolve several open questions from topological surface dynamics and continuous symplectic topology: 

1. We show that the group of Hamiltonian homeomorphisms of any compact surface with (possibly empty) boundary is not simple

2. We extend the Calabi homomorphism to the group of Hameomorphisms constructed by Oh-Müller.

3. We construct an infinite dimensional family of quasimorphisms on the group of area and orientation preserving homeomorphisms of the two-sphere.

July 30, 12 pm ET: Dusa McDuff (Columbia University), Nicki Magill (Cornell University), and Morgan Weiler (Rice University), Recursive staircase patterns in Hirzebruch surfaces

Abstract: We will explain the main results of our study of the ellipsoidal embedding capacity function for the family of Hirzebruch surfaces $H_b$ obtained by blowing up a fixed projective plane $\CP^2(1)$ by a ball of size $b<1$, where $1$ is the area of the line. We have found that there is an open dense subset BLOCK \subset [0,1), consisting of points $b$ for which the function $c_{H_b}$ is finitely determined.  On the other hand, at each endpoint of each of the infinitely many intervals in BLOCK the function $c_{H_b}$ has a staircase, in other words $c_{H_b}$ is affected by infinitely many obstructions.

Here $c_{H_b}(z)$ measures the maximum size of an ellipse with aspect ratio $z$ that embeds symplectically into  H_b$.  We will try to get the main ideas of the arguments across, without going into too many details.

The proofs use a variety of methods, including almost toric fibrations.

January 29, 12 pm ET: Shira Tanny (Tel-Aviv University) - The Poisson bracket invariant: elementary and hard approaches.

In 2006 Entov and Polterovich proved that functions forming a partition of unity with displaceable supports cannot commute with respect to the Poisson bracket. In 2012 Polterovich conjectured a quantitative version of this theorem. I will discuss three interconnected topics: a solution of this conjecture in dimension two (with Lev Buhovsky and Alexander Logunov), a link between this problem and Grothendieck's theorem from functional analysis (with Efim Gluskin), and new results related to the Floer-theoretical approach to this conjecture (with Yaniv Ganor).

February 5, 12 pm ET: Penka Georgieva (Jussieu Institute of Mathematics) - Klein TQFT and real Gromov-Witten invariants

In this talk I will explain how the Real Gromov-Witten theory of local 3-folds with base a Real curve gives rise to an extension of a 2d Klein TQFT. The latter theory is furthermore semi-simple which allows for complete computation from the knowledge of a few basic elements which can be calculated explicitly. As a consequence of the explicit expressions we find in the Calabi-Yau case we obtain the expected GV formula and relation to SO/Sp Chern-Simons theory. 

February 12, 12 pm ET: Yu-Shen Lin (Boston University) - SYZ Mirror Symmetry on P^2 and enumerative geometry

In this talk I will explain the existence of SYZ fibrations on P^2 relative to smooth elliptic curves and the dual fibration on the corresponding mirrors. As an application, the special Lagrangian fibrations leads to the tropical/holomorphic correspondence of holomorphic discs. In particular, one can achieve the folklore conjecture as the equivalence between certain open Gromov-Witten invariants and the log Gromov-Witten invariants with maximal tangency in algebraic geometry for P^2 . Part of the talk will be based on the joint work with T. Collins, A. Jacob and S.-C. Lau, T.-J. Lee. 

February 19, 12 pm ET: Honghao Gao (Michigan State University) - Augmentations, fillings, and clusters

The augmentation variety of a Legendrian link consists of the functor of points in the Chekanov-Eliashberg differential graded algebra. In the study of low dimensional contact and symplectic topology, one important subject is to understand exact Lagrangian fillings of a given Legendrian link. For a positive braid link, we introduce a cluster K2 structure on its augmentation variety. Using the perspective of Ekholm-Honda-Kalman theory, we prove that admissible exact Lagrangian fillings, a subset of decomposable ones, induce cluster seeds in the cluster K2 augmentation variety. These cluster seeds can be used as invariants for exact Lagrangian fillings. We provide an algorithm to compute these cluster seeds. We also use the cluster Donaldson-Thomas transformation to produce infinitely many Lagrangian fillings for Legendrian links of infinite types. This is joint work with L. Shen and D. Weng.

February 26, 12 pm ET: Simon Allais  (ENS Lyon) - Periodic points of Hamiltonian diffeomorphisms and generating functions

Ginzburg and Gürel recently showed that a hamiltonian diffeomorphism of CP^d a hyperbolic periodic point have infinitely many periodic points whereas fixed points of a pseudo-rotation are isolated as an invariant set. In 2019, Shelukhin proved a homology version of the Hofer-Zehnder conjecture in a large class of symplectic manifolds M that includes CP^d: a Hamiltonian diffeomorphism with more homologically visible fixed points than the dimension of the homology of M has infinitely many periodic points. These results rely on the quantum structure of the Floer homology.

In this talk, I will explain how the study of sublevel sets of generating functions can replace the use of J-holomorphic curves and Floer theory in the study of periodic points of CP^d, based on ideas of Givental and Théret in the 90s.

February 26, 3 pm ET: Yael Karshon (Toronto University) - Non-linear Maslov index on lens spaces

Let L be a lens space with its standard contact structure. We use generating functions to construct a "non-linear Maslov index", which associates an integer to any contact isotopy of  L  that starts at the identity, and whose properties allow us to prove rigidity properties of  L  as a contact manifold.

This is joint work with Gustavo Granja, Milena Pabiniak, and Sheila (Margherita) Sandon, and it follows earlier work of Givental and Theret that applied to real and complex projective spaces.

March 5, 12 pm ET: Arnav Tripathy (Harvard University) - K3 metrics and disk counting

I'll discuss an approach to producing explicit Ricci-flat metrics on K3 surfaces before indicating the utility thereof for a symplectic geometry problem: explicitly computing counts of open disks in K3 surfaces. This is all joint work with M. Zimet.

March 12, 12 pm ET: Joint talk - Orsola Capovilla-Searle (Duke) and Angela Wu (UCL) - Weinstein handlebodies for complements of smoothed toric divisors

In this talk, we are concerned with two important classes of symplectic manifolds: toric manifolds, which are equipped with an effective Hamiltonian action of the torus, and Weinstein manifolds, which come with handle decompositions compatible with their symplectic structures. We prove that the complements of a class of smoothed toric divisors which we call "centered" support a Weinstein structure, and we describe an algorithm which produces the specific Weinstein handlebody diagram such complements. This is based on joint work with Acu, Gadbled, Marinković, Murphy, and Starkston.

March 19, 12 pm ET: Noémie Legout (Uppsala University) - A-infinity category of Lagrangian cobordisms

In this talk I will define a Floer complex associated to a pair of Lagrangian cobordisms in the symplectization of a contact manifold, by counting pseudo-holomorphic disks in the SFT setting. I will then explain how to construct a product on this complex and A-infinity maps in general, and show the existence of a continuation element, leading to the definition of an A-infinity category via localization.

March 26, 12 pm ET: Francisco Presas (ICMAT) - The homotopy type of the contactomorphism group of a contact $3$-fold

We show that for any tight $3$-fold, there is an almost injective $h$—principle between the space of contactomorphisms of a tight $3$-fold relative to a point and the space of diffeomorphisms of the $3$-fold relative to a point. The injection fails at $\pi_0$-level and the failure can be computed in formal terms. We will show how surprising this looks from a historical perspective. The surjectivity is completely controlled by the obvious necessary condition: the family of diffeomorphisms should admit a formal contactomorphism structure. The previous result has plenty of corollaries:

- the connected component of any Legendrian knot, in the standard $3$ sphere, has the homotopy type of $U(2)\times K(G,1), where $G$ is the fundamental group of the associated long smooth knot (computed by Hatcher and Budney in general).

- Similar statements are proven for transverse knots in the sphere and non-parametrized Legendrian and transverse knots.

- The computation of any contactomorphism group of a tight $3$-fold as long as the diffeomorphism group is given (that is usually the case given Hatcher’s research in the last 30 years).

We will sketch a proof of the main result that is very much based on a multi-parametric convex surfaces theory. In turn, this theory is based on the contractibility of the space of compactly supported contactomorphisms of the ball: a result announced by Eliashberg around 30 years ago. If time allows, we will provide an alternative to Eliashberg's approach: this new proof of Eliashberg's theorem is work in progress. 

This is joint work with Edu Fernández and Xabi Martínez.

April 2, 12 pm ET: Melissa Liu (Columbia University) - Topological Recursion and Crepant Transformation Conjecture

The Crepant Transformation Conjecture (CTC), first proposed by Yongbin Ruan and later refined/generalized by others, relates Gromov-Witten (GW) invariants of K-equivalent smooth varieties or smooth Deligne-Mumford stacks. We will outline a proof of all-genus open and closed CTC for symplectic toric Calabi-Yau 3-orbifolds based on joint work with Bohan Fang, Song Yu, and Zhengyu Zong. Our proof relies on the Remodeling Conjecture (proposed by Bouchard-Klemm-Marino-Pasquetti and proved in full generality by Fang, Zong and the speaker) relating open and closed GW invariants of a symplectic toric Calabi-Yau 3-orbifold to invariants of its mirror curve defined by Chekhov-Eynard-Orantin Topological Recursion.

April 9, 12 pm ET: Susan Tolman (Illinois) - Beyond semitoric

A compact four dimensional completely integrable system $f : M \to R^2$ is semitoric if it has only non-degenerate singularities, without hyperbolic blocks, and one of the components of $f$ generates a circle action.  Semitoric systems have been extensively studied and have many nice properties: for example, the preimages $f^{-1}(x)$  are all connected.  Unfortunately, although there are many interesting examples of semitoric systems, the class has some limitation.  For example, there are blowups of $S^2 \times S^2$ with Hamiltonian circle actions which cannot be extended to semitoric systems.  We expand the class of semitoric systems by allowing certain degenerate singularities, which we call ephemeral singularities.  We prove that the preimage $f^{-1}(x)$ is still connected for this larger class.  We hope that this class will be large enough to include not only all compact four manifolds with Hamiltonian circle actions, but more generally all complexity one spaces. Based on joint work with D. Sepe.

April 16, 12 pm ET: Tobias Ekholm (Uppsala University) - Skein module curve counts and recursion

Counting holomorphic curves in a Calabi-Yau 3-fold X with Maslov zero Lagrangian boundary condition L by their boundaries in the framed skein module of L gives a deformation invariant quantity. We review this construction briefly and compare the resulting invariants to real Gromov-Witten invariants when there is an involution. We then study the toric brane in complex 3-space and knot conormals. Here we show that holomorphic curves on the Legendrian that is the ideal boundary of the Lagrangian stores the information of the curve count in terms of a skein valued recursion relation, which is comparatively easy to compute. If time permits we will also describe analogous recursion relations for conormals of knots. The talk reports on joint works with Penka Geoergieva, Lenhard Ng, and Vivek Shende.

April 23, 12pm ET: Bulent Tosun (Alabama) - On embedding problems for 3-manifolds in 4-space

The problem of embedding one manifold into another has a long, rich history, and proved to be tremendously important for development of geometric topology since the 1950s. In this talk I will focus on the 3-manifold embedding problem in 4-space. Given a closed, orientable 3-manifold Y, it is of great interest but often a difficult problem to determine whether Y may be smoothly embedded in R^4. This is the case even for integer homology spheres, and restricting to special classes such as Seifert manifolds, the problem is open in general, with positive answers for some such manifolds and negative answers in other cases. On the other hand, under additional geometric considerations coming from symplectic geometry (such as hypersurfaces of contact type) and complex geometry (such as the boundaries of holomorphically or rationally or polynomially convex Stein domains), the problems become tractable and in certain cases a uniform answer is possible. For example, recent work shows for Brieskorn homology spheres: no such 3-manifold admits an embedding as a hypersurface of contact type in R^4. This implies restrictions on the topology of rationally and polynomially convex domains in C^2. In this talk I will provide further context and motivations for these results, and give some details of the proof. This is joint work with Tom Mark. 

April 30, 12pm ET: Heather Lee (University of Washington) - Global homological mirror symmetry for genus two curves

As a complex manifold, a genus-2 curve is a hypersurface in its Jacobian variety which is an abelian surface.  Cannizzo's thesis proved a homological mirror symmetry (HMS) result for genus two curves, with the mirror being a Landau-Ginzburg model (Y, W), where Y is a locally toric Calabi-Yau 3-fold and $W: Y\to \mathbb C$ is a symplectic fibration with a singular fiber above 0.  The critical locus is a “banana” configuration of three 2-spheres with the same symplectic area; this gives rise to a 1-parameter family of symplectic structures which is mirror to a 1-parameter family of complex structures on the genus-2 curve. We extend this construction to prove a global HMS result to cover the 3-dimensional moduli space of complex structures on the genus-2 curve.  Some ingredients involved include the construction of more general symplectic structures where the areas of the three 2-spheres may vary independently, as well as the computation of the monodromy of a fiber around the singularity of the fibration.  This is a joint work with Haniya Azam, Catherine Cannizzo, and Chiu-Chu Melissa Liu. 

May 7, 12pm ET:  Olga Plamenevskaya (Stony Brook University) - Links of complex surface singularities: symplectic vs algebraic fillings

A link of an isolated complex surface singularity is a 3-manifold obtained by intersecting the surface with a small sphere centered at the singular point. The link carries a canonical contact structure, given by the complex tangencies. Milnor fibers of possible smoothings of the singular point give Stein fillings for this contact structure; the resolution of the singularity also gives a filling. An important question is whether all Stein fillings come from this algebraic construction: this is known to hold in some simple cases (eg for lens spaces). We will show that even in the "next simplest" case, for many rational singularities, there is a plethora of Stein fillings that do not arise from Milnor fibers. The ingredients of this story include (1) T.de Jong-D.van Straten's description of Milnor fibers of certain rational surface singularities in terms of deformations of associated singular plane curves, (2) its symplectic analog that allows to describe Stein fillings of the links via certain arrangements of curves, and (3) comparison of the arrangements of curves in both settings.

Joint work with L. Starkston, arXiv:2006.06631. 

Note: Laura Starkston will give a talk on the same topic in the Symplectic Zoominar, Friday May 7, 9:15-10:45 ET. The two talks will be independent but complementary to one another. Each will include the necessary background and an outline of the work, but  the speakers will focus on different aspects in more detail. 

Informal symplectic tea and gather (first slot), special URL below

We will host an informal symplectic tea on gather.town from 12 pm ET - 1 pm ET / 9 am PT - 10 am PT, at the following URL:  https://gather.town/app/abBT3vhSL9vD0zbL/virtualsymplectictea. For those of you who haven't used it, gather.town simulates the experience of a large group gathering by allowing you to "walk up to people" in its interface and video chat with the people you are close to. 

We are looking forward to seeing you there!

September 18: Eric Zaslow (Northwestern): Legendrian Weaves (second slot, 9/18/20 noon PT)

I will describe work with Roger Casals.  We show how planar diagrams called N-graphs encode Legendrian surfaces which cover the plane N-to-1.  These N-graphs can be used to express Reidemeister moves, surgeries, and connect sums; to describe a Markov move à la braids; to construct large classes of examples of any genus; to define moduli spaces which can be used to distinguish surfaces up to Legendrian isotopy; to discuss cluster charts and mutations; to construct exact Lagrangian fillings; and to define a planar algebra.

September 25: Jonny Evans (Lancaster): A Lagrangian Klein bottle you can't squeeze (second slot, 9/25/20 noon PT)

Given a nonorientable Lagrangian surface L in a symplectic 4-manifold, how far can you deform the symplectic form before there is no Lagrangian surface isotopic to L? I will discuss this problem in general and explain the solution in a particular case.

October 2: WHVSS seminar + SYZ virtual workshop joint event (special meeting times and Zoom URLs below)

Part of Floer-theoretic and algebro-geometric aspects of SYZ mirror symmetry, September 29-October 2nd, 2020

Talk 1: Nick Sheridan (Edinburgh): The Gamma and SYZ conjectures (8:00-9:15am PDT / 11:00-12:15pm EDT/ 16:00-17:15 GMT+1 / 17:00-18:15 GMT+2; NOTE DIFFERENT TALK TIME)

Abstract: In the first part of the talk I will give some background on the Gamma Conjecture, which says that mirror symmetry does *not* respect integral cycles: rather, the integral cycles on a complex manifold correspond to integral cycles on the symplectic mirror, multiplied by a certain transcendental characteristic class called the Gamma class. In the second part of the talk I will explain a new geometric approach to the Gamma Conjecture, which is based on the SYZ viewpoint on mirror symmetry. We find that the appearance of ζ(k) in the asymptotics of period integrals arises from the codimension-k singular locus of the SYZ fibration. This is based on joint work with Abouzaid, Ganatra, and Iritani.

Zoom Link: https://harvard.zoom.us/j/91642158286?pwd=U0hJNmY3aFVEejJyaEdabWZXMG01UT09  (NOTE DIFFERENT ZOOM LINK)

Meeting ID: 916 4215 8286, Password: syz2020

Talk 2: Zack Sylvan (Columbia): Mirrors to points near SYZ singularities (9:30-10:45am PDT / 12:30-1:45pm EDT / 17:30-18:45 GMT+1 / 18:30-19:45 GMT+2; NOTE DIFFERENT TALK TIME)

Abstract: Family Floer theory is a tool introduced by K. Fukaya and extended by M. Abouzaid for studying non-Hamiltonian perturbations of a Lagrangian. In recent years, it has been one of the main approaches for relating SYZ and homological mirror symmetry. In the first half, I’ll discuss some of the philosophy of this approach, and I’ll explain the main difficulty in applying it to singlular SYZ fibrations. In the second half, I’ll explain a resolution to this difficulty for the spaces ∏xi = 1 + ∑yj ⊂ ℂ^m×(ℂ*)^n, which appear as neighborhoods of SYZ singularities. In particular, I’ll explain how to cook up a torus-like closed Lagrangian brane for every point of the mirror. This is joint work in progress with M. Abouzaid.

Zoom Link: https://harvard.zoom.us/j/91642158286?pwd=U0hJNmY3aFVEejJyaEdabWZXMG01UT09   (NOTE DIFFERENT ZOOM LINK)

Meeting ID: 916 4215 8286, Password: syz2020

DISCUSSION/TEA PERIOD (11:00-11:30am PDT / 2:00-2:30pm EDT/ 19:00-19:30 GMT+1 / 20:00-20:30 GMT+2; NOTE DIFFERENT TIME)

Discussion/tea will take place on gather.town at the following link: https://gather.town/app/abBT3vhSL9vD0zbL/virtualsymplectictea

October 9: Chris Woodward (Rutgers): Towards a Lagrangian minimal model program (second slot, 10/09/20 noon PT)

This was originally scheduled as an informal talk over summer which was meant to cover the idea that the Fukaya category of a symplectic manifold can be understood via a version of the minimal model program; this is related to  Lagrangian mean curvature flow and the Thomas-Yau conjecture but there are many interesting aspects that do not require a Lagrangian mcf with surgery. In particular I will discuss a recent result with Venugopalan and Xu on Fukaya categories of blow-ups, and one ingredient in this result which is a version of an unpublished theorem of Abouzaid-Fukaya-Oh-Ohta-Ono that summands in the Fukaya category corresponding to different values of the disk potential map to orthogonal summands in quantum cohomology under the open closed map. 

October 16: Marcy Robertson (Melbourne): Expansions, completions and automorphisms of welded tangled foams (second slot, 10/16/20 noon PT)

Welded tangles are knotted surfaces in R^4. Bar-Natan and Dancso described a class of welded tangles which have “foamed vertices” where one allows surfaces to merge and split. The resulting welded tangled foams carry an algebraic structure, similar to the planar algebras of Jones, called a circuit algebra. In joint work with Dancso and Halacheva we provide a one-to-one correspondence between circuit algebras and a form of rigid tensor category called ``wheeled props.'' This is a higher dimensional version of the well-known algebraic classification of planar algebras as certain pivotal categories.  

This classification allows us to connect these ``welded tangled foams,'' to the Kashiwara-Vergne conjecture in Lie theory. In work in progress, we show that the group of homotopy automorphisms of the (rational completion of) the wheeled prop of welded foams is isomorphic to the group of symmetries KV, which acts on the solutions to the Kashiwara-Vergne conjecture. Moreover, we explain how this approach illuminates the close relationship between the group KV and the pro-unipotent Grothendieck--Teichmüller group.

This talk is aimed at a topological audience and as such will not assume an extensive categorical background. Joint work with Zsuzsanna Dancso and Iva Halacheva. 

October 23: Yusuf Barış Kartal (Princeton): Iterates of symplectomorphisms and p-adic analytic actions (second slot, 10/23/20 noon PT)

Let M be a (negatively) monotone, symplectic manifold satisfying some strong non-degeneracy assumption and let \phi be a symplectomorphism isotopic to identity. In this talk, we prove that the rank of HF(\phi^k(L),L') is constant in k with finitely many possible exceptions. The key tool is inspired by the work of Bell on dynamical Mordell-Lang conjecture: namely we construct "p-adic analytic actions" on a version of the Fukaya category, and use this to construct a coherent sheaf over the p-adic unit disc whose rank at k gives the rank of HF(\phi^k(L),L').

October 30: Artem Kotelskiy (Indiana): Khovanov homology via immersed curves (second slot, 10/30/20 noon PT)

Finding geometric origins of Khovanov homology has been one of the central questions in the field. Work of many people [Seidel, Smith, Manolescu, Abouzaid] culminated in a Floer-theoretic interpretation, where the topological input is a k-bridge decomposition of a knot, and the symplectic moduli space is a subset of a Hilbert scheme of a Milnor fiber. In this talk we will describe a novel approach, where the topological input is a Conway two-sphere S intersecting a knot K in 4 points, and the moduli space is the 4-punctured sphere S itself. Namely, we will show that Khovanov homology Kh(K), and its deformation BN(K) due to Bar-Natan, are isomorphic to the wrapped Floer homology of a pair of specifically constructed immersed curves on the dividing 4-punctured sphere S. The key ingredients will be embedding of deformed arc algebra into the wrapped Fukaya category of the 4-punctured sphere, as well as invoking a theorem of Haiden-Katzarkov-Kontsevich to obtain immersed curves as invariants. 

We will also describe a connection of our framework to homological mirror symmetry statement for the three-punctured sphere, via the matrix factorization framework of Khovanov-Rozansky. This will restrict the geometry of curves-invariants, making it easier to develop applications of our framework. To describe the latter, we will prove that Ramussen's s-invariant is preserved by mutation, and also discuss mutation invariance of Kh(K;Q). After that we will describe a criterion for when a knot is Khovanov-thin, in the presence of dividing Conway sphere. Time-permitting, we will also discuss some work-in-progress on the generalized cosmetic crossing conjecture.

This is joint work with Liam Watson and Claudius Zibrowius.

November 6 (first slot): Yuhan Sun (Rutgers): Some computations of relative symplectic cohomology (first slot, 11/06/20 noon ET)

Relative symplectic cohomology, constructed by U.Varolgunes, provides a useful tool to study topological and dynamical properties of closed subsets in a symplectic manifold. I will discuss several computational aspects about it, with a focus on index bounded Liouville domains in Calabi-Yau manifolds. In particular, a spectral sequence will be defined in this case. If time permits, some thought of the homologically index bonded case will also be explained.

November 13 (second slot): Ben Wormleighton (Washington University): Asymptotics of ECH capacities via algebraic positivity

Connections to algebraic geometry have emerged at various points in the study of symplectic embeddings and have offered novel insight into both algebraic and symplectic problems. I will describe a collection of ideas relating certain obstructions to embeddings of symplectic 4-manifolds introduced by Hutchings — ECH capacities — to rich invariants in algebraic positivity called algebraic capacities. This perspective yields new sharp embedding results for closed toric surfaces and resolves questions about the asymptotics of ECH capacities. Time permitting, I will discuss generalisations to higher dimensions and next steps in four dimensions.

November 20 (second slot): Mohan Swaminathan (Princeton): Super-rigidity and bifurcations of embedded curves in Calabi-Yau 3-folds

By building on Wendl's recent result on the genericity of super-rigidity in dimensions >= 6, we can study bifurcations of moduli spaces of embedded holomorphic curves in Calabi-Yau 3-folds. In this talk, I will describe a particular instance of this bifurcation analysis. As an application, we can define an integer-valued refinement of Gromov-Witten invariants in some special cases beyond the primitive cases. This is based on joint work in progress with Shaoyun Bai.

December 4: no seminar (ongoing mirror symmetry workshop)

There will be no seminar this week, on account of the online workshop Current Advances in Mirror Symmetry (December 4-5), organized by the Simons Collaboration on Homological Mirror Symmetry.

December 11 (first slot): Cecilia Karlsson (Oslo): Legendrian contact homology for Weinstein handle attachments in higher dimensions

The symplectic homology of a Weinstein manifold is encoded in the Legendrian contact homology of the attaching spheres of the top index handles. In this talk I describe a geometric set-up where we can calculate the Legendrian contact homology of the attaching spheres from Legendrians in 1-jet spaces. Since the latter is well-studied this simplifies the calculations. If time permits I will use these techniques to calculate the singular homology of the free loop space of the complex projective plane.

This is a generalization of similar work by Ekholm-Ng to higher dimensions.

Egor Shelukhin (University of Montreal): Smith theory in filtered Floer homology and Hamiltonian diffeomorphisms (7/24/20 7pm GMT)

We describe how Smith theory applies in the setting of Hamiltonian Floer homology filtered by the action functional, and provide applications to questions regarding Hamiltonian diffeomorphisms, including the Hofer-Zehnder conjecture on the existence of infinitely many periodic points. 

Eva Miranda (Universitat Politècnica de Catalunya): The singular Weinstein conjecture (7/24/20 4pm GMT)

The purpose of this talk is to present some recent results concerning Reeb dynamics on a $b^m$-contact manifolds. b-Contact manifolds (and more generally b^m´-contact manifolds) are the odd-dimensional counterpart to $b^m$-symplectic manifolds which have been a center of attention in Poisson Geometry. The study of $b^m$-Reeb dynamics is motivated by well-known problems in fluid dynamics (Beltrami fields) and celestial mechanics, where those geometric structures naturally appear.  

The first part of the talk will focus on the singular Weinstein conjecture following https://arxiv.org/abs/2005.09568  (joint work with Cédric Oms). We prove that in dimension 3 there are always infinite periodic orbits on the critical set (if compact). In particular, we will prove that the dynamics on positive energy level-sets in the restricted planar circular three-body problem are described by the Reeb vector field of a $b^3$-contact form that admits an infinite number of periodic orbits at the critical set.  This investigation goes hand-in-hand with the Weinstein conjecture on non-compact manifolds having compact ends of convex type. In particular, we extend Hofer's arguments to open overtwisted contact manifolds that are $\R^+$-invariant in the open ends, obtaining as a corollary the existence of periodic $b^m$-Reeb orbits away from the critical set.

At the end of the talk, we will focus on singular Reeb orbits (joint work with Cédric Oms and Daniel Peralta-Salas). Inspired by Poincaré's orbits going to infinity in the (restricted) three-body problem, we investigate the existence of singular Reeb orbits emanating from/going to the critical set and we prove their existence for generic Melrose b-contact structures. In the proof, we use the correspondence between b-Beltrami vector fields and b-contact structures.

Cheuk Yu Mak (Cambridge): Symplectic annular Khovanov homology (7/17/20 7pm GMT)

Annular Khovanov homology is an invariant of annular links (links in a solid torus) introduced by Asaeda-Przytycki-Sikora as an analogue of Khovanov homology for links. Auroux-Grigsby-Wehrli showed that the first piece of the annular Khovanov homology can be identified with the Hochschild homology of the Fukaya-Seidel category of A_n Milnor fibers with coefficients in braid bimodules. In this talk, we will introduce a symplectic version of annular Khovanov homology using Hochschild homology of the Fukaya-Seidel category of more general type A nilpotent slices. Building on the work of Abouzaid-Smith and Beliakova-Putyra-Wehrli, we show that the symplectic version is isomorphic to the ordinary version. Finally, we will explain how to derive a spectral sequence from the symplectic annular Khovanov homology to the symplectic Khovanov homology directly using symplectic geometry. This is based on a joint work with Ivan Smith.

Jagna Wiśniewska (ETH Zurich): Rabinowitz Floer homology for tentacular hyperboloids (7/17/20 4pm GMT)

Rabinowitz Floer homology is an algebraic invariant of contact-type hypersurfaces in exact symplectic manifolds. It was first defined for compact hypersurfaces and in this talk we will show how to extend it to include examples of non-compact hypersurfaces. We will also present computational results of Rabinowitz Floer homology for a class of hyperboloids and show how it can be used to prove the Weinstein conjecture. This is joint work with Federica Pasquotto, Alexander Fauck and Will Merry.

Xin Jin (Boston College): Microlocal sheaf categories and the J-homomorphism (7/10/20 7pm GMT)

The theory of microlocal sheaves, developed by Kashiwara--Schapira, has found many applications in the study of symplectic topology. For a smooth Lagrangian L in a cotangent bundle of a smooth manifold and a commutative ring spectrum k, one can associate a sheaf of microlocal categories, which is locally constant with fiber equivalent to Mod(k). It admits a classifying map L--->BPic(k). We will show that the classifying map factors through the Gauss map L--->U/O and the delooping of the J-homomorphism U/O--->BPic(S), where S is the sphere spectrum. As an application, combining with previous results of Guillermou, we show that if L is a compact smooth exact Lagrangian, then the classifying map is homotopically trivial, recovering a result of Abouzaid--Kragh. 

Aliakbar Daemi (WUSTL): Lagrangians, SO(3)-instantons and the Atiyah-Floer Conjecture (7/10/20 4pm GMT)

A useful tool to study a 3-manifold is the space of representations of its fundamental group into a Lie group. Any 3-manifold can be decomposed as the union of two handlebodies. Thus representations of the 3-manifold group into a Lie group can be obtained by intersecting representation varieties of the two handlebodies. Casson utilized this observation to define his celebrated invariant. Later Taubes introduced an alternative approach to define Casson invariant using more geometrical objects. By building on Taubes' work, Floer refined Casson invariant into a 3-manifold invariant which is known as instanton Floer homology. The Atiyah-Floer conjecture states that Casson's original approach can be also used to define a graded vector space and the resulting invariant of 3-manifolds is isomorphic to instanton Floer homology. In this talk, I will discuss a variation of the Atiyah-Floer conjecture, which states that framed Floer homology (defined by Kronheimer and Mrowka) is isomorphic to symplectic framed Floer homology (defined by Wehrheim and Woodward). I will explain how the closed-open string map is related to framed Floer homology. Finally I comment on how earlier works of Seidel and Smith might provide useful computational tools for framed Floer homology. This talk is based on a joint work with Kenji Fukaya and Maksim Lipyanskyi. 

Dan Cristofaro-Gardiner (UCSC): Obstructing infinite staircases (7/3/20, 7 pm GMT)
Talk 3 of 3 in 7/3 Ellipsoid Day joint with the Montreal, PU/IAS, Paris & Tel-Aviv Symplectic Zoominar

Abstract: A landmark result, due to McDuff and Schlenk, asserts that in determining when a four-dimensional symplectic ellipsoid can be symplectically embedded into a four-dimensional ball, the answer is given by an “infinite staircase” determined by the odd-index Fibonacci numbers and the Golden Mean. There has recently been considerable interest in better understanding this phenomenon for more general embedding problems.  I will explain a theorem showing that for any four-dimensional convex toric domain of finite type, if an infinite staircase occurs, then its singular points must accumulate at a unique point, characterized by an explicit quadratic equation.  I will then explain how to apply this theorem to prove that when the target is a rational ellipsoid, there is an infinite staircase in precisely three cases -- when the target has "eccentricity" 1, 2, or 3/2; interestingly, in each of these cases, the corresponding embeddings can be constructed explicitly using polytope mutation.  Part of this is joint work with Holm, Mandini and Pires, but will not overlap with their talk.

Roger Casals (UC Davis): Sharp Ellipsoid Embeddings and Toric Mutations (7/3/20, 4 pm GMT)
Talk 2 of 3 in 7/3 Ellipsoid Day joint with the Montreal, PU/IAS, Paris & Tel-Aviv Symplectic Zoominar

Abstract: In this talk we will explain how to construct volume-filling symplectic embeddings of 4-dimensional ellipsoids by employing polytope mutations in almost-toric varieties. The construction uniformly recovers the sharp embeddings in the Fibonacci Staircase of McDuff-Schlenk, the Pell Staircase of Frenkel-Muller and the Cristofaro-Gardiner-Kleinman's Staircase, and also adds new infinite sequences. I will explain the intuition behind this construction and introduce the two main ingredients for the proof: polytope mutations, following M. Symington and Akhtar-Coates-Galkin-Kasprzyk, and our study of symplectic tropical curves in almost-toric fibrations. This is joint work with R. Vianna.

Umut Varolgunes (Stanford): Mirror symmetry for symplectic cluster manifolds (6/26/20 7pm GMT)

Abstract: I will start by explaining a general framework for constructing non-archimedean analytic mirrors of symplectic manifolds with a Lagrangian fibration using relative symplectic cohomology (including some expected concrete relationships of the A and B-sides). Then I will define symplectic cluster manifolds (conjecturally “half” hyperkahler rotations of smooth Looijenga interiors), which admit a Lagrangian fibration over a topological plane with only focus-focus singularities. These symplectic manifolds are open and geometrically bounded, but not necessarily exact or have contact boundary. Using a general locality result and computations for two local models, I will construct analytic mirrors of symplectic cluster manifolds. Finally, I will describe a conjecture reinterpreting these mirrors as analytifications of certain cluster varieties over the Novikov field (with the same seed data as the Looijenga interior). Joint work with Yoel Groman.

Ivan Smith (Cambridge): Fukaya categories of surfaces and the mapping class group (6/26/20 4pm GMT)

Abstract: I will explain how to build the classical mapping class group of a closed surface of genus at least two starting from the derived Fukaya category of the surface. The proof illustrates numerous different Floer-theoretic technologies in a concrete case. This talk reports on joint work with Denis Auroux.

Kenji Fukaya (Simons Center): SYZ and KAM (6/19/20 7pm GMT) 

Abstract: SYZ (Strominger-Yau-Zaslow) fibration is a torus fibration which appears in the study of Mirror symmetry.  The fibers are Lagrangian tori so it can be regarded as an integrable system.

KAM (Kolmogorov-Arnold-Moser) theory is a classical important theory in Hamiltonian dynamics which describes a behavior of a dynamical system that is close to (but is not) an integrable system.

In this talk I propose that certain symplectic diffeomorphisms can be regarded as dynamical systems which are close to one related to SYZ fibration and explain how we can apply KAM theory to such dynamical systems.

I explain several assumptions which KAM theory assumes, in this case, can be justified by an idea related to homological Mirror symmetry.

I also describe some explicit examples such as pencils of cubic curves and quartic surfaces.

Vera Vertesi (Universität Wien): Cut and Paste Techniques for Open Books (6/19/20 4pm GMT)

Abstract: Due to their combinatorial nature, open books have been one of the major tools of research for 3-dimensional contact manifolds. In this talk I will introduce a new technique to do cut and paste arguments for open books, and on the way I will introduce a generalisation for open books for contact 3-manifolds with a fixed characteristic foliation on their boundary. These objects are called foliated open books. I will explain that, although more complicated, foliated open books are still combinatorial. I will illustrate their use by proving a result about the additivity of the support norm for tight contact structures. I finish with another application, and show that foliated open books are the natural objects to define the contact invariant in bordered Floer homology. Some of the work presented are joint work with Akram Alishahi, Vikt\'oria F\”oldv\'ari, Kristen Hendricks, Joan Licata and Ina Petkova.

Zhengyi Zhou (IAS): (RP^{2n-1}, xi_std) is not exactly fillable for n != 2^k (6/12/20 7pm GMT)

Abstract: I will show that the 2n-1 dimensional real projective space with the standard contact structure is not exactly fillable when n is not a power of 2. Then I will prove that there exist strongly fillable but not exactly fillable contact manifolds in all dimensions greater than 3. Time permitting, I will explain how a similar approach can be used to obtain uniqueness results on symplectically spherical/Calabi-Yau fillings of Y = \partial(V x D) for any Liouville domain V.

Sushmita Venugopalan (IMS Chennai): Tropical Fukaya Algebras (6/12/20 4pm GMT)

Abstract : A multiple cut operation on a symplectic manifold produces a collection of cut spaces, each containing relative normal crossing divisors. We explore what happens to curve count-based invariants when a collection of cuts is applied to a symplectic manifold. The invariant we consider is the Fukaya algebra of a Lagrangian submanifold that is contained in the complement of relative divisors. The ordinary Fukaya algebra in the unbroken manifold is homotopy equivalent to a `broken Fukaya algebra' whose structure maps count `broken disks' associated to rigid tropical graphs. Via a further degeneration, the broken Fukaya algebra is homotopy equivalent to a `tropical Fukaya algebra' whose structure maps are sums of products over vertices of tropical graphs. This is joint work with Chris Woodward.

Kristen Hendricks (Rutgers): A surgery exact triangle for involutive Heegaard Floer homology, and consequences (6/5/20 7pm GMT)

Abstract: We construct a surgery exact triangle in the involutive variant of Ozsvath and Szabo's Heegaard Floer homology, and give an application to the structure of the integer homology cobordism group. This is joint work in progress with J. Hom, M. Stoffregen, and I. Zemke.

Daniel Álvarez-Gavela (Princeton): The nearby Lagrangian conjecture from the K-theoretic viewpoint (6/5/20 4pm GMT)

In this talk I will explain some connections between the nearby Lagrangian conjecture and the algebraic K-theory of spaces. These connections have opened up as a consequence of the recent existence result for (twisted) generating functions due to Abouzaid, Courte, Guillermou and Kragh. In work in progress joint with Abouzaid, Courte and Kragh we find that a certain geometric description of the splitting map for the algebraic K-theory of a point due to Waldhausen and Bökstedt gives a new restriction on the framed bordism class of nearby Lagrangians. In particular I will show that if L is any Lagrangian homotopy sphere in the cotangent bundle of the standard sphere, then the connected sum of L with itself bounds a parallelizable manifold. This extends known constraints for the possible class of L in the group of homotopy spheres modulo those which bound a parallelizable manifold. I will also touch on joint work with Igusa concerning the higher torsion of Legendrians in 1-jet spaces and make some speculations about the higher torsion of nearby Lagrangians.

Jeff Hicks (Cambridge): Lagrangian Surgery and Lagrangian Cobordism (5/29/20 7pm GMT)

Lagrangian cobordisms form an equivalence relation on Lagrangian submanifolds of a symplectic manifold X. In the monotone setting, the work of Biran and Cornea show that cobordant Lagrangian submanifolds have equivalent Floer homology. However, to date the only known 2-ended monotone Lagrangian cobordisms are those constructed as the suspension of a Hamiltonian isotopy. This talk will explain how Lagrangian cobordism can be decomposed into models based on Haug antisurgeries. I will also speculate about the relation between Biran and Cornea's work on iterated exact triangles and the Fukaya, Oh, Ohta, and Ono surgery exact triangle in the context of this decomposition.

Ailsa Keating (Cambridge): Homological mirror symmetry for log Calabi-Yau surfaces (5/29/20 4pm GMT)

Given a log Calabi-Yau surface Y with maximal boundary D, I'll explain how to construct a mirror Landau-Ginzburg model, and sketch a proof of homological mirror symmetry for these pairs when (Y,D) is distinguished within its deformation class (this is mirror to an exact manifold). I'll explain how to relate this to the total space of the SYZ fibration predicted by Gross-Hacking-Keel, and, time permitting, explain ties with earlier work of Auroux-Katzarkov-Orlov and Abouzaid. Joint work with Paul Hacking.

Ko Honda (UCLA): Convex hypersurface theory in higher-dimensional contact topology (5/22/20 7pm GMT)

Convex surface theory and bypasses are extremely powerful tools for analyzing contact 3-manifolds. In particular they have been successfully applied to many classification problems. After briefly reviewing convex surface theory in dimension three, we explain how to generalize many of their properties to higher dimensions. This is joint work with Yang Huang.

Jack Smith (Cambridge University): Towards Berglund-Hübsch mirror symmetry (5/22/20 4pm GMT)

An nxn non-negative integer matrix encodes an n-term polynomial in n variables, by using each column to define the exponents in one monomial. Berglund and Hübsch predicted that the polynomials associated to transpose matrices should have mirror Landau-Ginzburg models; precisely, the Fukaya-Seidel category of one polynomial should be equivalent to graded matrix factorisations of the other. I'll describe a new strategy for attacking this conjecture, based on joint work in progress with Benjamin Gammage.

Oleg Lazarev (Columbia University): Weinstein geometry of cotangent bundles (5/15/20 7pm GMT) 

Abstract: Although the cotangent bundle of a sphere T*S^n has very few closed exact Lagrangians (conjecturally only one), we will explain that it has many singular Lagrangians in the form of Weinstein subdomains. We first produce flexible subdomains of T*S^n, which in high-dimensions yield exotic Weinstein presentations for T^*S^n as the standard ball with a single handle attached along an exotic Legendrian knot. The algebraic side of the story for the wrapped Fukaya category is closely connected to a result of Thomason in algebraic K-theory. Then we discuss joint work with Z. Sylvan that associates to every finite collection of prime integers a (non-flexible) Weinstein subdomain of T*S^n whose Fukaya category is localized at those primes and show that the Fukaya category of any Weinstein subdomain is one of these prime localizations.

Xujia Chen (Stony Brook University): Lifting cobordisms and Kontsevich-type recursions for counts of real curves (5/15/20 4pm GMT)

Abstract: Kontsevich's recursion, proved by Ruan-Tian in the early 90s, is a recursion formula for genus 0 Gromov-Witten invariants. For symplectic fourfolds and sixfolds with a real structure (i.e. an anti-symplectic involution, analogue of the usual conjugation map on C^n), signed invariant counts of real rational pseudo-holomorphic curves were defined by Welschinger in 2003. In 2006-07, Solomon re-interpreted Welschinger's invariants, proposed Kontsevich-type recursion formulas for them, and suggested a potential adaptation of the proof in the complex case for confirming them. For many symplectic fourfolds and sixfolds, these recursions determine all invariants from basic inputs. We establish Solomon's recursions by a different approach: lifting cobordisms from the moduli spaces of real domains to the moduli space of real maps and incorporating the wall-crossing corrections from the walls obstructing relative-orientability.

Melissa Zhang (University of Georgia): An annular filtration on Sarkar-Seed-Szabó's spectral sequence
5/8/20 7pm GMT

Abstract: Khovanov homology is a combinatorial invariant of links in the three-sphere borne from structures in representation theory. Nevertheless, there are many spectral sequences relating Khovanov homology to geometrically-defined invariants, such as Ozsváth-Szabó's Heegaard Floer homology. Seidel-Smith defined its geometric counterpart, symplectic Khovanov homology, which Abouzaid-Smith showed is indeed isomorphic to combinatorial Khovanov homology over characteristic 0. Inspired by symplectic Khovanov homology's O(2) action, Sarkar-Seed-Szabó extended Szabó's geometric spectral sequence, which is a combinatorial spectral sequence conjectured to be isomorphic to Ozsváth-Szabó's spectral sequence (from the Khovanov homology of a knot to the Heegaard Floer homology of the branched double cover of its mirror knot). A bifiltered version of this complex admits a family of Rasmussen-type link invariants.

In joint work with Linh Truong, we show that for links in a solid torus (i.e. annular links), Sarkar-Seed-Szabó's complex admits third filtration. This annular filtration allows us to define a 2-parameter family of annular concordance invariants s_{r,t} analogous to Grigsby-Licata-Wehrli's annular Rasmussen invariants d_t (from annular Khovanov-Lee theory). The two families share many properties, including applications to 3D contact geometry and smooth knot concordance. 

Semon Rezchikov (Columbia University): Generalizations of Hodge-de-Rham degeneration for Fukaya categories
5/8/20 4pm GMT

Abstract: Hodge theory shows that the Hodge-de-Rham spectral sequence associated to a compact Kahler manifold degenerates. Kaledin showed that the non-commutative Hodge-de-Rham spectral sequence associated to a smooth proper dg-category over a field of characteristic zero degenerates as well. When the category is just smooth or just proper, Kontsevich conjectured that certain weaker statements, which are true for smooth or proper varieties, should continue to hold in the categorical setting. Recently, counterexamples to Kontsevich's conjectures were found by Efimov. I will discuss the background to this story, and then I will explain why the conjectures of Kontsevich do hold, for analytic reasons, when the category is a Fukaya category. The argument suggests interesting directions to explore regarding the homological algebra of PROPs of surfaces.

Jo Nelson & Morgan Weiler (Rice): ECH of prequantization bundles and lens spaces
5/1/20 7 pm GMT

Abstract: In 2011, Farris provided an expected dictionary between counts of pseudoholomorphic cylinders and Z_2-graded embedded contact homology (ECH) of prequantization bundles over Riemann surfaces.  We upgrade to a full Z-grading, and in combination with the domain dependent methods introduced by Farris in his thesis, make use of the direct limits for filtered ECH established in Hutchings-Taubes proof of the Arnold-Chord conjecture to extend the Morse-Bott methods for prequantization bundles to the realm of ECH.  In particular, we establish that the ECH of a prequantization bundle over a Riemann surface is isomorphic to the exterior algebra of the homology of this base.   We comment on future work, which relates the U map in ECH to Gromov-Witten invariants of the base, permitting computations of the associated ECH capacities and an expected stabilization result purely in the context of ECH. 

Tim Large (MIT): Floer K-theory and exotic Liouville manifolds
5/1/20 4 pm GMT

Abstract: In this talk, we will explain how to construct Liouville manifolds which have vanishing symplectic cohomology but non-vanishing symplectic K-theory. In particular, we construct an exotic symplectic structure on Euclidean space which is not distinguished by traditional Floer homology invariants. Instead, it is detected by a module spectrum for complex K-theory, built as a variant of Cohen-Jones-Segal’s Floer homotopy type. The proof involves passage through (wrapped) Fukaya categories with coefficients in a ring spectrum, rather than an ordinary ring; we will outline the construction of such "spectral Fukaya categories" in the setting of exact symplectic manifolds. 

Yu Pan (MIT): Augmentations and exact Lagrangian surfaces
4/24/20 7pm GMT

Abstract: Augmentations are some algebraic invariants of Legendrians that are tightly related to both embedded and immersed exact Lagrangian fillings. We will talk about various relations between embedded and immersed exact Lagrangian surfaces using tools related to augmentations.

Abigail Ward (Harvard): Homological mirror symmetry for elliptic Hopf surfaces
4/24/20 4pm GMT

Abstract: One can produce non-Kähler complex surfaces by performing logarithmic transformations on projective elliptic surfaces; for example, elliptic Hopf surfaces (including the classical Hopf surface S^1 x S^3) can be obtained by performing such operations to the product of the projective plane with an elliptic curve. In situations where the original surface has a mirror symplectic space, one can ask if there is a "mirror operation" to the logarithmic transformation, i.e. a way of producing a mirror to the logarithmically transformed surface from the original mirror space. We will discuss an answer to this question in the case of elliptic Hopf surfaces. For each such surface S, we will produce a mirror "non-algebraic Landau-Ginzburg model" with an associated Fukaya category. We will relate objects of this Fukaya category to coherent analytic sheaves on S.

Andrew Hanlon (Simons Center): On Fukaya-Seidel categories mirror to toric varieties
4/17/20 7pm GMT

Abstract: We will discuss one way of defining a Fukaya-Seidel category mirror to a toric variety and use it to understand homological mirror symmetry in this setting. Along the way, we will see how this Fukaya-Seidel category relates to more traditional definitions. This is partly based on joint work in progress with Jeff Hicks.

Hiro Lee Tanaka (Texas State University): Formalisms of gluing Fukaya categories
4/17/20 4pm GMT

Abstract: In the Weinstein setting, we know that wrapped Fukaya categories glue, so it behooves us to understand a general framework that captures the properties of this gluing. After explaining a few approaches of how to formalize gluing procedures in the 2-dimensional setting, we'll explain why we think a framework inspired by factorization homology seems most promising to capture the general behavior of local-to-global invariants of Weinstein sectors. Setting up this formalism sheds insights into things like the following: (a) We can classify all local-to-global invariants of 2-dimensional Liouville sectors. (b) We see that the Floer theory of Lagrangian cobordisms in R^oo recovers the higher K theory of the integers. (We have been unable to compute this higher K theory for decades, and a computation would yield powerful results in arithmetic.) (c) Wrapped Floer theory can shed insight into higher homotopy groups of Liouville embedding spaces. 

Hang Yuan (Stony Brook): Family Floer program and non-archimedean SYZ mirror construction
4/10/20 7pm GMT

Abstract: Given a Lagrangian fibration, we construct a mirror Landau-Ginzburg model consisting of a rigid analytic space and a global potential, based on Fukaya’s family Floer theory and non-archimedean tropical geometry. This may be thought of as a symplectic version of Gross-Seibert’s program.
Locally, the potential is given by counting Maslov-two disks. A local chart is the zero locus of weak Maurer-Cartan equations of an A_∞ algebra in the non-archimedean torus fibration.
We will explain how an A_∞ homomorphism gives us a wall-crossing formula, which leads to a transition map between two local charts. If time allows, I will explain why the transition map only depends on the homotopy class of this A_∞ homomorphism.

Chris Wendl (Humboldt-Universität zu Berlin): Equivariant transversality, super-rigidity and all that
4/10/20 4pm GMT

Abstract: As everyone knows, you can't have transversality and symmetry at the same time.  If you could, then multiply covered holomorphic curves would not be such a headache, and many popular symplectic invariants would be much easier to define.  My goal in this talk is to clarify precisely what obstructs equivariant transversality, and just how happy you should be when those obstructions vanish.  I will sketch applications in three distinct settings: (1) Generic functions on a finite-dimensional orbifold are Morse; (2) Generic 1-parameter deformations of geodesible line fields undergo only birth-death and period-doubling bifurcations; (3) Generic almost complex structures on a symplectic Calabi-Yau 3-fold make the Cauchy-Riemann operator intersect the zero-section cleanly (i.e. all holomorphic curves are "super-rigid").  I will also try to explain what technical lemmas need to be proved (e.g. "Petri's condition") in order for this machinery to work in any given setting, though I will probably not explain how to prove them.

Ezra Getzler (Northwestern): New results on Batalin-Vilkovisky geometry
4/3/20 7pm GMT

Abstract: In the theory of supermanifolds, there is an odd analogue of symplectic geometry, where the parity of the symplectic form is fermionic. Liouville's theorem breaks down in this case: there is no canonical integral. The replacement for this is Batalin-Vilkovisky geometry, which has been brought into its modern form by Khudaverdian. The replacements for integrals are pairs made up of a half-density on the odd symplectic supermanifold satisfying a certain second-order linear differential equation and a Lagrangian submanifold. In this talk, I will describe recent work with Sean Pohorence (see https://arxiv.org/abs/1911.11269), in which we generalize this formalism to allow distinct Lagrangians on different charts, joined by isotopies on the intersections of charts, which are themselves joined by isotopies of isotopies in higher dimensions, and so on. This formalism has proved useful in understanding some situations in quantum field theory where there is no global choice of gauge (the Gribov ambiguity).
In this talk, I will focus on the mathematical side of the story, leaving the quantum field theory for another day.

Michael Sullivan (UMass Amherst): The persistence of the Legendrian contact homology algebra
4/3/20 4pm GMT

Abstract: The well-studied displacement energy of a Lagrangian submanifold is the minimum Hamiltonian oscillation needed so that the Lagrangian does not intersect its image under the Hamiltonian isotopy. For dimension reasons, the analogous Legendrian displacement energy definition requires the contact Hamiltonian image of the Legendrian to not intersect the Reeb flow of the original Legendrian. In this contact setting, I will discuss how to apply the persistent homology barcodes of a filtered Legendrian contact (Chekanov-Eliashberg) algebra to get lower bounds on the number of such intersections. The numerical bounds vary with the size of the Hamiltonian oscillation and hold when a Legendrian has a large loose chart, in contrast to a Legendrian with a small loose chart which sometimes has a small displacement energy. I will also discuss a related Legendrian non-squeezing result. This is joint work with Georgios Dimitroglou Rizell.

Laura Starkston (UC Davis): Weinstein Trisections
3/27/20 7pm GMT

Abstract: Gay and Kirby proved that every smooth 4-manifold admits a trisection--a decomposition into three pieces, each of which is a 1-handlebody. A Weinstein trisection is a trisection which is nicely compatible with a symplectic structure on the 4-manifold. We will explain this structure and show that every symplectic 4-manifold admits a Weinstein trisection. This is joint work with Peter Lambert-Cole and Jeffrey Meier.

Paul Seidel (MIT): Quantum Steenrod operations
3/27/20 4pm GMT

Abstract: Quantum Steenrod operations play a role in both Hamiltonian dynamics and mirror symmetry. I will explain a bit about their structure, give a sample dynamical application, and then tell you what we know about their mirror-symmetric meaning. Part of this is joint work with Nicholas Wilkins.