Time: Fridays, 9:15–10:45 a.m. (Montréal/Princeton hour)
(Meeting ID: 971 1614 7750 ; Passcode: 816898)

• ### Yash Deshmukh (Columbia)

Title: Moduli spaces of nodal curves from homotopical algebra

Abstract: I will discuss how the Deligne-Mumford compactification of curves arises from the uncompactified moduli spaces of curves as a result of some algebraic operations related to (pr)operadic structures on the moduli spaces. I will describe how a variation of this naturally gives rise to another new partial compactification of moduli spaces curves. Time permitting, I will indicate how it is related to secondary operations on symplectic cohomology and discuss some ongoing work in this direction.

• ### Lea Kenigsberg (Columbia)

Title: Coproduct structures, a tale of two outputs

Abstract: I will motivate the study of coproducts and describe a new coproduct structure on the symplectic cohomology of Liouville manifolds. Time permitting, I will indicate how to compute it in an example to show that it's not trivial. This is based on my thesis work, in progress.

• ### Thomas Massoni (Princeton)

Title: Non-Weinstein Liouville domains and three-dimensional Anosov flows

Abstract: Weinstein domains and their symplectic invariants have been extensively studied over the last 30 years. Little is known about non-Weinstein Liouville domains, whose first instance is due to McDuff. I will describe two key examples of such domains in dimension four, and then explain how they fit into a general construction based on Anosov flows on three-manifolds. The symplectic invariants of these “Anosov Liouville domains” constitute new invariants of Anosov flows. The algebraic structure of their wrapped Fukaya categories is in stark contrast with the Weinstein case.

This is mostly based on joint work arXiv:2211.07453 with Kai Cieliebak, Oleg Lazarev and Agustin Moreno.

1. Regular research talks are of 60 minutes. There will be 30 min at the end of each talk reserved for discussion. The first 15 minutes (roughly) are, mainly, for questions addressed to the speaker. After that, questions and answers may involve different participants.
2. Once in several sessions we intend to have a seminar consisting of three 20min talks (followed each by 10min of discussion time) given by young researchers/recent PhD's. Suggestions, nominations, and volunteers (including a title and short abstract) should be sent to Egor Shelukhin at egorshel@gmail.com (with cc to octav.cornea@gmail.com).
3. It is intended that all talks be accessible to a global community in symplectic geometry/topology and beyond (thus, they should contain an introduction of interest to a broad audience).
5. We post links to the slides of the talks as well as links to recordings of the talks.
6. In addition to this webpage, announcements are posted on researchseminars.org and sent out via the virtual symplectic seminars group.

The Zoominar promotes an atmosphere of collegiality, equity and respect and is committed to creating a welcoming and inclusive environment for all participants, enabling them to fully focus on mathematics.

Current Zoominar organizers: Octav Cornea (Montréal), Helmut Hofer (IAS), Vincent Humilière (Paris), Agustin Moreno (IAS), Leonid Polterovich (Tel Aviv), Egor Shelukhin (Montréal), Shira Tanny (IAS), Sara Tukachinsky (Tel Aviv), Claude Viterbo (Paris).

## Future talks

• Dec. 2: Cancelled on account of the Symplectix seminar http://symplectix.blogspot.com/
• Dec. 9: Robert Cardona (ICMAT),  Periodic orbits and Birkhoff sections of stable Hamiltonian structures, (abstract)
In this talk, we start by reviewing recent results on the dynamics of Reeb vector fields defined by contact forms on three-dimensional manifolds, and then introduce Reeb fields defined by stable Hamiltonian structures. These are more general and arise, for instance, in stable regular energy level sets of Hamiltonian systems. We give a characterization of Reeb fields that are aperiodic or that have finitely many periodic orbits (under a certain nondegeneracy assumption). Finally, we give sufficient conditions for the existence of an adapted broken book decomposition or the existence of a Birkhoff section. This is joint work with A. Rechtman.
• Dec. 16: TBD

## Past talks

### --Fall 2022--

• Nov. 18: Cancelled on account of the Symplectix seminar http://symplectix.blogspot.com/
and the conference Floer homotopical methods in low dimensional and symplectic topology, https://www.msri.org/workshops/1024
• Nov. 11: Roger Casals (UC Davis), A microlocal invitation to Lagrangian fillings, (video), (slides), (abstract)
We present recent developments in symplectic geometry and explain how they motivated new results in the study of cluster algebras. First, we introduce a geometric problem: the study of Lagrangian surfaces in the standard symplectic 4-ball bounding Legendrian knots in the standard contact 3-sphere. Thanks to results from the microlocal theory of sheaves, which we will survey, we then show that this geometric problem gives rise to an interesting moduli space. In fact, we establish a bridge translating geometric operations, such as Lagrangian disk surgeries, into algebraic properties of this moduli space, such as the existence of cluster algebra structures. The talk is intended for a broad symplectic audience and all key ideas will be introduced and motivated.
• Nov. 4: Ipsita Datta (IAS), Lagrangian cobordisms, enriched knot diagrams, and algebraic invariants, (video), (abstract)
We introduce new invariants to the existence of Lagrangian cobordisms in $$\mathbb{R}^4$$. 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.
• Oct. 28: Three 20min research talks
• Pierre-Alexandre Mailhot (UdeM), The spectral diameter of a Liouville domains and its applications, (video), (slides), (abstract)
The spectral norm provides a lower bound to the Hofer norm. It is thus natural to ask whether the diameter of the spectral norm is finite or not. During this short talk, I will give a sketch of the proof that, in the case of Liouville domains, the spectral diameter is finite if and only if the symplectic cohomology of the underlying manifold vanishes. With that relationship in hand, we will explore applications to symplecticaly aspherical symplectic manifolds and Hofer geometry.
• Nicole Magill (Cornell), A correspondence between obstructions and constructions for staircases in Hirzebruch surfaces, (video), (slides), (abstract)
The ellipsoidal embedding function of a symplectic four manifold M measures how much the symplectic form on M must be dilated in order for it to admit an embedded ellipsoid of some eccentricity. It generalizes the Gromov width and ball packing numbers. In most cases, finitely many obstructions besides the volume determine the function. If there are infinitely many obstructions determining the function, M is said to have an infinite staircase. This talk will give a classification of which Hirzebruch surfaces have infinite staircases. We will focus on explaining the correspondence between the obstructions coming from exceptional classes and the constructions from almost toric fibrations. We define a way to mutate triples of exceptional classes to produce new triples of exceptional classes, which corresponds to mutations in almost toric fibrations. This is based on various joint work with Dusa McDuff, Ana Rita Pires, and Morgan Weiler.
• Ofir Karin (Tel Aviv), Approximation of Generating Function Barcode for HamiltonianDiffeomorphisms, (video), (slides), (abstract)
Persistence modules and barcodes are used in symplectic topology to define new invariants of Hamiltonian diffeomorphisms, however methods that explicitly calculate these barcodes are often unclear. In this talk I will define one such invariant called the GF-barcode of compactly supported Hamiltonian diffeomorphisms of $$\mathbb{R}^{2n}$$ by applying Morse theory to generating functions quadratic at infinity associated to such Hamiltonian diffeomorphisms and provide an algorithm (i.e a finite sequence of explicit calculation steps) that approximates it along with a few computation examples. This is joint work with Pazit Haim-Kislev.
• Oct. 21: Jean Gutt (Albi and Toulouse), Symplectic convexity? (an ongoing story...), (video), (slides), (abstract)
What is the symplectic analogue of being convex? We shall present different ideas to approach this question. Along the way, we shall present recent joint results with J.Dardennes and J.Zhang on monotone toric domains non-symplectomorphic to convex domains and with M.Pereira and V.Ramos on cube-normalized capacities.
• Oct. 14: Igor Uljarević (Belgrade), Contact non-squeezing via selective symplectic homology, (video), (slides), (abstract)
I will introduce a new version of symplectic homology that resembles the relative symplectic homology and that is related to the symplectic homology of a Liouville sector. This version, called selective symplectic homology, is associated with a Liouville domain and an open subset of its boundary. The selective symplectic homology is obtained as the direct limit of the Floer homology groups for Hamiltonians whose slopes tend to infinity on the open subset but remain close to 0 and positive on the rest of the boundary. As an application, I will prove a contact non-squeezing phenomenon on homotopy spheres that are fillable by Liouville domains with infinite dimensional symplectic homology: there exists a smoothly embedded closed ball in such a sphere that cannot be made arbitrarily small by a contact isotopy. These homotopy spheres include examples that are diffeomorphic to standard spheres and whose contact structures are homotopic to standard contact structures.

### --Spring 2022--

• Jun. 24: Julian Chaidez (IAS/Princeton), The Ruelle invariant and convexity in higher dimensions, (video), (slides), (abstract)
I will explain how to construct the Ruelle invariant of a symplectic cocycle over an arbitrary measure preserving flow. I will provide examples and computations in the case of Hamiltonian flows and Reeb flows (in particular, for toric domains). As an application of this invariant, I will construct toric examples of dynamically convex domains that are not symplectomorphic to convex ones in any dimension.

This talk is based on joint works arXiv:2012.12869 and arXiv:2205.00935 with Oliver Edtmair.
• Jun. 17: Yoel Groman (HUJI), Locality and deformations in relative symplectic cohomology, (video), (slides), (abstract)
Relative symplectic cohomology is a Floer theoretic invariant associated with compact subsets K of a closed or geometrically bounded symplectic manifold M. The motivation for studying it is that it is often possible to reduce the study of global Floer theory of M to the Floer theory of a handful of local models covering M which one hopes will be easier to compute (Varolgunes’ spectral sequence). As an example, it is expected that at least in the setting of the Gross-Siebert program, the mirror can be pieced together from the relative symplectic cohomologies of neighborhoods of fibers of an SYZ fibration (singular or not). However, even when K is a well understood model, such as the Weinstein neighborhood of a Lagrangian torus, the construction of relative SH is rather unwieldy. In particular, it is not entirely obvious how to relate the symplectic cohomology of K relative to M with Floer theoretic invariants intrinsic to K. I will discuss a number of results, most of them in preparation, which aim to alleviate this difficulty in the setting Lagrangian torus fibrations with singularities. Partly joint with U. Varolgunes.
• Jun. 3: Guangbo Xu (Texas A&M), Integer-valued Gromov-Witten type invariants, (video), (slides), (abstract)
Gromov-Witten invariants for a general target are rational-valued but not necessarily integer-valued. This is due to the contribution of curves with nontrivial automorphism groups. In 1997 Fukaya and Ono proposed a new method in symplectic geometry which can count curves with a trivial automorphism group. While ordinary Gromov-Witten invariants only use the orientation on the moduli spaces, this integer-valued counts are supposed to also use the (stable) complex structure on the moduli spaces. In this talk I will present the recent joint work with Shaoyun Bai in which we rigorously defined the integer-valued Gromov-Witten type invariants in genus zero for a symplectic manifold. This talk is based on the preprint https://arxiv.org/abs/2201.02688.
• May 27: Three 20min research talks
• Daniel Rudolf (Bochum), Viterbo‘s conjecture for Lagrangian products in $$\mathbb{R}^4$$, (video), (slides), (abstract)
We show that Viterbo‘s conjecture (for the EHZ-capacity) for convex Lagrangian products in $$\mathbb{R}^4$$ holds for all Lagrangian products (any trapezoid in $$\mathbb{R}^2$$)x(any convex body in $$\mathbb{R}^2$$). Moreover, we classify all equality cases of Viterbo’s conjecture within this configuration and show which of them are symplectomorphic to a Euclidean ball. As by-product, we conclude sharp systolic Minkowski billiard inequalities for geometries which have trapezoids as unit balls. Finally, we show that the flows associated to the above mentioned equality cases (which are polytopes) satisfy a weak Zoll property, namely, that every characteristic that is almost everywhere away from lower-dimensional faces is closed, runs over exactly 8 facets, and minimizes the action.
• Miguel Pereira (Augsburg), The Lagrangian capacity of toric domains, (video), (slides), (abstract)
In this talk, I will state a conjecture giving a formula for the Lagrangian capacity of a convex or concave toric domain. First, I will explain a proof of the conjecture in the case where the toric domain is convex and 4-dimensional, using the Gutt-Hutchings capacities as well as the McDuff-Siegel capacities. Second, I will explain a proof of the conjecture in full generality, but assuming the existence of a suitable virtual perturbation scheme which defines the curve counts of linearized contact homology. This second proof makes use of Siegel's higher symplectic capacities.
• Maksim Stokić (Tel Aviv), $$C^0$$ contact geometry of isotropic submanifolds, (video), (slides), (abstract)
Homeomorphism is called contact if it can be written as $$C^0$$-limit of contactomorphisms. The contact version of Eliashberg-Gromov rigidity theorem states that smooth contact homeomorphisms preserve contact structure. Submanifold $$L$$ of a contact manifold $$(Y,\xi)$$ is called isotropic if $$\xi|_{TL}=0$$. Isotropic submanifolds of maximal dimension are called Legendrian, otherwise we call them subcritical isotropic.
In this talk, we will try to answer whether the isotropic property is preserved by contact homeomorphisms. It is expected that subcritical isotropic submanifolds are flexible, while we expect that Legendrians are rigid. We show that subcritical isotropic curves are flexible, and we give a new proof of the rigidity of Legendrians in dimension 3. Moreover, we provide a certain type of rigidity of Legendrians in higher dimensions.
• May 20: Claude Viterbo (Paris), Gamma-support, gamma-coisotropic subsets and applications, (video), (slides), (abstract)
To an element in the completion of the set of Lagrangians for the spectral distance we associate a support. We show that such a support is $$\gamma$$-coisotropic (a notion we shall define in the talk) and we shall give examples and counterexamples of $$\gamma$$-coisotorpic sets that can be (or cannot be) $$\gamma$$-supports. Finally we give some applications of these notions to singular support of sheaves (joint work with S. Guillermou) and dissipative dynamics, allowing us to extend the notion of Birkhoff attractor (joint with V. Humilière).
• May 6: Kevin Ruck (Augsburg), Tate homology and powered flybys, (video), (slides), (abstract)
In this talk I want to show that in the planar circular restricted three body problem there are infinitely many symmetric consecutive collision orbits for all energies below the first critical energy value. By using the Levi-Civita regularization we will be able to distinguish between two different orientations of these orbits and prove the above claim for both of them separately. In the first part of the talk I want to explain the motivation behind this result, especially its connection to powered flybys. Afterwards I will introduce the main technical tools, one needs to prove the above statement, like Lagrangian Rabinowitz Floer Homology and its $$G$$-equivariant version. To be able to effectively calculate this $$G$$-equivariant Lagrangian RFH, we will relate it to the Tate homology of the group $$G$$. With this tool at hand we will then finally be able to prove that there are infinitely many consecutive collision orbits all facing in a specific direction.
• Apr. 29: Joel Fine (ULB), Knots, minimal surfaces and J-holomorphic curves, (video), (slides), (abstract)
Let $$K$$ be a knot or link in the 3-sphere, thought of as the ideal boundary of hyperbolic 4-space, $$H^4$$. The main theme of my talk is that it should be possible to count minimal surfaces in $$H^4$$ which fill $$K$$ and obtain a link invariant. In other words, the count doesn’t change under isotopies of $$K$$. When one counts minimal disks, this is a theorem. Unfortunately there is currently a gap in the proof for more complicated surfaces. I will explain “morally” why the result should be true and how I intend to fill the gap. In fact, this (currently conjectural) invariant is a kind of Gromov–Witten invariant, counting $$J$$-holomorphic curves in a certain symplectic 6-manifold diffeomorphic to $$S^2\times H^4$$. The symplectic structure becomes singular at infinity, in directions transverse to the $$S^2$$ fibres. These singularities mean that both the Fredholm and compactness theories have fundamentally new features, which I will describe. Finally, there is a whole class of infinite-volume symplectic 6-manifolds which have singularities modelled on the above situation. I will explain how it should be possible to count $$J$$-holomorphic curves in these manifolds too, and obtain invariants for links in other 3-manifolds.
• Apr. 22: Jack Smith (Cambridge), From Floer to Hochschild via matrix factorisations, (video), (slides), (abstract)
The Hochschild cohomology of the Floer algebra of a Lagrangian $$L$$, and the associated closed-open string map, play an important role in the generation criterion for the Fukaya category and in deformation theory approaches to mirror symmetry. I will explain how, in the monotone setting, one can build a map from the Floer cohomology of $$L$$ with certain local coefficients to (a version of) Hochschild cohomology. This map makes things much more geometric, by transferring the algebraic complexity to the world of matrix factorisations, and is an isomorphism when $$L$$ is a torus.
• Apr. 15: Kyler Siegel (USC), Singular plane curves and stable nonsqueezing phenomena, (video), (slides), (abstract)
The existence of rational plane curves of a given degree with prescribed singularities is a subtle and active area in algebraic geometry. This problem turns out to be closely related to difficult enumerative problems which arise in symplectic field theory, which in turn play a central role in the theory of high dimensional symplectic embeddings. In this talk I will discuss various perspectives on these enumerative problems and present a new closed formula for relevant curve counts as a sum over decorated trees.
• Apr. 8: Yann Rollin (Nantes), Lagrangians, symplectomorphisms and zeroes of moment maps, (video), (slides), (abstract)
I will present two constructions of Kähler manifolds, endowed with Hamiltonian torus actions of infinite dimension. In the first example, zeroes of the moment map are related to isotropic maps from a surfaces in $$\mathbb{R}^{2n}$$. In the second example, which is actually a hyperKähler moment map, the zeroes are related to symplectic maps of the torus $$T^4$$. The corresponding modified moment map flows have short time existence. Polyhedral analogues of these constructions can be used to investigate piecewise linear symplectic geometry.

### --Winter 2022--

• Mar. 25: Three 20min research talks
• Benoît Joly (Bochum), Barcodes for Hamiltonian homeomorphisms of surfaces, (video), (slides), (abstract)
In this talk, we will study the Floer Homology barcodes from a dynamical point of view. Our motivation comes from recent results in symplectic topology using barcodes to obtain dynamical results. We will give the ideas of new constructions of barcodes for Hamiltonian homeomorphisms of surfaces using Le Calvez's transverse foliation theory. The strategy consists in copying the construction of the Floer and Morse Homologies using dynamical tools like Le Calvez's foliations.
• Marco Castronovo (Columbia), Polyhedral Liouville domains, (video), (slides), (abstract)
I will explain the construction of a new class of Liouville domains that live in a complex torus of arbitrary dimension, whose boundary dynamics encodes information about the singularities of a toric compactification. The primary motivation for this work is to find a symplectic interpretation of some curious Laurent polynomials that appear in mirror symmetry for Fano manifolds; it also potentially opens a path to bound symplectic capacities of polarized projective varieties from below.
• Agniva Roy (Georgia Tech), Constructions of high dimensional Legendrians and isotopies, (video), (slides), (abstract)
I will talk about an ongoing project that explores the construction of high dimensional Legendrian spheres from supporting open books and contact structures. The input is a Lagrangian disk filling of a Legendrian knot in the binding. We try to understand the relationship between different constructions from the same input, and suggest parallels, in the $$S^{2n+1}$$ case, to a construction defined by Ekholm for $$\mathbb{R}^{2n+1}$$.
• Mar. 18: Anton Izosimov (Arizona), Dimers, networks, and integrable systems, (video), (slides), (abstract)
I will review two combinatorial constructions of integrable systems: Goncharov-Kenyon construction based on counting perfect matchings in bipartite graphs, and Gekhtman-Shapiro-Tabachnikov-Vainshtein construction based on counting paths in networks. After that I will outline my proof of equivalence of those constructions. The talk is based on my recent preprint arXiv:2108.04975.
• Mar. 4: Marie-Claude Arnaud (Paris), Invariant submanifolds for conformal dynamics, (video), (slides), (abstract)
In a work with Jacques Fejoz, we consider the conformal dynamics on a symplectic manifold , i.e. for which the symplectic form is transformed colinearly to itself. In the non-symplectic case, we study the problem of isotropy and uniqueness of invariant submanifolds. More precisely, in this talk, I will explain a relation between topological entropy and isotropy and some uniqueness results.
• Feb. 25: Erman Çineli (Paris), Topological entropy of Hamiltonian diffeomorphisms: a persistence homology and Floer theory perspective, (video), (slides), (abstract)
In this talk I will introduce barcode entropy and discuss its connections to topological entropy. The barcode entropy is a Floer-theoretic invariant of a compactly supported Hamiltonian diffeomorphism, measuring, roughly speaking, the exponential growth under iterations of the number of not-too-short bars in the barcode of the Floer complex. The topological entropy bounds from above the barcode entropy and, conversely, the barcode entropy is bounded from below by the topological entropy of any hyperbolic locally maximal invariant set. As a consequence, the two quantities are equal for Hamiltonian diffeomorphisms of closed surfaces. The talk is based on a joint work with Viktor Ginzburg and Başak Gürel.
• Feb. 18: Umut Varolgunes (Boğaziçi), Reynaud models from relative Floer theory, (video), (slides), (abstract)
I will start by explaining the construction of a formal scheme starting with an integral affine manifold $$Q$$ equipped with a decomposition into Delzant polytopes. This is a weaker and more elementary version of degenerations of abelian varieties originally constructed by Mumford. Then I will reinterpret this construction using the corresponding Lagrangian torus fibration $$X\to Q$$ and relative Floer theory of its canonical Lagrangian section. Finally, I will discuss a conjectural generalization of the story to decompositions of CY symplectic manifolds into symplectic log CY's whose boundaries are "opened up".
• Feb. 11: Ely Kerman (UIUC), On symplectic capacities and their blind spots, (video), (slides), (abstract)
In this talk I will discuss a joint project with Yuanpu Liang in which we establish several properties of the sequence of symplectic capacities defined by Gutt and Hutchings for star-shaped domains using $$S^1$$-equivariant symplectic homology. Among the results discussed will be the fact that, unlike the first of these capacities, the others all fail to satisfy the symplectic version of the Brunn Minkowski established by Artstein-Avidan and Ostrover. We also show that the Gutt-Hutchings capacities, together with the volume, do not constitute a complete set of symplectic invariants even for convex bodies with smooth boundary. The examples constructed to prove these results are not exotic. They are convex and concave toric domains. The main new tool used is a significant simplification of the formulae of Gutt and Hutchings for the capacities of such domains, that holds under an additional symmetry assumption. This allows us to compute the capacities in new examples and to identify and exploit blind spots that they sometimes share.
• Jan. 28: Three 20min research talks
• Dustin Connery-Grigg (UdeM), Geometry and topology of Hamiltonian Floer complexes in low-dimension, (video), (slides), (abstract)
In this talk, I will present two results relating the qualitative dynamics of non-degenerate Hamiltonian isotopies on surfaces to the structure of their Floer complexes.

The first will be a topological characterization of those Floer chains which represent the fundamental class in $$CF_*(H,J)$$ and which moreover lie in the image of some chain-level PSS map. This leads to a novel symplectically bi-invariant norm on the group of Hamiltonian diffeomorphisms, which is both $$C^0$$-continuous and computable in terms of the underlying dynamics. The second result explains how certain portions of the Hamiltonian Floer chain complex may be interpreted geometrically in terms of positively transverse singular foliations of the mapping torus, with singular leaves given by certain maximal collections of unlinked orbits of the suspended flow. This construction may be seen to provide a Floer-theoretic construction of the `torsion-low’ foliations which appear in Le Calvez’s theory of transverse foliations for surface homeomorphisms, thereby establishing a bridge between the two theories.
• Pazit Haim-Kislev (Tel Aviv), Symplectic capacities of p-products, (video), (slides), (abstract)
A generalization of the cartesian product and the free sum of two convex domains is the p-product operation.
We investigate the behavior of symplectic capacities with respect to symplectic p-products, and we give applications related to Viterbo's volume-capacity conjecture and to p-decompositions of the symplectic ball.
• Thibaut Mazuir (Paris), Higher algebra of A-infinity algebras in Morse theory, (video), (slides), (abstract)
In this short talk, I will introduce the notion of n-morphisms between two A-infinity algebras. These higher morphisms are such that 0-morphisms correspond to standard A-infinity morphisms and 1-morphisms correspond to A-infinity homotopies. Their combinatorics are then encoded by new families of polytopes, which I call the n-multiplihedra and which generalize the standard multiplihedra.

Elaborating on works by Abouzaid and Mescher, I will then explain how this higher algebra of A-infinity algebras naturally arises in the context of Morse theory, using moduli spaces of perturbed Morse gradient trees.
• Jan. 21: Susan Tolman (UIUC), Beyond semitoric, (video), (slides), (abstract)
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.
• Jan. 14: Michael Sullivan (UMass Amherst), Quantitative Legendrian geometry, (video), (slides), (abstract)
I will discuss some quantitative aspects for Legendrians in a (more or less) general contact manifold. These include lower bounds on the number of Reeb chords between a Legendrian and its contact Hamiltonian image, the non-degeneracy of the Chekanov/Hofer/Shelukhin Legendrian metric, and some 3-dimensional non-squeezing results. The main tool is the barcode of a relative Rabinowitz Floer theory. This is joint work with Georgios Dimitroglou Rizell.

### --Fall 2021--

• Dec. 17: Three 20min research talks
• Wenyuan Li (Northwestern), Estimating Reeb chords using microlocal sheaf theory, (video), (slides), (abstract)
We show that, for closed Legendrians in 1-jet bundles, when there is a sheaf with singular support on the Legendrian, then (1) its self Reeb chords are bounded from below by half the sum of Betti numbers, and (2) the Reeb chords between itself and its Hamiltonian push off is bounded from below by Betti numbers when the $$C^0$$-norm of the Hamiltonian is small. I will show how to visualize Reeb chords/Lagrangian intersections in sheaf theory, and then explain the duality exact triangle and the persistence structure used in the proof. If time permits, I will state a conjecture on the relative Calabi-Yau structure that arises from the duality exact triangle.
• Jakob Hedicke (Bochum), Lorentzian distance functions on the group of contactomorphisms, (video), (slides), (abstract)
The notion of positive (non-negative) contact isotopy, defined by Eliashberg and Polterovich, leads to two relations on the group of contactomorphisms.
These relations resemble the causal relations of a Lorentzian manifold.
In this talk we will introduce a class of Lorentzian distance functions on the group of contactomorphisms, that are compatible with these relations.
The Lorentzian distance functions turn out to be continuous with respect to the Hofer-norm of a contactomorphism defined by Shelukhin.
• Johan Asplund (Uppsala), Simplicial descent for Chekanov-Eliashberg dg-algebras, (video), (slides), (abstract)
In this talk we introduce a type of surgery decomposition of Weinstein manifolds we call simplicial decompositions. We will discuss the result that the Chekanov-Eliashberg dg-algebra of the attaching spheres of a Weinstein manifold satisfies a descent (cosheaf) property with respect to a simplicial decomposition. Simplicial decompositions generalize the notion of Weinstein connected sum and there is in fact a one-to-one correspondence (up to Weinstein homotopy) between simplicial decompositions and so-called good sectorial covers. The motivation behind these results is the sectorial descent result for wrapped Fukaya categories by Ganatra-Pardon-Shende.
• Dec. 10: Urs Frauenfelder (Augsburg), GIT quotients and symplectic data analysis, (video), (abstract)
This is joint work with Agustin Moreno and Dayung Koh. The restricted three-body problem is invariant under various antisymplectic involutions. These real structures give rise to the notion of symmetric periodic orbits which simultaneously have a closed string interpretation namely as a periodic orbit as well as an open string interpretation as Hamiltonian chords. This makes the bifurcation analysis of symmetric periodic orbits very intriguing since under bifurcations two local Floer homologies are invariant, the periodic one as well as the Lagrangian one. In this talk we explain how methods from symmetric space theory can help to extract efficiently datas from reduced monodromy matrices of periodic orbits helping to analyse the possible bifurcation patterns.
• Nov. 26: Mohammed Abouzaid (Columbia), Complex cobordism and Hamiltonian fibrations, (video), (slides), (abstract)
I will discuss joint work with McLean and Smith, lifting the results of Seidel, Lalonde, McDuff, and Polterovich concerning the topology of Hamiltonian fibrations over the 2-sphere from rational cohomology to complex cobordism. In addition to the use of Morava K-theory (as in the recent work with Blumberg on the Arnold Conjecture), the essential new ingredient is the construction of global Kuranishi charts for genus 0 pseudo-holomorphic curves; i.e. their realisation as quotients of zero loci of sections of equivariant vector bundles on manifolds.
• Nov. 19: Fabio Gironella (HU Berlin), Exact orbifold fillings of contact manifolds, (video), (slides), (abstract)
The topic of the talk will be Floer theories on exact symplectic orbifolds with smooth contact boundary. More precisely, I will first describe the construction, which only uses classical transversality techniques, of a symplectic cohomology group on such symplectic orbifolds. Then, I will give some geometrical applications, such as restrictions on possible singularities of exact symplectic fillings of some particular contact manifolds, and the existence, in any odd dimension at least 5, of a pair of contact manifolds with no exact symplectic (smooth) cobordisms in either direction. This is joint work with Zhengyi Zhou.
• Nov. 5: Three 20min research talks
• Rohil Prasad (Princeton), The smooth closing lemma for area-preserving surface diffeomorphisms, (video), (slides), (abstract)
In this talk, I will discuss recent joint work with D. Cristofaro-Gardiner and B. Zhang showing that a generic area-preserving diffeomorphism of a closed surface has a dense set of periodic points. This follows from a result called a “smooth closing lemma” for area-preserving surface diffeomorphisms; this answers in the affirmative Smale’s 10th problem in the setting of area-preserving surface diffeomorphisms. The proof uses quantitative analysis of spectral invariants from periodic Floer homology via various estimates in Seiberg-Witten theory.
• Alex Pieloch (Columbia), Sections and unirulings of families over the projective line, (video), (slides), (abstract)
We will discuss the existence of rational (multi)sections and unirulings for projective families $$f: X \rightarrow P^1$$ with at most two singular fibres. Specifically, we will discuss two ingredients for constructing 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.
• Jimmy Chow (CUHK), Hofer geometry of coadjoint orbits and Peterson's theorem, (video), (slides), (abstract)
We will discuss a complete computation of Savelyev's homomorphism associated to any coadjoint orbit of a compact Lie group $$G$$, where the domain is restricted to the based loop homology of $$G$$. This gives at the same time some applications to the Hamiltonian groups of these spaces and a geometric proof of an unpublished theorem of Peterson. This theorem tells us explicitly how the multiplicative structure constants of the based loop homology of $$G$$ determine those of the quantum cohomology of its coadjoint orbits.
• Oct. 29: Yakov Eliashberg (Stanford), Detecting non-trivial elements in the spaces of Legendrian knots via Algebraic K-theory, (video), (slides), (abstract)
This talk is based on a joint work with Thomas Kragh.
Using the generating function theory we split inject homotopy groups of pseudo-isotopy and/or h-cobordism spaces into various spaces of Legendrian manifolds, e.g. the space of Legendrian unknots in $${\mathbb R}^{2n+1}$$ for a sufficiently large $$n$$. For instance, there is a non-trivial element in $$\pi_2$$ of the space of Legendrian unknots in $${\mathbb R}^{2n+1}$$ for $$n\geq 12$$.
• Oct. 22: Yaniv Ganor (Technion), Big fiber theorems and ideal-valued measures in symplectic topology, (video), (slides), (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.
• Oct. 15: Umberto Hryniewicz (RWTH Aachen), Results on abundance of global surfaces of section, (video), (slides), (abstract)
One might ask if global surfaces of section (GSS) for Reeb flows in dimension 3 are abundant in two different senses. One might ask if GSS are abundant for a given Reeb flow, or if Reeb flows carrying some GSS are abundant in the set of all Reeb flows. In this talk, answers to these two questions in specific contexts will be presented. First, I would like to discuss a result, obtained in collaboration with Florio, stating that there are explicit sets of Reeb flows on $$S^3$$ which are right-handed in the sense of Ghys; in particular, for such a flow all finite (non-empty) collections of periodic orbits spans a GSS. Then, I would like to discuss genericity results, obtained in collaboration with Colin, Dehornoy and Rechtman, for Reeb flows carrying a GSS; as a particular case of such results, we prove that a $$C^\infty$$-generic Reeb flow on the tight 3-sphere carries a GSS.
• Oct. 8: Three 20min research talks
• Jean-Philippe Chassé (UdeM), Convergence and Riemannian bounds on Lagrangian submanifolds, (video), (slides), (abstract)
Recent years have seen the appearance of a plethora of possible metrics on spaces of Lagrangian submanifolds. Indeed, on top of the better-known Lagrangian Hofer metric and spectral norm, Biran, Cornea, and Shelukhin have constructed families of so-called weighted fragmentation metrics on these spaces. I will explain how — under the presence of bounds coming from Riemannian geometry — all these metrics behave well with respect to the set-theoretic Hausdorff metric.
• Leo Digiosia (Rice), Cylindrical contact homology of links of simple singularities, (video), (slides), (abstract)
In this talk we consider the links of simple singularities, which are contactomoprhic to $$S^3/G$$ for finite subgroups $$G$$ of $$SU(2,C)$$. We explain how to compute the cylindrical contact homology of $$S^3/G$$ by means of perturbing the canonical contact form by a Morse function that is invariant under the corresponding rotation subgroup. We prove that the ranks are given in terms of the number of conjugacy classes of $$G$$, demonstrating a form of the McKay correspondence. We also explain how our computation realizes the Seifert fiber structure of these links.
• Rima Chatterjee (Cologne), Cabling of knots in overtwisted contact manifolds, (video), (slides), (abstract)
Knots associated to overtwisted manifolds are less explored. There are two types of knots in an overtwisted manifold – loose and non-loose. Non-loose knots are knots with tight complements whereas loose knots have overtwisted complements. While we understand loose knots, non-loose knots remain a mystery. The classification and structure problems of these knots vary greatly compared to the knots in tight manifolds. Especially we are interested in how satellite operations on a knot in overtwisted manifold changes the geometric property of the knot. In this talk, I will discuss under what conditions cabling operation on a non-loose knot preserves non-looseness. This is a joint work with Etnyre, Min and Mukherjee.

### --Spring 2021--

• Jul. 16: Joint with Berlin Mathematical School https://www.math-berlin.de/academics/bms-fridays
• Helmut Hofer (IAS), The Floer Jungle: 35 years of Floer Theory, (video), (slides-ppt), (slides-pdf), (abstract)
An exceptionally gifted mathematician and an extremely complex person, Floer exhibited, as one friend put it, a "radical individuality." He viewed the world around him with a singularly critical way of thinking and a quintessential disregard for convention. Indeed, his revolutionary mathematical ideas, contradicting conventional wisdom, could only be inspired by such impetus, and can only be understood in this context.

Poincaré's research on the Three Body Problem laid the foundations for the fields of dynamical systems and symplectic geometry. From whence the ancestral trail follows Marston Morse and Morse theory, Vladimir Arnold and the Arnold conjectures, through to breakthroughs by Yasha Eliashberg. Likewise, Charles Conley and Eduard Zehnder on the Arnold conjectures, Mikhail Gromov's theory of pseudoholomorphic curves, providing a new and powerful tool to study symplectic geometry, and Edward Witten's fresh perspective on Morse theory. And finally, Andreas Floer, who counter-intuitively combined all of this, hitting the "jackpot" with what is now called Floer theory.

https://math-berlin.de/images/poster/Hofer_2021-07.pdf
• Jul. 9: Laurent Côté (IAS/Harvard), Action filtrations associated to smooth categorical compactifications, (video), (slides), (abstract)
There is notion of a smooth categorical compactification of dg/A-infinity categories: for example, a smooth compactification of algebraic varieties induces a smooth categorical compactification of the associated bounded dg categories of coherent sheaves. In symplectic topology, wrapped Fukaya categories of Weinstein manifolds admit smooth compactifications by partially wrapped Fukaya categories. The goal of this talk is to explain how to associate an "action filtration" to a smooth categorical compactifications, which is invariant (up to appropriate equivalence) under zig-zags of smooth compactifications. I will then discuss applications to symplectic topology and categorical dynamics. This talk reports on joint work with Y. Baris Kartal.
• Jul. 2: Joint with Institut Henri Poincaré https://indico.math.cnrs.fr/event/5767/
• Felix Schlenk (UniNE), Symplectically knotted cubes, (video), (slides), (abstract)
While by a result of McDuff the space of symplectic embeddings of a closed 4-ball into an open 4-ball is connected, the situation for embeddings of cubes $$C^4 = D^2 \times D^2$$ is very different. For instance, for the open ball $$B^4$$ of capacity 1, there exists an explicit decreasing sequence $$c_1, c_2, \dots \to 1/3$$ such that for $$c \lt c_k$$ there are at least $$k$$ symplectic embeddings of the closed cube $$C^4(c)$$ of capacity $$c$$ into $$B^4$$ that are not isotopic. Furthermore, there are infinitely many non-isotopic symplectic embeddings of $$C^4(1/3)$$ into $$B^4$$.

A similar result holds for several other targets, like the open 4-cube, the complex projective plane, the product of two equal 2-spheres, or a monotone product of such manifolds and any closed monotone toric symplectic manifold.

The proof uses exotic Lagrangian tori.

This is joint work with Joé Brendel and Grisha Mikhalkin.
• Jun. 25: Three 20min research talks
• Mohan Swaminathan (Princeton), Super-rigidity and bifurcations of embedded curves in Calabi-Yau 3-folds, (video), (slides), (abstract)
I will describe my recent work, joint with Shaoyun Bai, which studies a class of bifurcations of moduli spaces of embedded pseudo-holomorphic curves in symplectic Calabi-Yau 3-folds and their associated obstruction bundles. As an application, we are able to give a direct definition of the Gopakumar-Vafa invariant in a special case.
• Ben Wormleighton (WashU), Lattice formulas for rational SFT capacities of toric domains, (video), (slides), (abstract)
Siegel has recently defined ‘higher’ symplectic capacities using rational SFT that obstruct symplectic embeddings and behave well with respect to stabilisation. I will report on joint work with Julian Chaidez that relates these capacities to algebro-geometric invariants, which leads to computable, combinatorial formulas for many convex toric domains.
• Jonathan Zung (Princeton), Reeb flows transverse to foliations, (video), (slides), (abstract)
Eliashberg and Thurston showed that $$C^2$$ taut foliations on 3-manifolds can be approximated by tight contact structures. I will explain a new approach to this theorem which allows one to control the resulting Reeb flow and hence produce many hypertight contact structures. Along the way, I will explain how harmonic transverse measures may be used to understand the holonomy of foliations.
• Jun. 18: Agustin Moreno (Uppsala), On the spatial restricted three-body problem, (video), (slides), (abstract)
In his search for closed orbits in the planar restricted three-body problem, Poincaré’s approach roughly reduces to:

(1) Finding a global surface of section;
(2) Proving a fixed-point theorem for the resulting return map.

This is the setting for the celebrated Poincaré-Birkhoff theorem. In this talk, I will discuss a generalization of this program to the spatial problem.

For the first step, we obtain the existence of global hypersurfaces of section for which the return maps are Hamiltonian, valid for energies below the first critical value and all mass ratios. For the second, we prove a higher-dimensional version of the Poincaré-Birkhoff theorem, which gives infinitely many orbits of arbitrary large period, provided a suitable twist condition is satisfied. Time permitting, we also discuss a construction that associates a Reeb dynamics on a moduli space of holomorphic curves (a copy of the three-sphere), to the given dynamics, and its properties.

This is based on joint work with Otto van Koert.
• Jun. 11: Francisco Presas (ICMAT), The homotopy type of the space of tight contact structures and the overtwisted mirage, (video), (slides), (abstract)
We compute the homotopy type of any connected component of the space of tight contact structures on a 3-fold. In fact, we actually prove a partial h-principle for the inclusion of the contactomorphism group into the diffeomorphism group. The basic building block is the homotopy equivalence induced by the inclusion of the contactomorphism group of the sphere relative to a point and the diffeomorphism group relative to a point result recently proven by Elisahberg and Mishachev.

Then, we wonder how these sets of techniques work for overtwisted manifolds? i.e. we just try to prove the same theorem than in the tight case, assuming that the triangulation is very small, we easily obtain that all the cells are tight and then, everything looks like working, so however, there must be something wrong because we find several contradictions: the overtwisted mirage. Once the mistake is understood, we proceed to compute the homotopy type of the space of contact structures/contactomorphisms by using just Mishachev-Eliiashberg result, i.e. we reprove the 3-dimensional overtiwsted h-principle as a corollary. We will compute the space of embeddings of overtwisted disks in some particular manifolds. Finally we end by explaining the conjecture tight $$\Longleftrightarrow$$ overtwisted.

This is j/w with Dahyana Farias, Eduardo Fernández and Xabi Martínez
• Jun. 4: Simion Filip (Chicago), Degenerations of Kahler forms on K3 surfaces, and some dynamics, (video), (slides), (abstract)
K3 surfaces have a rich geometry and admit interesting holomorphic automorphisms. As examples of Calabi-Yau manifolds, they admit Ricci-flat Kähler metrics, and a lot of attention has been devoted to how these metrics degenerate as the Kähler class approaches natural boundaries. I will discuss how to use the full automorphism group to analyze the degenerations and obtain certain canonical objects (closed positive currents) on the boundary. While most of the previous work was devoted to degenerating the metric along an elliptic fibration (motivated by the SYZ picture of mirror symmetry) I will discuss how to analyze all the other points. Time permitting, I will also describe the construction of canonical heights on K3 surfaces (in the sense of number theory), generalizing constructions due to Silverman and Tate.
Joint work with Valentino Tosatti.
• May 28: Three 20min research talks
• Oğuz Şavk (Boğaziçi University), Classical and new plumbings bounding contractible manifolds and homology balls, (video), (slides), (abstract)
A central problem in low-dimensional topology asks which homology 3-spheres bound contractible 4-manifolds and homology 4-balls. In this talk, we address this problem for plumbed 3-manifolds and we present the classical and new results together. Along the way, we touch symplectic geometry by using the classical results of Eliashberg and Gompf. Our approach is based on Mazur’s famous argument which provides a unification of all results.
• Irene Seifert (Heidelberg), Periodic delay orbits and the polyfold IFT, (video), (slides), (abstract)
Differential delay equations arise very naturally, but they are much more complicated than ordinary differential equations. Polyfold theory, originally developed for the study of moduli spaces of pseudoholomorphic curves, can help to understand solutions of certain delay equations. In my talk, I will show an existence result about periodic delay orbits with small delay. If time permits, we can discuss possible further applications of polyfold theory to the differential delay equations. This is joint work with Peter Albers.
• Hang Yuan (Stony Brook), Disk counting via family Floer theory, (video), (slides), (abstract)
Given a family of Lagrangian tori with full quantum corrections, the non-archimedean SYZ mirror construction uses the family Floer theory to construct a non-archimedean analytic space with a global superpotential. In this talk, we will first briefly review the construction. Then, we will apply it to the Gross’s fibrations. As an application, we can compute all the non-trivial open GW invariants for a Chekanov-type torus in $$CP^n$$ or $$CP^r \times CP^{n-r}$$. When $$n=2$$, $$r=1$$, we retrieve the previous results of Auroux an Chekanov-Schlenk without finding the disks explicitly. It is also compatible with the Pascaleff-Tonkonog’s work on Lagrangian mutations.
• May 21: Cancelled on account of Advances in Symplectic Topology, https://indico.math.cnrs.fr/event/5787/
• May 14: Daniel Álvarez-Gavela (MIT), Caustics of Lagrangian homotopy spheres with stably trivial Gauss map, (video), (slides), (abstract)
The h-principle for the simplification of caustics (i.e. Lagrangian tangencies) reduces a geometric problem to a homotopical problem. In this talk I will explain the solution to this homotopical problem in the case of spheres. More precisely, I will show that the stably trivial elements of the $$n$$th homotopy group of the Lagrangian Grassmannian $$U_n/O_n$$, which lies in the metastable range, admit representatives with only fold type tangencies. By the h-principle, it follows that if $$D$$ is a Lagrangian distribution defined along a Lagrangian homotopy sphere $$L$$, then there exists a Hamiltonian isotopy which simplifies the tangencies between $$L$$ and $$D$$ to consist only of folds if and only if $$D$$ is stably trivial. I will give two applications of this result, one to the arborealization program and another to the study of nearby Lagrangian homotopy spheres. Joint work with David Darrow (in the form of an undergraduate research project).
• May 7: Laura Starkston (UC Davis), Unexpected fillings, singularities, and plane curve arrangements, (video), (slides), (abstract)
I will discuss joint work with Olga Plamenevskaya studying symplectic fillings of links of certain complex surface singularities, and comparing symplectic fillings with complex smoothings. We develop characterizations of the symplectic fillings using planar Lefschetz fibrations and singular braided surfaces. This provides an analogue of de Jong and van Straten's work which characterizes the complex smoothings in terms of decorated complex plane curves. We find differences between symplectic fillings and complex smoothings that had not previously been found in rational complex surface singularities.
• Apr. 30: Cancelled on account of the conference From Hamiltonian Systems to Symplectic Topology and Beyond, https://indico.math.cnrs.fr/event/5786/
• Apr. 23: Cancelled on account of the Spring School on Symplectic and Contact Topology, https://conferences.cirm-math.fr/2329.html
• Apr. 16: Three 20min research talks
• Maxim Jeffs (Harvard), Mirror symmetry and Fukaya categories of singular varieties, (video), (slides), (abstract)
In this talk I will explain Auroux' definition of the Fukaya category of a singular hypersurface and two results about this definition, illustrated with some examples. The first result is that Auroux' category is equivalent to the Fukaya-Seidel category of a Landau-Ginzburg model on a smooth variety; the second result is a homological mirror symmetry equivalence at certain large complex structure limits. I will also discuss ongoing work on generalizations.
• Côme Dattin (Nantes), Wrapped sutured Legendrian homology and the conormal of braids, (video), (slides), (abstract)
In this talk we will discuss invariants of sutured Legendrians. A sutured contact manifold can be seen as either generalizing the contactisation of a Liouville domain, or as a presentation of a contact manifold with convex boundary. Using the first point of view, we define the wrapped sutured homology of Legendrians with boundary, employing ideas coming from Floer theory. To illustrate the second aspect, we apply the unit conormal construction to braids with two strands, which yields a sutured Legendrian. We will show that, if the conormals of two 2-braids are Legendrian isotopic, then the braids are equivalent.
• Bingyu Zhang (Institut Fourier, Université Grenoble Alpes), Capacities from the Chiu-Tamarkin complex, (video), (slides), (abstract)
In this talk, we will discuss the Chiu-Tamarkin complex. It is a symplectic/contact invariant that comes from the microlocal sheaf theory. I will explain how to define some capacities using the Chiu-Tamarkin complex in both symplectic and contact situations. The main result is the structure theorem of the Chiu-Tamarkin complex of convex toric domains. Consequently, we can compute the capacities of convex toric domains.
• Apr. 9: Sara Tukachinsky (IAS), Relative quantum cohomology and other stories, (video), (slides), (abstract)
We define a quantum product on the cohomology of a symplectic manifold relative to a Lagrangian submanifold, with coefficients in a Novikov ring. The associativity of this product is equivalent to an open version of the WDVV equations for an appropriate disk superpotential. Both structures — the quantum product and the WDVV equations — are consequences of a more general structure we call the tensor potential, which will be the main focus of this talk. This is joint work with Jake Solomon.
• Apr. 2: Sheel Ganatra (USC), Categorical non-properness in wrapped Floer theory, (video), (slides), (abstract)
In all known explicit computations on Weinstein manifolds, the self-wrapped Floer homology of non-compact exact Lagrangian is always either infinite-dimensional or zero. We will explain why a global variant of this observed phenomenon holds in broad generality: the wrapped Fukaya category of any positive-dimensional Weinstein (or non-degenerate Liouville) manifold is always either non-proper or zero, as is any quotient thereof. Moreover any non-compact connected exact Lagrangian is always either a "non-proper object" or zero in such a wrapped Fukaya category, as is any idempotent summand thereof. We will also examine where the argument could break if one drops exactness, which is consistent with known computations of non-exact wrapped Fukaya categories which are smooth, proper, and non-vanishing (e.g., work of Ritter-Smith).

### --Winter 2021--

• Mar. 26: Three 20min research talks
• Jesse Huang (UIUC), Variation of FLTZ skeleta, (video), (slides), (abstract)
In this short talk, I will discuss an interpolation of FLTZ skeleta mirror to derived equivalent toric varieties. This is joint work with Peng Zhou.
• Shaoyun Bai (Princeton), SU(n)–Casson invariants and symplectic geometry, (video), (slides), (abstract)
In 1985, Casson introduced an invariant of integer homology 3-spheres by counting SU(2)-representations of the fundamental groups. The generalization of Casson invariant by considering Lie groups SU(n) has been long expected, but the original construction of Casson encounters some difficulties. I will present a solution to this problem, highlighting the equivariant symplectic geometry and Atiyah-Floer type result entering the construction.
• Thomas Melistas (UGA), The Large-Scale Geometry of Overtwisted Contact Forms, (video), (slides), (abstract)
Inspired by the symplectic Banach-Mazur distance, proposed by Ostrover and Polterovich in the setting of non-degenerate starshaped domains of Liouville manifolds, we define a distance on the space of contact forms supporting a given contact structure on a closed contact manifold and we use it to bi-Lipschitz embed part of the 2-dimensional Euclidean space into the space of overtwisted contact forms supporting a given contact structure on a smooth closed manifold.
• Mar. 19: Egor Shelukhin (UdeM), Lagrangian configurations and Hamiltonian maps, (video), (slides), (abstract)
We study configurations of disjoint Lagrangian submanifolds in certain low-dimensional symplectic manifolds from the perspective of the geometry of Hamiltonian maps. We detect infinite-dimensional flats in the Hamiltonian group of the two-sphere equipped with Hofer's metric, showing in particular that this group is not quasi-isometric to a line. This answers a well-known question of Kapovich-Polterovich from 2006. We show that these flats in $$Ham(S^2)$$ stabilize to certain product four-manifolds, prove constraints on Lagrangian packing, find new instances of Lagrangian Poincare recurrence, and present a new hierarchy of normal subgroups of area-preserving homeomorphisms of the two-sphere. The technology involves Lagrangian spectral invariants with Hamiltonian term in symmetric product orbifolds. This is joint work with Leonid Polterovich.
• Mar. 12: Oleg Lazarev (Harvard), Inverting primes in Weinstein geometry, (video), (slides), (abstract)
A classical construction in topology associates to a space $$X$$ and prime $$p$$, a new "localized" space $$X_p$$ whose homotopy and homology groups are obtained from those of $$X$$ by inverting $$p$$. In this talk, I will discuss a symplectic analog of this construction, extending work of Abouzaid-Seidel and Cieliebak-Eliashberg on flexible Weinstein structures. Concretely, I will produce prime-localized Weinstein subdomains of high-dimensional Weinstein domains and also show that any Weinstein subdomain of a cotangent bundle agrees Fukaya-categorically with one of these special subdomains. The key will be to classify which objects of the Fukaya category of $$T^*M$$ – twisted complexes of Lagrangians – are quasi-isomorphic to actual Lagrangians. This talk is based on joint work with Z. Sylvan.
• Mar. 5: Sobhan Seyfaddini (Paris), Periodic Floer homology and the large-scale geometry of Hofer's metric on the sphere, (video), (slides), (abstract)
The group of Hamiltonian diffeomorphisms of a symplectic manifold admits a remarkable bi-invariant metric, called Hofer’s metric. My talk will be about a recent joint work with Dan Cristofaro-Gardiner and Vincent Humilière resolving the following two open-questions related to the large-scale geometry of this metric. The first, due to Kapovich and Polterovich, asks whether the two-sphere, equipped with Hofer’s metric, is quasi-isometric to the real line; we show that it is not. The second, due to Fathi, asks whether the group of area and orientation preserving homeomorphisms of the two-sphere is a simple group; we show that it is not. Key to our proofs is a new sequence of spectral invariants defined via Hutchings’ Periodic Floer Homology.
• For two somewhat related talks by Rémi Leclercq and Vincent Humilière on Mar. 6, see the link: https://dms.umontreal.ca/~cornea/MicroC0.
• Feb. 26: Generating Functions Day
• 9:15am EST: Sylvain Courte (Université Grenoble Alpes), Twisted generating functions and the nearby Lagrangian conjecture, (video), (slides), (abstract)
I will explain the notion of twisted generating function and show that a closed exact Lagrangian submanifold L in the cotangent bundle of M admits such a thing. The type of function arising in our construction is related to Waldhausen's tube space from his manifold approach to algebraic K-theory of spaces. Using the rational equivalence of this space with BO, as proved by Bökstedt, we conclude that the stable Lagrangian Gauss map of L vanishes on all homotopy groups. In particular when M is a homotopy sphere, we obtain the triviality of the stable Lagrangian Gauss map and a genuine generating function for L. This is a joint work with M. Abouzaid, S. Guillermou and T. Kragh.
• 12pm EST, at the WHVSS: Simon Allais (ENS Lyon), Periodic points of Hamiltonian diffeomorphisms and generating functions, (abstract)
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.
• 3pm EST, at the WHVSS: Yael Karshon (Toronto University), Non-linear Maslov index on lens spaces, (abstract)
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.
• Feb. 19: Daniel Pomerleano (UMass Boston), Intrinsic mirror symmetry and categorical crepant resolutions, (video), (slides), (abstract)
Gross and Siebert have recently proposed an "intrinsic" programme for studying mirror symmetry. In this talk, we will discuss a symplectic interpretation of some of their ideas in the setting of affine log Calabi-Yau varieties. Namely, we describe work in progress which shows that, under suitable assumptions, the wrapped Fukaya category of such a variety $$X$$ gives an intrinsic "categorical crepant resolution" of $$Spec(SH^0(X))$$. No background in mirror symmetry will be assumed for the talk.
• Feb. 12: Cheuk Yu Mak (Edinburgh), Non-displaceable Lagrangian links in four-manifolds, (video), (slides), (abstract)
One of the earliest fundamental applications of Lagrangian Floer theory is detecting the non-displaceablity of a Lagrangian submanifold. Many progress and generalisations have been made since then but little is known when the Lagrangian submanifold is disconnected. In this talk, we describe a new idea to address this problem. Subsequently, we explain how to use Fukaya-Oh-Ohta-Ono and Cho-Poddar theory to show that for every $$S^2 \times S^2$$ with a non-monotone product symplectic form, there is a continuum of disconnected, non-displaceable Lagrangian submanifolds such that each connected component is displaceable. This is a joint work with Ivan Smith.
• Feb. 5: Yusuf Barış Kartal (Princeton), Algebraic torus actions on Fukaya categories, (video), (slides), (abstract)
The purpose of this talk is to explore how Lagrangian Floer homology groups change under (non-Hamiltonian) symplectic isotopies on a (negatively) monotone symplectic manifold $$(M,\omega)$$ satisfying a strong non-degeneracy condition. More precisely, given two Lagrangian branes $$L,L',$$ consider family of Floer homology groups $$HF(\phi_v(L),L')$$, where $$v\in H^1(M,\mathbb R)$$ and $$\phi_v$$ is the time-1 map of a symplectic isotopy with flux $$v$$. We show how to fit this collection into an algebraic sheaf over the algebraic torus $$H^1(M,\mathbb G_m)$$. The main tool is the construction of an "algebraic action" of $$H^1(M,\mathbb G_m)$$ on the Fukaya category. As an application, we deduce the change in Floer homology groups satisfy various tameness properties, for instance, the dimension is constant outside an algebraic subset of $$H^1(M,\mathbb G_m)$$. Similarly, given closed $$1$$-form $$\alpha$$, which generates a symplectic isotopy denoted by $$\phi_\alpha^t$$, the Floer homology groups $$HF(\phi_\alpha^t(L),L')$$ have rank that is constant in $$t$$, with finitely many possible exceptions.