Information up to (and including) January 2021 can be found on the original seminar webpage at https://dms.umontreal.ca/~cornea/Seminar.html
Next session – Jan. 28: Three 20min research talks

Dustin ConneryGrigg (UdeM)
Title: Geometry and topology of Hamiltonian Floer complexes in lowdimension
In this talk, I will present two results relating the qualitative dynamics of nondegenerate 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 chainlevel PSS map. This leads to a novel symplectically biinvariant 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 Floertheoretic construction of the `torsionlow’ foliations which appear in Le Calvez’s theory of transverse foliations for surface homeomorphisms, thereby establishing a bridge between the two theories. 
Pazit HaimKislev (Tel Aviv)
Title: Symplectic capacities of pproducts
A generalization of the cartesian product and the free sum of two convex domains is the pproduct operation.
We investigate the behavior of symplectic capacities with respect to symplectic pproducts, and we give applications related to Viterbo's volumecapacity conjecture and to pdecompositions of the symplectic ball. 
Thibaut Mazuir (Paris)
Title: Higher algebra of Ainfinity algebras in Morse theory
In this short talk, I will introduce the notion of nmorphisms between two Ainfinity algebras. These higher morphisms are such that 0morphisms correspond to standard Ainfinity morphisms and 1morphisms correspond to Ainfinity homotopies. Their combinatorics are then encoded by new families of polytopes, which I call the nmultiplihedra and which generalize the standard multiplihedra.
Elaborating on works by Abouzaid and Mescher, I will then explain how this higher algebra of Ainfinity algebras naturally arises in the context of Morse theory, using moduli spaces of perturbed Morse gradient trees.
About the seminar
 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.
 Once a month 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).
 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).
 Please do not hesitate to ask questions.
 We post links to the slides of the talks as well as links to recordings of the talks.
 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

Feb. 11: Ely Kerman (UIUC),
On symplectic capacities and their blind spots,
(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 starshaped 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 ArtsteinAvidan and Ostrover. We also show that the GuttHutchings 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.
 Feb. 18: Umut Varolgunes (Boğaziçi), TBA
 Feb. 25: Erman Cineli (Paris), TBA
 Mar. 4: MarieClaude Arnaud (Paris), TBA
 Mar. 18: TBD

Mar. 25: Three 20min research talks
 Benoît Joly (Bochum), TBA
 TBD
 TBD
 Apr. 8: TBD
 Apr. 15: TBD
 Apr. 22: TBD
 Apr. 29: TBD
 May 6: TBD
 May 20: TBD
 May 27: TBD
 Jun. 10: TBD
 Jun. 17: TBD
 Jun. 24: TBD
 Jul. 8: TBD
 Jul. 15: TBD
Past talks
Winter 2022

Jan. 21: Susan Tolman (UIUC),
Beyond semitoric,
(video),
(abstract)
A compact four dimensional completely integrable system \(f:M \to R^2\) is semitoric if it has only nondegenerate 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 nondegeneracy of the Chekanov/Hofer/Shelukhin Legendrian metric, and some 3dimensional nonsqueezing 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 1jet 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 CalabiYau 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 (nonnegative) 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 Hofernorm of a contactomorphism defined by Shelukhin. 
Johan Asplund (Uppsala),
Simplicial descent for ChekanovEliashberg dgalgebras,
(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 ChekanovEliashberg dgalgebra 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 onetoone correspondence (up to Weinstein homotopy) between simplicial decompositions and socalled good sectorial covers. The motivation behind these results is the sectorial descent result for wrapped Fukaya categories by GanatraPardonShende.

Wenyuan Li (Northwestern),
Estimating Reeb chords using microlocal sheaf theory,
(video),
(slides),
(abstract)

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 threebody 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 2sphere from rational cohomology to complex cobordism. In addition to the use of Morava Ktheory (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 pseudoholomorphic 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 areapreserving surface diffeomorphisms,
(video),
(slides),
(abstract)
In this talk, I will discuss recent joint work with D. CristofaroGardiner and B. Zhang showing that a generic areapreserving diffeomorphism of a closed surface has a dense set of periodic points. This follows from a result called a “smooth closing lemma” for areapreserving surface diffeomorphisms; this answers in the affirmative Smale’s 10th problem in the setting of areapreserving surface diffeomorphisms. The proof uses quantitative analysis of spectral invariants from periodic Floer homology via various estimates in SeibergWitten 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.

Rohil Prasad (Princeton),
The smooth closing lemma for areapreserving surface diffeomorphisms,
(video),
(slides),
(abstract)

Oct. 29: Yakov Eliashberg (Stanford),
Detecting nontrivial elements in the spaces of Legendrian knots via Algebraic Ktheory,
(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 pseudoisotopy and/or hcobordism 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 nontrivial 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 idealvalued 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 idealvalued 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 righthanded in the sense of Ghys; in particular, for such a flow all finite (nonempty) 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 3sphere carries a GSS.

Oct. 8: Three 20min research talks

JeanPhilippe 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 betterknown Lagrangian Hofer metric and spectral norm, Biran, Cornea, and Shelukhin have constructed families of socalled 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 settheoretic 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 nonloose. Nonloose knots are knots with tight complements whereas loose knots have overtwisted complements. While we understand loose knots, nonloose 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 nonloose knot preserves nonlooseness. This is a joint work with Etnyre, Min and Mukherjee.

JeanPhilippe Chassé (UdeM),
Convergence and Riemannian bounds on Lagrangian submanifolds,
(video),
(slides),
(abstract)
Spring 2021

Jul. 16: Joint with Berlin Mathematical School https://www.mathberlin.de/academics/bmsfridays

Helmut Hofer (IAS),
The Floer Jungle: 35 years of Floer Theory,
(video),
(slidesppt),
(slidespdf),
(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 counterintuitively combined all of this, hitting the "jackpot" with what is now called Floer theory.
https://mathberlin.de/images/poster/Hofer_202107.pdf

Helmut Hofer (IAS),
The Floer Jungle: 35 years of Floer Theory,
(video),
(slidesppt),
(slidespdf),
(abstract)

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/Ainfinity 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 zigzags 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 4ball into an open 4ball 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 nonisotopic symplectic embeddings of \(C^4(1/3)\) into \(B^4\).
A similar result holds for several other targets, like the open 4cube, the complex projective plane, the product of two equal 2spheres, 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.

Felix Schlenk (UniNE),
Symplectically knotted cubes,
(video),
(slides),
(abstract)

Jun. 25: Three 20min research talks

Mohan Swaminathan (Princeton),
Superrigidity and bifurcations of embedded curves in CalabiYau 3folds,
(video),
(slides),
(abstract)
I will describe my recent work, joint with Shaoyun Bai, which studies a class of bifurcations of moduli spaces of embedded pseudoholomorphic curves in symplectic CalabiYau 3folds and their associated obstruction bundles. As an application, we are able to give a direct definition of the GopakumarVafa 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 algebrogeometric 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 3manifolds 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.

Mohan Swaminathan (Princeton),
Superrigidity and bifurcations of embedded curves in CalabiYau 3folds,
(video),
(slides),
(abstract)

Jun. 18: Agustin Moreno (Uppsala),
On the spatial restricted threebody problem,
(video),
(slides),
(abstract)
In his search for closed orbits in the planar restricted threebody problem, Poincaré’s approach roughly reduces to:
(1) Finding a global surface of section;
(2) Proving a fixedpoint theorem for the resulting return map.
This is the setting for the celebrated PoincaréBirkhoff 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 higherdimensional version of the PoincaréBirkhoff 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 threesphere), 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 3fold. In fact, we actually prove a partial hprinciple 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 MishachevEliiashberg result, i.e. we reprove the 3dimensional overtiwsted hprinciple 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 CalabiYau manifolds, they admit Ricciflat 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 lowdimensional topology asks which homology 3spheres bound contractible 4manifolds and homology 4balls. In this talk, we address this problem for plumbed 3manifolds 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 nonarchimedean SYZ mirror construction uses the family Floer theory to construct a nonarchimedean 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 nontrivial open GW invariants for a Chekanovtype torus in \(CP^n\) or \(CP^r \times CP^{nr}\). When \(n=2\), \(r=1\), we retrieve the previous results of Auroux an ChekanovSchlenk without finding the disks explicitly. It is also compatible with the PascaleffTonkonog’s work on Lagrangian mutations.

Oğuz Şavk (Boğaziçi University),
Classical and new plumbings bounding contractible manifolds and homology balls,
(video),
(slides),
(abstract)
 May 21: Cancelled on account of Advances in Symplectic Topology, https://indico.math.cnrs.fr/event/5787/

May 14: Daniel ÁlvarezGavela (MIT),
Caustics of Lagrangian homotopy spheres with stably trivial Gauss map,
(video),
(slides),
(abstract)
The hprinciple 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 hprinciple, 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.cirmmath.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 FukayaSeidel category of a LandauGinzburg 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 2braids are Legendrian isotopic, then the braids are equivalent.

Bingyu Zhang (Institut Fourier, Université Grenoble Alpes),
Capacities from the ChiuTamarkin complex,
(video),
(slides),
(abstract)
In this talk, we will discuss the ChiuTamarkin complex. It is a symplectic/contact invariant that comes from the microlocal sheaf theory. I will explain how to define some capacities using the ChiuTamarkin complex in both symplectic and contact situations. The main result is the structure theorem of the ChiuTamarkin complex of convex toric domains. Consequently, we can compute the capacities of convex toric domains.

Maxim Jeffs (Harvard),
Mirror symmetry and Fukaya categories of singular varieties,
(video),
(slides),
(abstract)

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 nonproperness in wrapped Floer theory,
(video),
(slides),
(abstract)
In all known explicit computations on Weinstein manifolds, the selfwrapped Floer homology of noncompact exact Lagrangian is always either infinitedimensional or zero. We will explain why a global variant of this observed phenomenon holds in broad generality: the wrapped Fukaya category of any positivedimensional Weinstein (or nondegenerate Liouville) manifold is always either nonproper or zero, as is any quotient thereof. Moreover any noncompact connected exact Lagrangian is always either a "nonproper 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 nonexact wrapped Fukaya categories which are smooth, proper, and nonvanishing (e.g., work of RitterSmith).
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 3spheres 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 AtiyahFloer type result entering the construction.

Thomas Melistas (UGA),
The LargeScale Geometry of Overtwisted Contact Forms,
(video),
(slides),
(abstract)
Inspired by the symplectic BanachMazur distance, proposed by Ostrover and Polterovich in the setting of nondegenerate 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 biLipschitz embed part of the 2dimensional Euclidean space into the space of overtwisted contact forms supporting a given contact structure on a smooth closed manifold.

Jesse Huang (UIUC),
Variation of FLTZ skeleta,
(video),
(slides),
(abstract)

Mar. 19: Egor Shelukhin (UdeM),
Lagrangian configurations and Hamiltonian maps,
(video),
(slides),
(abstract)
We study configurations of disjoint Lagrangian submanifolds in certain lowdimensional symplectic manifolds from the perspective of the geometry of Hamiltonian maps. We detect infinitedimensional flats in the Hamiltonian group of the twosphere equipped with Hofer's metric, showing in particular that this group is not quasiisometric to a line. This answers a wellknown question of KapovichPolterovich from 2006. We show that these flats in \(Ham(S^2)\) stabilize to certain product fourmanifolds, prove constraints on Lagrangian packing, find new instances of Lagrangian Poincare recurrence, and present a new hierarchy of normal subgroups of areapreserving homeomorphisms of the twosphere. 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 AbouzaidSeidel and CieliebakEliashberg on flexible Weinstein structures. Concretely, I will produce primelocalized Weinstein subdomains of highdimensional Weinstein domains and also show that any Weinstein subdomain of a cotangent bundle agrees Fukayacategorically 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 quasiisomorphic to actual Lagrangians. This talk is based on joint work with Z. Sylvan.

Mar. 5: Sobhan Seyfaddini (Paris),
Periodic Floer homology and the largescale geometry of Hofer's metric on the sphere,
(video),
(slides),
(abstract)
The group of Hamiltonian diffeomorphisms of a symplectic manifold admits a remarkable biinvariant metric, called Hofer’s metric. My talk will be about a recent joint work with Dan CristofaroGardiner and Vincent Humilière resolving the following two openquestions related to the largescale geometry of this metric. The first, due to Kapovich and Polterovich, asks whether the twosphere, equipped with Hofer’s metric, is quasiisometric 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 twosphere 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 Ktheory 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 pseudorotation are isolated as an invariant set. In 2019, Shelukhin proved a homology version of the HoferZehnder 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),
Nonlinear Maslov index on lens spaces,
(abstract)
Let L be a lens space with its standard contact structure. We use generating functions to construct a "nonlinear 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.

9:15am EST: Sylvain Courte (Université Grenoble Alpes),
Twisted generating functions and the nearby Lagrangian conjecture,
(video),
(slides),
(abstract)

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 CalabiYau 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),
Nondisplaceable Lagrangian links in fourmanifolds,
(video),
(slides),
(abstract)
One of the earliest fundamental applications of Lagrangian Floer theory is detecting the nondisplaceablity 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 FukayaOhOhtaOno and ChoPoddar theory to show that for every \(S^2 \times S^2\) with a nonmonotone product symplectic form, there is a continuum of disconnected, nondisplaceable 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 (nonHamiltonian) symplectic isotopies on a (negatively) monotone symplectic manifold \((M,\omega)\) satisfying a strong nondegeneracy 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 time1 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.
Earlier
 March 27, 2020 – January 29, 2021: https://dms.umontreal.ca/~cornea/Seminar.html