## GeNeSys: Geneva-Neuchâtel Symplectic geometry seminar

**2022, October 18, Université de Genève, salle 6-13**

Ilia Itenberg (Sorbonne) Real enumerative invariants and their refinement Salle 6-13, 14h15

**Abstract:**

The talk is devoted to several real and tropical enumerative problems. We suggest new invariants of the projective plane (and, more generally, of toric surfaces) that arise as results of an appropriate enumeration of real elliptic curves. These invariants admit a refinement (according to the quantum index) similar to the one introduced by Grigory Mikhalkin in the rational case. We discuss the combinatorics of tropical counterparts of the elliptic invariants under consideration and establish a tropical algorithm allowing one to compute them. This is a joint work with Eugenii Shustin.

**2022, September 27, Tuesday, Université de Neuchâtel**

Richard Hind (University of Notre Dame) Obstructing Lagrangian isotopies Room B107, 14:00

**Abstract:**

I will describe some obstructions to the existence of Lagrangian tori in subsets of Euclidean space, and also to isotopies between the tori. The obstructions come from holomorphic curves and In simple situations are sharp. As a consequence we can derive obstructions to certain 4 dimensional symplectic embeddings, which turn out not to be especially strong, but the analysis does lead to precise statements about stabilized ellipsoid embeddings. Results are taken from joint works with Emmanuel Opshtein, Jun Zhang and Kyler Siegel and Dan Cristofaro-Gardiner.

Joé Brendel (Université de Neuchâtel and Tel Aviv University) Lagrangian tori in S^2 x S^2 Room E213, 16:00

**Abstract:**

There is an obvious family of Lagrangian tori in $S^2 \times S^2$, namely those obtained as a product of circles in the factors. We discuss the classification of such product tori up to symplectomorphisms and note that the non-monotone case is qualitatively very different from the monotone one. In the proof, we use a symmetric version of McDuff's probes. The resulting classification can be used to tackle many related questions: Which of the above tori are the image of a product torus in a ball under a Darboux embedding? What is the Hamiltonian monodromy group of the product tori? How many disjoint copies (up to Hamiltonian isotopy) of a given product torus can be packed into the ambient space? Why does the Lagrangian analogue of the flux conjecture fail so badly? If time permits we will say something about exotic tori, i.e. tori which are not symplectomorphic to product tori. This is partially based on joint work with Joontae Kim.

2022, July 15, Friday, Université de Genève

Kyler Siegel (University of Southern California) "On the symplectic complexity of affine varieties"

Abstract

Symplectic topology is a framework for studying global features of spaces, lying somewhere between differential topology and algebraic geometry in terms of flexibility versus rigidity. In this talk we introduce a new notion of “symplectic complexity” for smooth complex affine varieties. This captures purely symplectic features which are different from classical topological invariants such as homology, and it also goes beyond the standard usage of Floer theory. As our main application, we study symplectic embeddings between divisor complements in complex projective space, giving a complete characterization in many cases.

Welcome and lunch. 11h-14h

Lecture 1. 14h-15h

Lecture 2. 15h30.

2022, April 29, Friday, Université de Neuchâtel

Yakov Eliashberg (Stanford, visiting ITS-Zürich) "Honda-Huang's work on contact convexity revisited"

Abstract

Two years ago Ko Honda and Yang Huang proved a series of remarkable results concerning contact convexity in high dimension. Unfortunately, their proof is extremely involved and not easy to follow. I will explain in the talk another proof, joint with Dishant Pancholi. While it follows the same overall strategy as Honda-Huang’s proof, it is drastically simpler in its implementation.

Welcome and lunch. 12h-14h

Lecture 1. 14h-15h

Lecture 2. 15h30.

2022, January 24, Monday, Université de Genève, Rue du Conseil-Général 7-9 1205 Geneva, Room 6-13

12h00 — Welcome and lunch 14h00 — Umut Varolgunes (University of Edinburgh and Bogazici University) Symplectic degenerations, relative Floer theory and Reynaud models 15h00 — coffee/tea break 16h00 — John Alexander Cruz Morales (Universidad Nacional de Colombia and Max-Planck-Institut-fur-Mathematik) Towards a Dubrovin conjecture for Frobenius manifolds 17h00 - Discussions

Umut Varolgunes.
**Symplectic degenerations, relative Floer theory and Reynaud models**

I will start by explaining the construction of a formal scheme starting with an integral affine manifold Q equipped with a decomposition into convex 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 induced Lagrangian torus fibration X\to Q and the relative Floer theory of its canonical Lagrangian section. Finally, I will discuss a conjectural generalization of the story to symplectic degenerations of CY symplectic manifolds to normal crossing symplectic log CY varieties.

John Alexander Cruz Morales.
**Towards a Dubrovin conjecture for Frobenius manifolds**

In this talk we will report an ongoing work aiming to establish a Dubrovin conjecture for general Frobenius manifolds. Dubrovin conjecture was formulated in 1998 (with a very precise statement in 2018) as a relation between the Frobenius manifold coming from the quantum cohomology of a Fano manifold X and the derived category of coherent sheaves of X. We will give some speculations of how to extend that relation to a one between semisimple Frobenius manifolds and some derived categories We will sketch the situation in a particular example (3-point Ising model) which might be of interest for symplectic geometers.

2021, Nov 29, Monday, Université de Neuchâtel, Rue Emile Argand 11, Room B217

14:00 Dietmar Salamon (ETH Zürich) GIT from the differential geometric viewpoint 16:00 Grigory Mikhalkin (Université de Genève) Exoteric capacities, camels and cryptocamels

**GIT from the differential geometric viewpoint**

** Dietmar Salamon**

In this talk I will discuss some central results in geometric invariant theory, such as the Kempf-Ness Theorem and the Hilbert-Mumford Criterion for (semi/poly)stability from a differential geometric point of view. Stability is defined in terms of a moment map for the Hamiltonian action of a compact Lie groupon a closed Kähler manifold and a key resultis the moment-weight inequality, relating the Mumford weights to the infimum of the norm of the moment map on a complexified group orbit. Central ingredients in the proofs are the gradient flows of the square of the moment map and of the Kempf-Ness function. The motivation for this approach (which will not be discussed in the talk) lies in certain infinite-dimensional anlogues of GIT that arise naturally in various areas in geometry. The talk is based on joint work with Valentina Georgoulas and Joel Robbin.

**Exoteric capacities, camels and cryptocamels**

** Grigory Mikhalkin**

Exoteric capacities are based on pseudoholomorphic curves extendable to the outside of symplectic domains. In particular, the inside (esoteric) part of the curves cannot be of finite type. Nevertheless their definition (and in some cases also the computation) is completely straightforward. In the talk we introduce and discuss exoteric capacities as well as their applications to the symplectic camel problem and its generalizations.

2021, November 8, Monday, Université de Genève, Rue du Conseil-Général 7-9 1205 Geneva, Room 6-13

12h00 — Welcome and lunch 14h00 — Jeff Hicks (University of Edinburgh) The Support of the Lagrangian lift of a Tropical Hypersurface 15h00 — coffee/tea break 16h00 — Alex Oancea (Université de Strasbourg) Bialgebra structures in Floer theory 17h00 - Discussions

Jeff Hicks.
**The Support of the Lagrangian lift of a Tropical Hypersurface**

Given a tropical hypersurface in n-dimensional real space, there exists a Lagrangian submanifold in the algebraic n-torus whose projection under the log map approximates the given tropical hypersurface. We call this the Lagrangian realization of the tropical hypersurface. The support of a Lagrangian submanifold L is the set of Lagrangian torus fibers whose Lagrangian intersection Floer cohomology with L is non-vanishing. Under mirror symmetry, this corresponds to the support of the sheaf which is mirror to L. We discuss how to construct some Lagrangian realizations of tropical hypersurfaces, and when those realizations are exact we will compute their support. Time permitting, we'll make some connections with dimers and mutations.

Alex Oancea.
**Bialgebra structures in Floer theory**

I will explain how a certain type of bialgebra structure arises in Floer theory, in particular in the context of Rabinowitz Floer homology. When applied to cotangent bundles this explains a number of mysterious relations from string topology.

2021, May 27, Thursday, 13h00

Université de Neuchâtel, Rue Emile Argand 11

**The Geometry and Algebra of Lagrangian Cobordism**

**Paul Biran (ETH Zürich)**

In this talk we will survey the theory of Lagrangian cobordism and discuss the type of structures arising from it: a geometric viewpoint on derived Fukaya categories, new invariants, as well as metric structures on the space of Lagrangians. Based on joint works with O. Cornea and J. Zhang.

2021, April 14, Wednesday 12h30 — Welcome and lunch (room 06-13) 14h30 — Alexandre Jannaud (Neuchâtel) Dehn-Seidel twist, $C^0$ symplectic geometry and barcodes (room 01-05) 15h30 — coffee/tea break (room 06-13) 16h — Discussions (room 01-05)

**Dehn-Seidel twist, $C^0$ symplectic geometry and barcodes**

In this talk I will present my work initiating the study of the $C^0$ symplectic mapping class group, i.e. the group of isotopy classes of symplectic homeomorphisms, and present the proofs of the first results regarding the topology of the group of symplectic homeomorphisms. For that purpose, we will introduce a method coming from Floer theory and barcodes theory.

Applying this strategy to the Dehn-Seidel twist, a symplectomorphism of particular interest when studying the symplectic mapping class group, we will generalize to $C^0$ settings a result of Seidel concerning the non-triviality of the mapping class of this symplectomorphism. We will indeed prove that the generalized Dehn twist is infinite order in the group of the connected component of symplectic homeomorphisms. Doing so, we prove the non-triviality of the $C^0$ symplectic mapping class group of some Liouville domains.

2021, March 11, Thursday, Virtual Session Will Merry (ETH Zürich) 16:00-17:00 Meeting ID: 934 017 7640 Passcode: (the number of lines on a cubic surface)

**Symplectic cohomology of magnetic cotangent bundles **

Joint work with Y. Groman (and, time-permitting, Seongchan Kim). We construct a family version of symplectic Floer cohomology for magnetic cotangent bundles.

2021, February 25, Thursday, Virtual Session Sergey Galkin (PUC-Rio & HSE Moscow) 16:00-17:00 Meeting ID: 934 017 7640 Passcode: (the number of lines on a cubic surface)

**Topological field theories via mirror symmetry for moduli spaces**

Categorifying the Atiyah-Floer conjecture, Donaldson invented an extended topological field theory that assigns a category to a surface, an object to a handlebody, a chain complex to a Heegard decomposition, etc. The categories of Donaldson-Floer theory come from the symplectic geometry of the moduli spaces of flat $SU(2)$ connections on a surface, and the objects are associated to Lagrangian subvarieties. Fukaya developed further the foundations of such categories associated to arbitrary symplectic varieties. Mirror symmetry functors map Fukaya categories to the categories of matrix factorizations of the functions known as Ginzburg-Landau potentials, which are functorially obtained as Floer potentials, the generating functions for Maslov index 2 pseudo-holomorphic discs with boundary on a Lagrangian. Nishinou-Nohara-Ueda proposed a way to construct a monotone Lagrangian torus and simultaneously to compute its Floer potential starting from a small toric degeneration. A class of degenerations for the moduli spaces was constructed by Manon based on Tsuchiya-Ueno-Yamada construction of the sheaves of conformal blocks for Wess-Zumino-Witten $SU(2)$ conformal field theory and Faltings equivalence between conformal blocks over the interior of the moduli space of curves and non-abelian theta-functions. The toric degenerations correspond to the corners of Deligne-Mumford compactification, i.e. decomposition of a surface into trinions,and are enumerated by trivalent graphs. The resulting Ginzburg-Landau potentials can be described entirely in terms of the graph, as the sums over the vertices of generating functions for quantum Clebsch-Gordan inequalities, the class of functions that we call graph potentials. In the end of the day, the graph potentials give a new foundationfor the construction of Donaldson-Floer theories, which will be free of analysis. In the first part of the talk I will review and compare different constructions of integrable systems on the moduli space of flat $SU(2)$ connections and its relatives, such as moduli of Euclidean polytopes or moduli of Higgs bundles. Then I will focus on the relation between Manon-Nishinou-Nohara-Ueda integrable systems and graph potentials and will show how starting from graph potentials one constructs a 2-dimensional topological field theory.The talk is based on a joint work `2009.05568`

with Pieter Belmans and Swarnava Mukhopadhyay.

2020, December 10, Thursday, Virtual Session Selman Akbulut (Gökova Geometry Topology Institute) 14:00-15:00 Richard Hind (University of Notre Dame) 15:30-16:30 Meeting ID: 934 017 7640 Passcode: (the number of lines on a cubic surface)

**Selman Akbulut: On the 4-dimensional Smale Conjecture**

Localizing exoticity of smooth structures of 4-manifolds gives rise to corks, and localizing corks gives new information about diffeomorphisms of $S^4.$ This talk is about my recent attempt to construct a counterexample to the 4-dimensional Smale Conjecture, which says that every self diffeomorphism of $S^4$ is isotopic to the identity.

**Richard Hind: Optimal embeddings of Lagrangian tori**

I will describe joint work with Emmanuel Opshtein and Jun Zhang determining which product Lagrangian tori in 4-dimensional Euclidean space can be mapped into a given polydisk or ellipsoid by a Hamiltonian diffeomorphism. To detect the borderline where full rigidity is lost we look at holomorphic curves of very high degree. A project with Ely Kerman also uses high degree curves to obstruct certain embeddings of pairs of Lagrangians.

2020, November 12, Virtual Session Felix Schlenk (Neuchâtel) 14h30-15h30 Joé Brendel (Neuchâtel) 16h-17h https://unige.zoom.us/j/5928729514 Meeting ID: 592 872 9514 Passcode: (the number of lines on a cubic surface)

**Felix Schlenk (Neuchâtel) Quadrilaterals and non-isotopic cubes**

Abstract: From the square representing the usual toric fibration of the monotone $S^2 \times S^2$ one can get an infinite tree of quadrilaterals representing almost toric fibrations by mutation. The centers of these quadrilaterals correspond to exotic Lagrangian tori, and for those quadrilaterals that are “fat enough” one obtains non-isotopic symplectic embeddings of 4-cubes into $S^2 \times S^2$. After explaining this, I will show how to find fat quadrilaterals.

This is based on joint work with Grisha and Joé, and of Thomas Buc-d'Alché.

**Joé Brendel (Neuchâtel) The Chekanov torus in toric manifolds**

Abstract: The first example of an exotic Lagrangian torus in $\mathbb{C}^n$ goes back to Chekanov. In order to distinguish it from the Clifford torus, Chekanov used the displacement energy of nearby Lagrangian tori. We will start by reviewing this classical construction and see that it can be carried out in any monotone toric manifold and that (under some mild assumptions) displacement energy proves that the resulting torus is exotic. At the end we will draw many pictures and notice that, in dimension four, displacement energy recovers the base of the almost toric fibration of the first mutation. However, our recipe works in arbitrary dimensions and so we can also construct higher-dimensional polytopes and wonder whether they are the base of some singular fibration.

2020, October 14, Neuchâtel

**Weronika Czerniawska (Université de Genève) Some old useful results on quadratic forms**

Abstract: Since Gauss's 1801 work the study of quadratic forms has stimulated many developments in Number theory. In my talk I will present Markoff's paper on the topic that may be a beginning of an unexploited path between Symplectic geometry and Number Theory.

**Gleb Smirnov (ETH Zürich) Seidel’s theorem via gauge theory**

Abstract: A theorem of Seidel states that the square of generalized Dehn twist is not symplectically isotopic to the identity for many closed algebraic surfaces. In this talk, we will discuss a new proof of this theorem which does not rely on any Floer-theoretic considerations but instead uses invariants derived from the Seiberg-Witten equation.

2020, May 6, Virtual Session

Join Zoom Meeting https://unige.zoom.us/j/5928729514

Meeting ID: 592 872 9514

Password: (the number of lines on a cubic surface)

**Schedule**

We start the Zoom session 15 minutes before the scheduled beginning of each talk.

17h, Central European time zone (CEST)

**Joontae Kim (KIAS Seoul)“Real Lagrangian submanifolds in S^2 x S^2”**

Abstract: A Lagrangian submanifold in a symplectic manifold is called real if it is the fixed point set of an antisymplectic involution. In this talk, we explore the topology of real Lagrangians in S^2 x S^2, which shows interesting real symplectic phenomena. In particular, we prove that any real Lagrangian in S^2 x S^2 is Hamiltonian isotopic to either the antidiagonal sphere or the Clifford torus.

20h, Central European time zone (CEST)

**Seongchan Kim (Neuchâtel) “Symmetric periodic orbits in Hamiltonian systems” **

Abstract: A symmetric periodic orbit of a Hamiltonian system is a special kind of periodic orbit, which can be regarded as a Lagrangian intersection point. It plays an important role in the planar circular restricted three-body problem. In this talk I will give a gentle introduction to symmetric periodic orbits in Hamiltonian systems and present some results on existence and on the systole. This is partly joint work with Myeonggi Kwon and Joontae Kim.

2020, April 22, Virtual session

Join Zoom Meeting https://unige.zoom.us/j/5928729514

Meeting ID: 592 872 9514

Password: (the number of lines on a cubic surface)

**Schedule**

We start the Zoom session 15 minutes before the scheduled beginning of each talk.

17h, Central European time zone (CEST)

**Georgios Dimitroglou Rizell (Uppsala) “Lagrangian tori from a quantitative viewpoint” **

Abstract: Certain quantitative conditions on a Lagrangian torus in the ball or the projective plane ensure that it is Hamiltonian isotopic to the standard Clifford torus. (Roughly speaking, we need the existence of a big symplectic ball in its complement.) We explain the mechanisms behind this and give examples and applications.

20h, Central European time zone (CEST) = 11h Stanford (PDT)

**Yakov Eliashberg (Stanford) “Symplectic cobordisms revisited”**

Abstract: I will explain in the talk a simplified proof of our joined theorem with Emmy Murphy about construction of symplectic cobordisms between contact manifolds.

2020, March 4, Battelle Villa (Geneva)

**Schedule**

12h15-12h45 Meeting and initial discussion

12h45 we leave for lunch from the villa

14:30-15:30

** Felix Schlenk (University of Neuchâtel) “Neck-stretching, and unknottedness of Lagrangian spheres in
S^2 x S^2 ” (Part II)**

The “stretching the neck” technique is a powerful method in symplectic topology and Hamiltonian dynamics, with applications to embedding problems, to Reeb dynamics, and to the study of Lagrangian submanifolds. I'll try to explain this technique by using it to prove that every Lagrangian sphere in S^2 x S^2 is Hamiltonian isotopic to the anti-diagonal, following Richard Hind's GAFA paper from 2004.

15:30-16h Discussion

16:00-17:00

** Tobias Ekholm (Uppsala University) “Skeins on branes” **

We give an enumerative interpretation of coefficients of the HOMFLY polynomial of a knot in the 3-sphere as a count of holomorphic curves with boundary on a version of the Lagrangian conormal of the knot in the resolved conifold. A key ingredient in the proof is the definition of counts of holomorphic curves with boundary in the skein module of the Lagrnagian brane where they have their boundary. The talk reports on joint work with Vivek Shende

17h- Discussion with open end.

2020, February 19, Battelle Villa (Geneva)

**Schedule**

12h15-12h45 Meeting and initial discussion

12h45 we leave for lunch from the villa

14:30-15:30

** Felix Schlenk (University of Neuchâtel) “Neck-stretching, and unknottedness of Lagrangian spheres in S^2 x S^2”**

The “stretching the neck” technique is a powerful method in symplectic topology and Hamiltonian dynamics, with applications to embedding problems, to Reeb dynamics, and to the study of Lagrangian submanifolds. I'll try to explain this technique by using it to prove that every Lagrangian sphere in S^2 x S^2 is Hamiltonian isotopic to the anti-diagonal, following Richard Hind's GAFA paper from 2004.

15:30-16h Discussion

16:00-17:00

** Ilia Itenberg (Sorbonne University) “Real aspects of enumerative geometry” **

We will discuss several recent developments in real enumerative geometry, in particular, those concerning Welschinger invariants and refined enumeration of curves on surfaces.

17h- Discussion with open end.