Time: Fridays, 9:15–10:45 a.m. (Montréal/Princeton hour)
Zoom link: https://theias.zoom.us/j/97116147750?pwd=L2Fud1Y4Z2xsT3dhU2NrV0ZXd3lUQT09
(Meeting ID: 971 1614 7750 ; Passcode: 816898)

Jun. 18: Agustin Moreno (Uppsala)

Title: On the spatial restricted three-body problem

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.

About the seminar

  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 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).
  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).
  4. Please do not hesitate to ask questions: First indicate your intention (or even the question) by chat, then, when invited by the host/organizer, use your microphone and video (if available).
  5. We post links to the slides of the talks as well as links to recordings of the talks.

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: Daniel Álvarez-Gavela (MIT), Octav Cornea (Montréal), Dan Cristofaro-Gardiner (IAS), Helmut Hofer (IAS), Yusuf Barış Kartal (Princeton), Leonid Polterovich (Tel Aviv), Egor Shelukhin (Montréal), Sara Tukachinsky (IAS), Claude Viterbo (Paris), Zhengyi Zhou (IAS)

Future talks

  • Jun. 25: Three 20min research talks
    • Mohan Swaminathan (Princeton), TBA
    • Ben Wormleighton (WashU), Lattice formulas for rational SFT capacities of toric domains, (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, (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.
  • Jul. 2: Cancelled on account of Convexity in Contact and Symplectic Topology, https://indico.math.cnrs.fr/event/5788/
  • Jul. 9: Laurent Côté (IAS/Harvard), TBA
  • Jul. 16: Helmut Hofer (IAS), The Floer Jungle: 35 years of Floer Theory

Past talks

  • 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).
  • 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 (IMJ-PRG), 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.
  • Earlier talks: https://dms.umontreal.ca/~cornea/Seminar.html