Persistence Homology in Symplectic and Contact Topology



Institut National Universitaire Champollion, Place de Verdun
    • 1
    • 2
      Interlevel persistence in Floer theory

      The usual filtered Floer homology groups are formal analogues of the homologies of the sublevel sets of a Morse function on a manifold. In the Morse setting,
      by instead considering interlevel sets (preimages of general intervals) one obtains an algebraic structure that is classified by a barcode that refines the usual
      sublevel persistence barcode. I will describe an algebraic formalism that allows one to adapt this to Floer-theoretic settings. In this case of Hamiltonian Floer
      theory this gives rise to a pairing between distinguished spectral invariants of a Hamiltonian flow and of its inverse that satisfies stability and duality theorems.

      Speaker: Michael Usher
    • 3
      Coarse topology, oscillations and persistence

      Studying topology of the zero set of a polynomial map is a classical topic in algebraic geometry. One may attempt to extend this study to less regular objects, such as linear combinations of Laplace-Beltrami eigenfunctions or entire maps in several complex variables. A phenomenon which ultimately breaks the analogy is the existence of highly oscillatory behaviour on small scales. We will explain how, using persistence modules and barcodes, it is possible to systematically discard small oscillations and prove coarse analogues of certain classical theorems, such as Bézout's theorem, in the above mentioned setups. The talk is based on joint works with L. Buhovsky, J. Payette, I. Polterovich, L. Polterovich and E. Shelukhin.

      Speaker: Vukasin Stojisalvjevic
    • 4
      Sheaves for the completion of the space of Lagrangians

      For a manifold $M$ it is known that we can associate a sheaf on $M\times \mathbb{R}$ to any exact Lagrangian submanifold of $T^*M$. The space of exact Lagrangians carries the Viterbo's metric and, following Humili`ere, we can consider its completion. We will see that the correspondence between sheaves and symplectic geometry extends to this framework and, moreover, natural notions defined for the objects on both sides coincide: to any element in the completion of the space of Lagrangians we can associate a sheaf; the notions of $\gamma$-support and microsupport coincide and the microsupport of any sheaf if $\gamma$-coisotropic.

      These are joint works with Viterbo and Asano, Humilière, Ike, Viterbo.

      Speaker: Stéphane Guillermou
    • 5
      Symplectic embeddings into convex toric domains (via Zoom)

      This is a report on joint work with Cristofaro-Gardiner and Magill, that investigates symplectic embeddings into toric domains with irrational convex boundary.

      Speaker: Dusa McDuff
    • 6
      A conjectural chain model for positive $S^1$-equivariant symplectic homology of star-shaped toric domains in $\mathbb{C}^2$ (via Zoom)

      For any star-shaped toric domain in $\mathbb{C}^2$, we define a filtered chain complex which conjecturally computes positive $S^1$-equivariant symplectic homology of the domain.
      Assuming this conjecture, we show that the sequence $(c^{\mathrm{GH}}_k(X)/k)_k$ has a limit as $k$ goes to $\infty$, where $c^{\mathrm{GH}}_k$ denotes the $k$-th Gutt-Hutchings capacity.

      Speaker: Kei Irie
    • 7
      Contact Hamiltonian Floer homology and its applications

      In this talk, we will discuss a Floer-theoretic approach to studying Hamiltonian dynamics on contact manifolds, called contact Hamiltonian Floer homology. On the one hand, it provides a characterization of the rigidity of positive loops in the contactomorphism group which is a peculiar object often investigated in contact geometry. Mysteriously, our characterization highly relates to certain exotic behavior of the Floer continuation map in this setting. On the other hand, from an algebraic perspective, it admits a persistence module type upgrade (we call a gapped module), where one can read off numerical data from the associated barcode. These data help to define contact spectral invariant, contact boundary depth, etc., which also initiates the discovery of novel rigidity phenomenon of subsets in a contact manifold. This talk is based on joint work with Igor Uljarević.

      Speaker: Jun Zhang
    • 8
      Sesquicuspidal curves

      A classic question in algebraic geometry asks what are the possible singularities for a plane curve of a given degree and genus. This turns out to be closely connected with the theory of (stabilized) symplectic embeddings of ellipsoids. In this talk I will describe a construction (joint with D. McDuff) of new families of rational plane curves with desirable singularities. Key ingredients include a generalization of Orevkov's birational transformation and scattering diagrams for tropical curves.

      Speaker: Kyler Siegel
    • 9
      The Toda lattice, billiards and the Viterbo conjecture (via Zoom)

      The Toda lattice is one of the earliest examples of non-linear completely integrable systems. Under a large deformation, the Hamiltonian flow can be seen to converge to a billiard flow in a simplex. In the 1970s, action-angle coordinates were computed for the standard system using a non-canonical transformation and some spectral theory. In this talk, I will explain how to adapt these coordinates to the situation of a large deformation and how this leads to new examples of symplectomorphisms of Lagrangian products with toric domains. In particular, we find a sequence of Lagrangian products whose symplectic systolic ratio is one and we prove that they are symplectic balls. This is joint work with Y. Ostrover and D. Sepe.

      Speaker: Vinicius Ramos
    • 10
      Spectral norm and conformally symplectic maps: two applications

      I'll present two applications of the conformal invariance of the spectral norm: 1) a generalization of the Birkhoff attractor to higher dimension, 2) a weak version of the Nearby Lagrangian Conjecture. The completion of the space of Lagrangians with respect to the spectral norm plays a role in both cases.
      This is joint work with Marie-Claude Arnaud and Claude Viterbo.

      Speaker: Vincent Humilière
    • 11
      On the Hofer-rigidity of iterations and continuous Hamiltonian torsion

      We discuss a new rigidity phenomenon for iterations of Hamiltonian maps in the Hofer metric and in the C^0 metric. Joint work in progress with Nicholas Wilkins

      Speaker: Egor Shelukhin
    • 12
      Guided visit of the town of Albi
    • 13
      Cocktail at the Hôtel de Ville
    • 15
      Symplectic Barriers

      In his seminal 2001 paper, Biran introduced the concept of Lagrangian Barriers, a symplectic rigidity phenomenon coming from obligatory intersections with Lagrangian submanifolds which doesn't come from mere topology.
      One simple example is that when removing a Lagrangian plane from the four dimensional ball, the symplectic capacity shrinks down to half of the original capacity of the ball.
      Considering this example for a Lagrangian barrier, we ask how would the capacity change if one removes a plane which is not necessarily Lagrangian, or several such planes.
      We show that indeed there are non-trivial obstructions which give the first example (as far as we know) of Symplectic Barriers: obligatory intersections of symplectic embeddings with symplectic submanifolds.
      This is a joint work with Richard Hind and Yaron Ostrover.

      Speaker: Pazit Haim-Kislev
    • 16
      On the strong Arnold chord conjecture for convex contact forms

      Arnold conjectured that every closed Legendrian submanifold in the standard contact sphere $S^{2n-1}$ with a contact form has a Reeb chord. This was confirmed by K. Mohnke in 2001. In fact, Arnold originally conjectured that a Reeb chord with distinct endpoints exists. I will give a proof of this strong version of the conjecture for convex contact forms, namely contact forms on $S^{2n-1}$ induced by convex embeddings into $\mathbb{R}^{2n}$. I will also present a counterexample to the conjecture for nonconvex contact forms due to M. Hutchings.

      Speaker: Jungsoo Kang
    • 17
      Completed signed barcodes and zeta functions for open domains

      We describe some invariants of open domains related to bar codes coming from ECH and contact homology. These can be used for example to show that some open domains are not toric, and to distinguish some open toric domains which are difficult to distinguish otherwise.

      Speaker: Michael Hutchings
    • 18
      The local strong Viterbo conjecture

      I will report on joint work in progress with Abbondandolo and Benedetti confirming the strong Viterbo conjecture for convex domains close to the ball in arbitrary dimension. The proof involves a quasi-invariant normal form for contact forms close to a Zoll one. I will explain how this normal form can be interpreted in terms of geodesics for symplectic Banach-Mazur type distances.

      Speaker: Oliver Edtmair
    • 19
      Elementary SFT Spectral Invariants And The Strong Closing Property

      In this talk, I will discuss ESFT spectral gaps: a new, general class of spectral invariants for large class of stable Hamiltonian manifolds and their cobordisms, in any dimension. They are built using a min-max construction applied to J-holomorphic curves in symplectic field theory. I will explain their formal properties and provide a holomorphic curve "closing criterion" for the ESFT gaps that implies a strong version of the smooth closing lemma. I will explain how to prove this criterion for flows that are "Hofer near periodic" and I will state some open problems.
      This is joint work with Shira Tanny (IAS).

      Speaker: Julian Chaidez
    • 20
      Inverse reduction inequalities

      Classically symplectic reduction yields an estimate from below for the spectral distance between Lagrangians (for example in a cotangent bundle) by the spectral distance between reductions.
      We shall show here how using a result by Kiselev-Shelukhin using some enriched structure on barcodes, one can obtain upper bounds on the spectral distance from above and give a number of applications and conjectures.

      Speaker: Claude Viterbo