Talks (titles and abstracts)
Michael Bialy: Locally maximizing orbits for Twist maps and Birkhoff billiards
Locally maximizing billiard configurations (orbits) are those which give local maxima for the Length functional between any two impact points. For example, rotational invariant curves and Aubry-Mather sets are filled by locally maximizing orbits. In the first part of the talk I will explain how to get a sharp estimates on the measure of the set of these orbits for planar Birkhoff billiards. These estimates provide an effective version of the Birkhoff conjecture for centrally symmetric billiard tables. In the second part of the talk I discuss the Twist maps and Birkhoff billiards in higher dimensions. In particular, I show that the class of locally-maximizing orbits does not depend on the choice of generating function, similar to the result by P. Bernard and M.Mazzucchelli-Sorrentino on Tonelli Hamiltonians.
Based on the joint works with Robert MacKay, Andrey E. Mironov, Sergei Tabachnikov and Daniel Tsodikovich.
Lev Buhovsky: Flexibility of the adjoint action of the group of Hamiltonian diffeomorphisms
On a closed and connected symplectic manifold, the group of Hamiltonian diffeomorphisms has the structure of an infinite dimensional Fréchet Lie group, where the Lie algebra is naturally identified with the space of smooth and zero-mean normalized functions, and the adjoint action is given by pushforwards. In my talk I will explain why the adjoint action is flexible and how this relates to the subject of uniqueness of the Hofer metric. Based on a joint work with Yaron Ostrover, and a recent joint work with Maksim Stokić.
Laurent Charles: Magnetic Laplacians on symplectic manifolds
I will discuss the relations between magnetic geodesic flows on closed manifolds and the corresponding quantum Hamiltonians. For hyperbolic surfaces with constant magnetic field, the magnetic flow is periodic up to some critical energy, and the corresponding eigenvalues of the magnetic Laplacian have high degeneracies. More generally, for a non degenerate magnetic field, in the semiclassical limit, the magnetic Laplacians have spectral clusters. The number of eigenvalues in each cluster is given by a Riemann-Roch number. And the dynamic and the eigenvalue distribution can be described in terms of Toeplitz operators.
Octav Cornea: Metrics on spaces of Lagrangians through persistence and Fukaya categories
The space of exact Lagrangians in a Liouville domain is endowed with a certain class of metrics that reflect symplectic rigidity properties. In this talk, based on joint talk with Paul Biran and Jun Zhang, I will explain how these metrics are constructed by combining ideas coming from persistence theory and Fukaya category machinery.
Yakov Eliashberg: tba
Viktor Ginzburg: Topological Entropy of Hamiltonian Systems and Persistence Modules
Topological entropy is a fundamental invariant of a dynamical system, measuring its complexity. In this talk, we will focus on connections between the topological entropy of a Hamiltonian dynamical system, e.g., a Hamiltonian diffeomorphism or a geodesic flow, and the underlying filtered Morse or Floer homology viewed as a persistence module. We will recall the definition of barcode entropy — a Morse/Floer theoretic counterpart of topological entropy — and show that barcode entropy is closely related to topological entropy and that, for Hamiltonian diffeomorphisms and geodesic flows in low dimensions, these invariants are equal. The talk is based on joint work with Erman Cineli, Basak Gurel and Marco Mazzucchelli.
Misha Gromov: tba
Louis Ioos: Quantization commutes with Reduction for singular circle actions
Given the Hamiltonian action of a Lie group G on a symplectic manifold M, the principle of Quantization commutes with Reduction, due to Guillemin and Sternberg, states that the space of G-invariants of the quantization of M coincides with the quantization of its symplectic reduction by G. This principle provides in particular a geometric approach to the representation theory of Lie groups. In this talk, I will consider the case where G is a circle and where the symplectic reduction is a compact singular symplectic space, then present an approach to establish this principle based on the Berline-Vergne localization formula and the asymptotics of the Witten integral. This is based on a joint work in collaboration with Benjamin Delarue and Pablo Ramacher.
Alessandra Iozzi: The real spectrum compactification of character varieties: characterizations and applications
We describe properties of a compactification of general character varieties with good topological properties and give various interpretations of its ideal points. We relate this to the Thurston-Parreau compactification and, if time permits, we apply our results to the theory of maximal representations.
This is a joint work with Marc Burger, Anne Parreau and Maria Beatrice Pozzetti.
Vadim Kaloshin: Marked Length Spectral determination of analytic chaotic billiards
A famous result independently obtained by Ottal and Croke says that a geodesic flow on a compact surface of negative curvature can be determined by its marked length spectrum up to isometry. We prove a similar result for dispersing/hyperbolic billiards. We consider billiards obtained by removing from the plane three strictly convex analytic obstacles satisfying the non-eclipse condition. The restriction of the dynamics to the set of non-escaping orbits is conjugated to a subshift, which provides natural labeling of periodic orbits. Jointly with J. De Simoi and M. Leguil, we show that under suitable symmetry and genericity assumptions, the Marked Length Spectrum determines the geometry of all obstacles. For obstacles without symmetry assumption, V. Otto recently showed that the Marked Length Spectrum along with information about two obstacles determines the geometry of all remaining obstacles.
Yael Karshon: Symplectic excision
Removing a properly embedded ray from a (noncompact) manifold does not affect the topology nor the diffeotype. What about the symplectic analogue of this fact? And can we go beyond rays? I will show how to use incomplete Hamiltonian flows to excise interesting subsets: the product of a ray with a Cantor set, a "box with a tail", and - more generally - epigraphs of lower semicontinuous functions. This is based on joint work with Xiudi Tang, in which we answer
a question of Alan Weinstein.
David Kazhdan: tba
Aleksandr Logunov: Sign of Laplace eigenfunctions
The functions sin(kx), cos(kx) are positive on half of the circle. This talk will concern a related phenomenon of quasi-symmetry for the sign of Laplace eigenfunctions on Riemannian manifolds. We will talk about the distribution of sign and the question of Nazarov, Polterovich and Sodin at which scale quasi-symmetry holds and at which scale quasi-symmetry breaks down. Based on a joint work in progress with Fedya Nazarov.
Cheuk Yu Mak: Lagrangian link quasimorphisms and the non-simplicity of Hameomorphism group of surfaces
In this talk, we will explain the construction of a sequence of homogeneous quasi-morphisms of the area-preserving homeomorphism group of the sphere using Lagrangian Floer theory for links. This sequence of quasi-morphisms has asymptotically vanishing defects, so it is asymptotically a homomorphism. It enables us to show that the Hameomorphism group is not the smallest normal subgroup of the area-preserving homeomorphism group.
If time permits, we will explain how to generalize it to all positive genus surfaces even though we no longer have quasi-morphisms.
The case of the sphere is joint work with Daniel Cristofaro-Gardiner, Vincent Humilière, Sobhan Seyfaddini, and Ivan Smith. The case of positive genus surfaces is joint work with Ibrahim Trifa.
Dusa McDuff: Unicuspidal curves and symplectic staircases
This talk will explain connections between unicuspidal curves in the complex plane and the structure of the ellipsoidal capacity functions for rational symplectic four manifolds. It is joint work with Kyler Siegel.
Iosif Polterovich: Leonid goes spectral
While Leonid Polterovich is widely known for his results in symplectic topology and dynamical systems, his mathematical achievements are not limited to these subjects. In this talk, we highlight Leonid's contributions to spectral geometry.
Sobhan Seyfaddini: \(C^0\) symplectic topology & area-preserving homeomorphisms
In an influential article from the 1970s, Albert Fathi, having proven that the group of compactly supported volume-preserving homeomorphisms of the n-ball is simple for n ≥ 3, asked if the same statement holds in dimension 2. In a joint work with Cristofaro-Gardiner and Humiliére, we proved that the group of compactly supported area-preserving homeomorphisms of the 2-disc is not simple. This answers Fathi's question and settles what is known as the simplicity conjecture in the affirmative.
After a brief introduction to the field of \(C^0\) symplectic topology, which plays a crucial role in our story, I will outline a solution to the above question which was inspired by old and new results of Leonid Polterovich.
This is based on joint works with Dan Cristofaro-Gardiner, Vincent Humilière, Cheuk Yu Mak and Ivan Smith.
Egor Shelukhin: Around persistence modules in symplectic topology
We describe a few applications of persistence modules in symplectic topology and spectral geometry, starting from joint work with Leonid Polterovich on Hamiltonian diffeomorphisms which are far from being autonomous in Hofer's metric.
Sergei Tabachnikov: Nonconventional billiards
I shall discuss recent work on some nonconventional billiards, including billiards in symplectic spaces, wire billiards, and an apparently new outer billiard system in the plane whose generating function is the perimeter of polygons circumscribed polygons. The talk will survey the known results and focus on open problems.
Shira Tanny: Closing lemmas and holomorphic curve measurements
Given a flow on a manifold, how to perturb it in order to create a periodic orbit passing through a given region? While this question was initially studied in the 60s, various facets of it remain largely open. Recently, several advances were made in the context of Hamiltonian and contact flows. I will discuss an approach to this problem that relies on area measurements of pseudoholomorphic curves, inspired by the works of McDuff-Siegel and Hutchings. This is based on a joint work in progress with Julian Chaidez.
Corinna Ulcigrai: Ergodic theory and ergodic integrals of locally Hamiltonian flows
This talk will focus on locally Hamiltonian flows on surfaces, namely smooth two-dimensional flows which preserve a symplectic form. We will present a survey of results concerning the ergodic properties of this class of flows. In particular, we will discuss the deviations phenomena exhibited by ergodic integrals of smooth functions. Recent results on this phenomenon and applications to ergodicity of extensions are based on joint works with K. Fraczek and with P. Berk and F. Trujillo.
Shmuel Weinberger: How many manifolds are there homotopy equivalent to a given one?
This is the question that surgery theory was created to answer. In this talk, inspired by speculations of Gromov, I would like to discuss quantitative aspects of this problem - based on joint work with (mainly) Fedya Manin and Geunho Lim.