Weekly Bulletin

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.

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 𝐶0 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.

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.

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.

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.

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.

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.

In statistical learning theory, determining the sample complexity of realizable binary classification for VC classes was a long-standing open problem. The results of Simon and Hanneke established sharp upper bounds in this setting. However, the reliance of their argument on the uniform convergence principle limits its applicability to more general learning settings such as multiclass classification. In this talk, we will discuss a simple technique that addresses this issue. We will present optimal high probability risk bounds through a framework that surpasses the limitations of uniform convergence arguments. In addition to binary classification, we will see applications in settings where uniform convergence is provably sub-optimal. For multiclass classification, we prove an optimal risk bound scaling with the one-inclusion hypergraph density of the class, addressing the suboptimality of the analysis by Daniely and Shalev-Shwartz. Based on joint work with Ishaq Aden-Ali, Yeshwanth Cherapanamjeri and Abhishek Shetty.

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.

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.

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.

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.

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.

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.

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, in the semiclassical limit, magnetic Laplacians have spectral clusters. In each cluster, the dynamic and the eigenvalue distribution can be described in terms of Toeplitz operators.

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.