# Geometry & Dynamics Seminar 2021-22

## Please check each announcement since this is sometimes changed.

#### Upcoming Talks        Previous Talks        Previous Years

 20.10.2021, 14:10 (Wednesday) Patrick Iglesias-Zemmour (CNRS, HUJI) Title: Every symplectic manifold is a (linear) coadjoint orbit Location: Zoom session, the link is available upon request by email Abstract: I will show how every symplectic manifold (Hausdorff and second countable) is a coadjoint orbit of the group of automorphisms of its integration bundle, for the linear coadjoint action, even when the symplectic form is not integral, i.e., when the group of periods is dense in the real line. In this case the integration bundle is not a manifold because its torus of period is not a circle but an "irrational torus". This theorem answers a question asked by a few students, in particular on MathOverFlow: Is there a universal model for symplectic manifolds? The answer is "yes".  They are all coadjoint orbits. 27.10.2021, 14:10 (Wednesday) Daniel Tsodikovich (Tel Aviv University) Title: Billiard Tables with rotational symmetry Location: Zoom session, the link is available upon request by email Abstract: Consider the following simple geometric fact: the only centrally symmetric convex curve of constant width is a circle. The condition of having constant width is equivalent for the (Birkhoff) billiard map to have a 1-parameter family of two periodic orbits. We generalize this statement to curves that are invariant under a rotation by angle \frac{2\pi}{k},  for which the billiard map has a 1-parameter family of k-periodic orbits. We will also consider a similar setting for other billiard systems: outer billiards, symplectic billiards, and (a special case of) Minkowski billiards. Joint work with Misha Bialy. 03.11.2021, 16:10 (Wednesday) Egor Shelukhin (University of Montreal) (PLEASE NOTE CHANGE IN TIME!) Title: Hamiltonian no-torsion Location: Zoom session, the link is available upon request by email Abstract: We generalize in several ways Polterovich's well-known theorem that the Hamiltonian group of a closed symplectically aspherical manifold admits no non-trivial elements of finite order. We prove an analogous statement for Calabi-Yau and negatively monotone manifolds. For positively monotone manifolds we prove that non-trivial torsion implies geometric uniruledness of the manifold, answering a question of McDuff-Salamon. Moreover, in this case the following symplectic Newman theorem holds: a small Hofer-ball around the identity contains no finite subgroup. This is joint work with Marcelo Atallah. 10.11.2021, 14:10 (Wednesday) Umut Varolgunes (Stanford University, University of Edinburgh) Title: Trying to quantify Gromov's non-squeezing theorem Location: Zoom session, the link is available upon request by email Abstract: Gromov's celebrated result says (colloquially) that one cannot symplectically embed a ball of radius 1.1 into a cylinder of radius 1. I will show that in 4d if one removes from this ball a Lagrangian plane passing through the origin, then such an embedding becomes possible. I will also show that this gives the smallest Minkowski dimension of a closed subset with this property. I will end with many questions. This is based on joint work with K. Sackel, A. Song and J. Zhu. 17.11.2021, 14:10 (Wednesday) Pazit Haim Kislev (Tel Aviv University) Title: Symplectic capacities of p-products Location: Zoom session, the link is available upon request by email Abstract: In this talk we discuss symplectic capacities of convex domains and their behavior with respect to symplectic p-products. One application, by using a "tensor power trick", is to show that it is enough to prove Viterbo's volume-capacity conjecture in the asymptotic regime when the dimension is sent to infinity. In addition, we introduce a conjecture about higher order capacities of p-products and show that if it holds then there are no non-trivial p-decompositions of the symplectic ball. 24.11.2021, 14:10 (Wednesday) Gerhard Knieper (Ruhr University Bochum) Title: Growth rate of closed geodesics on surfaces without conjugate points. Location: Zoom session, the link is available upon request by email Abstract: Let (M,g) be a closed Riemannian surface of of genus at least 2 and no conjugate points. By the uniformization theorem such a surface admits a metric of negative curvature and therefore the topological entropy h of the geodesic flow is positive. Denote by P(t)  the number of free homotopy classes  containing a closed geodesic of period $\le t$. We will show: P(t) is asymptotically equivalent to e^(ht)/(ht) =F(t), i.e. the ratio of P and F  converges to 1 as t tends to infinity. An important ingredient in the proof is a mixing flow invariant measure given by the unique measure of maximal entropy. Under suitable hyperbolicity assumptions this result carries over to closed Riemannian manifolds without conjugate and higher dimension. For closed manifolds of negative curvature the above estimate is well known and has been originally obtained by Margulis. In a recent preprint the estimate has been also obtained by Ricks for  certain closed manifolds (rank 1 mflds) of non-positive curvature. This is a joint work with Vaughn Climenhaga and Khadim War. 01.12.2021, 14:10 (Wednesday) Sara Tukachinsky (Tel Aviv University) Title: Bounding chains as a tool in open Gromov-Witten theory Location: Zoom session, the link is available upon request by email Abstract: Moduli spaces of J-holomorphic disks have boundary. This interferes with desirable structures, such as Lagrangian Floer theory or open Gromov-Witten invariants. One tool for balancing out boundary contributions is a bounding chain. In this talk I will give some background on the problem, then discuss in detail what bounding chains are, how they can be constructed, and how they are used to define invariants. The work of several people will be mentioned, among them a joint work with J. Solomon. 08.12.2021, 14:10 (Wednesday) Louis Ioos (Max Planck Institute) Title: Quantization in stages and canonical metrics Location: Zoom session, the link is available upon request by email Abstract: In this talk, I will introduce the notion of quantization in stages, which lies at the basis of fundamental physical set-ups such as the Stern-Gerlach experiment, and explain how it can be realized over compact symplectic phase spaces via the use of Berezin-Toeplitz quantization of vector bundles. In particular, I will introduce and show how to compute the associated quantum noise. I will then describe an application to Hermite-Einstein metrics on stable vector bundles over a projective manifold, and if time permits, I will show how a refinement of these results in the case of the trivial line bundle can be applied to Kähler metrics of constant scalar curvature. 15.12.2021, 14:10 (Wednesday) Philippe Charron (Technion) Title: TBA Location: Zoom session, the link is available upon request by email Abstract: TBA 22.12.2021, 17:10 (Wednesday) Simion Filip (University of Chicago) (PLEASE NOTE CHANGE IN TIME!) Title: TBA Location: Zoom session, the link is available upon request by email Abstract: TBA 05.01.2022, 14:10 (Wednesday) Igor Uljarević (University of Belgrade) Title: TBA Location: Zoom session, the link is available upon request by email Abstract: TBA 02.03.2022, 14:10 (Wednesday) Joé Brendel (University of Neuchâtel) Title: TBA Location: Zoom session, the link is available upon request by email Abstract: TBA 09.03.2022, 14:10 (Wednesday) TBA Title: TBA Location: Zoom session, the link is available upon request by email Abstract: TBA 16.03.2022, 14:10 (Wednesday) TBA Title: TBA Location: Zoom session, the link is available upon request by email Abstract: TBA 23.03.2022, 14:10 (Wednesday) TBA Title: TBA Location: Zoom session, the link is available upon request by email Abstract: TBA 30.03.2022, 14:10 (Wednesday) TBA Title: TBA Location: Zoom session, the link is available upon request by email Abstract: TBA 06.04.2022, 14:10 (Wednesday) Sheng-Fu Chiu (Institute of Mathematics, Academia Sinica, Taiwan) Title: From Energy-Time Uncertainty to Symplectic Displacement Energy Location: Zoom session, the link is available upon request by email Abstract: Heisenberg's Uncertainty Principle is one of the most celebrated features of quantum mechanics, which states that one cannot simultaneously obtain the precise measurements of two conjugated physical quantities such as the pair of position and momentum or the pair of electric potential and charge density. Among the different formulations of this fundamental quantum property, the uncertainty between energy and time has a special place. This is because the time is rather a variable parametrizing the system evolution than a physical quantity waiting for determination. Physicists working on the foundation of quantum theory have understood this energy-time relation by a universal bound of how fast any quantum system with given energy can evolve from one state to another in a distinguishable (orthogonal) way. Recently, there have been many arguing that this bound is not a pure quantum phenomenon but a general dynamical property of Hilbert space. In this talk, in contrast to the usual Hilbert space formalism, we will provide a homological viewpoint of this evolutional speed limit based on a persistence-like distance of the derived category of sheaves : during a time period what is the minimal energy needed for a system to evolve from one sheaf to a status that is distinguishable from a given subcategory? As an application, we will also discuss its geometric incarnation in the dynamics of classical mechanics, namely the notion of symplectic displacement. We will see how this categorical energy manages to characterize the symplectic energy for disjointing a Lagrangian from an open set. 27.04.2022, 14:10 (Wednesday) TBA Title: TBA Location: Zoom session, the link is available upon request by email Abstract: TBA 11.05.2022, 14:10 (Wednesday) TBA Title: TBA Location: Zoom session, the link is available upon request by email Abstract: TBA 18.05.2022, 14:10 (Wednesday) TBA Title: TBA Location: Zoom session, the link is available upon request by email Abstract: TBA 25.05.2022, 14:10 (Wednesday) TBA Title: TBA Location: Zoom session, the link is available upon request by email Abstract: TBA 01.06.2022, 14:10 (Wednesday) TBA Title: TBA Location: Zoom session, the link is available upon request by email Abstract: TBA 08.06.2021, 14:10 (Wednesday) TBA Title: TBA Location: Zoom session, the link is available upon request by email Abstract: TBA

Organized by Misha Bialy, Lev Buhovsky, Yaron Ostrover, and Leonid Polterovich