# Geometry & Dynamics Seminar 2020-21

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

#### Upcoming Talks        Previous Talks        Previous Years

 21.10.2020, 14:10 (Wednesday) Laurent Charles (Sorbonne University, Paris), Yohann Le Floch (University of Strasbourg) Title: Quantum propagation for Berezin-Toeplitz operators Location: Zoom session, the link is available upon request by email Abstract: The relationship between quantum propagation, defined from Schrödinger equation, and Hamiltonian dynamics is a classical theme of semi-classical analysis. More generally, Lagrangian manifolds play an important role in our understanding of quantum mechanics. We will present several results illustrating this in the context of quantization of a complex compact phase space. 28.10.2020, 14:10 (Wednesday) Andrew Lobb (Durham University) Title: The rectangular peg problem Location: Zoom session, the link is available upon request by email Abstract: For any smooth Jordan curve and rectangle in the plane, we show that there exist four points on the Jordan curve forming the vertices of a rectangle similar to the given one. Joint work with Josh Greene. 04.11.2020, 17:10 (Wednesday) (PLEASE NOTE CHANGE IN TIME!) Laszlo Lempert (Purdue University) Title: On the adjoint action of symplectomorphism groups Location: Zoom session, the link is available upon request by email Abstract: Motivated by constructions in Kähler geometry, in this talk we consider a compact symplectic manifold $(X,\omega)$ and the group $G$ of its symplectomorphisms. We study the action of $G$ on the Fréchet space $C^\infty(X)$ of smooth functions, by pullback, and describe properties of convex functions $p:C^\infty(X)\to\mathbb R$ that are invariant under this action. 11.11.2020, 14:10 (Wednesday) Bo Berndtsson (Chalmers University of Technology) Title: Complex integrals and Kuperberg's proof of the Bourgain-Milman theorem Location: Zoom session, the link is available upon request by email Abstract: I will show a function version of the Bourgain-Milman theorem: $$\int e^{-\phi}\int e^{-\phi^*}\geq \pi^n$$, if $\phi$ is a symmetric  convex function on $\R^n$ and $\phi^*$ is its Legendre transform. The proof is inspired by Kuperberg's proof of the Bourgain-Milman theorem but uses complex analytic techniques. 18.11.2020, 14:10 (Wednesday) Misha Bialy (Tel Aviv University) Title: The Birkhoff-Poritsky conjecture for centrally-symmetric billiard tables Location: Zoom session, the link is available upon request by email Abstract: In this talk (joint work with A.E. Mironov) I shall discuss a recent proof of the Birkhoff-Poritsky conjecture for centrally-symmetric C^2-smooth convex planar billiards. We assume that the domain  between the invariant curve of 4-periodic orbits and the boundary of the phase cylinder is foliated by C^0-invariant curves. Under this assumption we prove that the billiard curve is an ellipse. The main ingredients of the proof are : (1) the non-standard generating function for convex billiards; (2) the remarkable structure of the invariant curve consisting of 4-periodic orbits; and (3) the integral-geometry approach initiated for rigidity results of circular billiards. Surprisingly, our result yields a Hopf-type rigidity for billiard in ellipse. 25.11.2020, 14:10 (Wednesday) Frol Zapolsky (University of Haifa) Title: Relative symplectic cohomology and ideal-valued measures Location: Zoom session, the link is available upon request by email Abstract: In a joint work in progress together with A. Dickstein, Y. Ganor, and L. Polterovich we prove new symplectic rigidity results. First, we categorify the notion of a heavy subset of a symplectic manifold (due to Entov-Polterovich), and in particular provide a simple algebraic criterion which guarantees that two heavy sets intersect. Next, we treat involutive maps defined on a symplectic manifold M; a smooth map M -> B is involutive if pullbacks of smooth functions on B Poisson commute. For such maps we prove a refinement of Entov-Polterovich's nondisplaceable fiber theorem, as well as a symplectic Tverberg-type theorem, which roughly says that each involutive map into a manifold of sufficiently low dimension has a fiber which intersects a wide family of subsets of M. All of these results are proved using a generalized version of Gromov's notion of ideal-valued measures, which furnish an easily digestible way to package the relevant information. We construct such measures using relative symplectic cohomology, an invariant recently introduced by U. Varolgunes, who also proved the Mayer-Vietoris property for it, on which our work relies in a crucial manner. Our main technical innovation is the relative symplectic cohomology of a pair, whose construction is inspired by homotopy theory. 02.12.2020, 17:10 (Wednesday) (PLEASE NOTE CHANGE IN TIME!) Vinicius G. B. Ramos (IMPA, Brazil) Title: Examples around the strong Viterbo conjecture Location: Zoom session, the link is available upon request by email Abstract: The Viterbo conjecture states that the ball maximizes any normalized symplectic capacity within all convex sets in R^{2n} of a fixed volume and that it is the unique maximizer. A stronger conjecture says that all normalized capacities coincide for convex sets. In joint work with Gutt and Hutchings, we prove the stronger conjecture for a somewhat different class of 4-dimensional domains, namely toric domains with a dynamically convex toric boundary. In joint work with Ostrover and Sepe, we prove that a 4-dimensional Lagrangian product which is a maximizer of the Hofer-Zehnder capacity is non-trivially symplectomorphic to a ball giving further evidence to the uniqueness claim of Viterbo's conjecture. In this talk, I will explain the proof of these two results. 09.12.2020, 14:10 (Wednesday) Lev Birbrair (Universidade Federal do Ceará, Brazil) Title: Lipschitz geometry of surface germs in $\R^4$: metric knots Location: Zoom session, the link is available upon request by email Abstract: A link at the origin of an isolated singularity of a two-dimensional semialgebraic surface in $\R^4$ is a topological knot (or link) in $S^3$. We study the connection between the ambient Lipschitz geometry of semialgebraic surface germs in $\R^4$ and the knot theory. Namely, for any knot $K$, we construct a surface $X_K$ in $\R^4$ such that: $X_K$ has a trivial knot at the origin; the germs $X_K$ are outer bi-Lipschitz equivalent for all $K$; two germs $X_{K}$ and $X_{K'}$ are ambient bi-Lipschitz equivalent only if the knots $K$ and $K'$ are isotopic. 16.12.2020, 14:10 (Wednesday) Barak Weiss (Tel Aviv University) Title: Ergodicity of rel foliations on the space of holomorphic one forms Location: Zoom session, the link is available upon request by email Abstract: The rel foliation is a foliation of the moduli space of abelian differentials obtained by "moving the zeroes of the one form while keeping all absolute periods fixed". It has been studied in complex analysis and dynamics under different names (isoperiodic foliation, Schiffer variation, kernel foliation). Until recent years the question of its ergodicity was wide open. Recently partial results were obtained by Calsamiglia-Deroin-Francaviglia and by Hamenstadt. In our work we completely resolve the ergodicity question. Joint work in progress with Jon Chaika and Alex Eskin, based on a far-reaching extension of a celebrated result of Eskin and Mirzakhani.  All relevant notions will be explained in the lecture and no prior familiarity with dynamics on spaces of one forms will be assumed. 23.12.2020, 14:10 (Wednesday) Ood Shabtai (Tel Aviv University) Title: On polynomials in two spectral projections of spin operators Location: Zoom session, the link is available upon request by email Abstract: We discuss the semiclassical behavior of an arbitrary bivariate polynomial evaluated on a pair of spectral projections of spin operators, and compare it with its value on a pair of random projections. 30.12.2020, 14:10 (Wednesday) Shira Tanny (Tel Aviv University) Title: A max-inequality for spectral invariants of disjointly supported Hamiltonians Location: Zoom session, the link is available upon request by email Abstract: The relation between spectral invariants of disjointly supported Hamiltonians and that of their sum was studied by Humiliere, Le Roux and Seyfaddini on aspherical manifolds. We study this relation in a wider setting and derive applications to Polterovich's Poisson bracket invariant. This is a work in progress. 06.01.2021, 14:10 (Wednesday) Vincent Humilière (Sorbonne University) Title: Is the group of compactly supported area preserving homeomorphisms of the 2-disk simple? Location: Zoom session, the link is available upon request by email Abstract: This long standing open problem has been recently solved in joint work with Dan Cristofaro-Gardiner and Sobhan Seyfaddini. I will present some background and the main ideas that lead to the proof. It is based on tools from symplectic topology and more precisely on a theory due to Hutchings, called Periodic Floer Homology. 13.01.2021, 14:00 (Wednesday) Dan Mangoubi (The Hebrew University of Jerusalem) Title: A Local version of Courant's Nodal domain Theorem Location: Zoom session, the link is available upon request by email Abstract: Let u_k be an eigenfunction of a vibrating string (with fixed ends) corresponding to the k-th eigenvalue. It is not difficult to show that the number of zeros of u_k is exactly k+1. Equivalently, the number of connected components of the complement of $u_k=0$ is $k$. In 1923 Courant found that in higher dimensions (considering eigenfunctions of the Laplacian on a closed Riemannian manifold M) the number of connected components of the open set $M\setminus {u_k=0}$ is at most $k$. In 1988 Donnelly and Fefferman gave a bound on the number of connected components of $B\setminus {u_k=0}$, where $B$ is a ball in $M$. However, their estimate was not sharp (even for spherical harmonics). We describe the ideas which give the sharp bound on the number of connected components in a ball. The talk is based on a joint work with S. Chanillo, A. Logunov and E. Malinnikova, with a contribution due to F. Nazarov. 03.03.2021, 14:10 (Wednesday) Boaz Klartag (Weizmann Institute of Science) Title: Rigidity of Riemannian embeddings of discrete metric spaces Location: Zoom session, the link is available upon request by email Abstract: Let M be a complete, connected Riemannian surface and suppose that S is a discrete subset of M. What can we learn about M from the knowledge of all distances in the surface between pairs of points of S? We prove that if the distances in S correspond to the distances in a 2-dimensional lattice, or more generally in an arbitrary net in R^2, then M is isometric to the Euclidean plane. We thus find that Riemannian embeddings of certain discrete metric spaces are rather rigid. A corollary is that a subset of Z^3 that strictly contains a two-dimensional lattice cannot be isometrically embedded in any complete Riemannian surface. This is a joint work with M. Eilat. 10.03.2021, 14:10 (Wednesday) Daniel Cristofaro-Gardiner (IAS Princeton; University of California, Santa Cruz) Title: The subleading asymptotics of the ECH spectrum Location: Zoom session, the link is available upon request by email Abstract: Embedded contact homology can be used to associate a sequence of spectral invariants, called ECH spectral invariants, to any closed three-manifold with a contact form.  In previous joint work, we proved a “Volume Property” that recovers the volume of any such manifold from the asymptotics of its ECH spectral invariants.  I will discuss recent work aimed at better understanding the subleading asymptotics of this sequence.  The main subject of my talk will be a joint work with Nikhil Savale in which we prove a new bound on the growth rate of the subleading asymptotics.   I will also briefly mention a conjecture, due to Hutchings, concerning recovering the “contact Ruelle invariant” from the subleading asymptotics. 17.03.2021, 14:10 (Wednesday) Dmitry Faifman (Tel Aviv University) Title: Around the Funk metric and its billiards Location: Zoom session, the link is available upon request by email Abstract: The Funk metric in the interior of a convex body is a lesser known relative of the projectively-invariant Hilbert metric, yet in some ways simpler and more natural. Starting with a few simple observations, we will explore some Funk-inspired generalizations of well-known results in the geometry of normed spaces and Minkowski billiards, such as Schäffer's dual girth conjecture and the Gutkin-Tabachnikov duality. I will also offer a Funk approach to the integrability of the hyperbolic billiard in a conic. Time permitting, I will discuss the volume of metric balls in Funk geometry, leading to a generalization of the Blaschke-Santalo inequality. 24.03.2021, 14:10 (Wednesday) Daniel Peralta-Salas (ICMAT Madrid) Title: Turing completeness and universality of steady Euler flows Location: Zoom session, the link is available upon request by email Abstract: I will review recents results on the Turing completeness and universality of steady solutions to the Euler equations. In particular, I will show the existence of three-dimensional fluid flows exhibiting undecidable trajectories and discuss other universality features such as embeddability of diffeomorphisms into steady Euler states. These results are motivated by Tao's programme to address the blow-up problem for the Navier-Stokes equations based on the Turing completeness of the fluid flows. This is based on joint works with Robert Cardona, Eva Miranda and Francisco Presas. 07.04.2021, 14:10 (Wednesday) Daniel Rosen (Ruhr-Universität Bochum) Title: Random inscribed polytopes in Non-Euclidean Geometries Location: Zoom session, the link is available upon request by email Abstract: Random polytopes have a long history, going back to Sylvester's famous four-point problem. Since then their study has become a mainstream topic in convex and stochastic geometry, with close connection to polytopal approximation problems, among other things. In this talk we will consider random polytopes in constant curvature geometries, and show that their volume satisfies a central limit theorem. The proof uses Stein's method for normal approximation, and extends to general projective Finsler metrics. 21.04.2021, 14:10 (Wednesday) Thomas Melistas (University of Georgia) Title: The Large-Scale Geometry of Overtwisted Contact Forms Location: Zoom session, the link is available upon request by email 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. We compare it to a recently defined contact Banach-Mazur distance by Rosen and Zhang and we use it in order 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. 28.04.2021, 14:10 (Wednesday) Zvi Shem-Tov (The Hebrew University of Jerusalem) Title: Conjugation-invariant norms on arithmetic groups Location: Zoom session, the link is available upon request by email Abstract: A classical theorem of Ostrowski says that every absolute value on the field of rational numbers, or equivalently on the ring of integers, is equivalent to either the standard (real) absolute value, or a $p$-adic absolute value, for which the closure of the integers is compact. In this talk we will see a non-abelian analogue of this result for $SL(n\ge3,\Z)$, and related groups of arithmetic type. We will see a relation to the celebrated Margulis' normal subgroup theorem, and derive rigidity results for homomorphisms into certain non-locally compact groups -- those endowed with a bi-invariant metric. We will also discuss a relation to the deep work of Nikolov-Segal on profinite groups. This is a joint work with Leonid Polterovich and Yehuda Shalom. 05.05.2021, 14:10 (Wednesday) Otto van Koert (Seoul National University) Title: A generalization of the Poincare-Birkhoff fixed point theorem and the restricted three-body problem Location: Zoom session, the link is available upon request by email Abstract: In joint work with Agustin Moreno, we propose a generalization of the Poincare-Birkhoff fixed point theorem. We start with a construction of global hypersurfaces of section in the spatial three-body problem, describe some return maps and suggest some generalizations of the Poincare-Birkhoff fixed point theorem. We use symplectic homology in the proof of our theorem. 12.05.2021, 14:10 (Wednesday) Georgios Dimitroglou Rizell (Uppsala University) Title: Non-degeneracy of Legendrians from bifurcation of contact homology Location: Zoom session, the link is available upon request by email Abstract: We show that the invariance of Legendrian contact homology can be formulated in terms of a bifurcation analysis whose action properties are continuous with respect to the oscillatory norm of the contact Hamiltonian. (I.e. the barcode varies continuously with respect to the same.) Combined with work of Rosen-Zhang this implies non-degeneracy of the Shelukhin-Chekanov-Hofer metric on the space of Legendrian embeddings. We also explain how convex surface techniques in dimension three can be used to prove a statement related to the converse: a non-Legendrian knot cannot be approximated by the image of a Legendrian knot under a sequence of C0-converging contactomorphisms. This is joint work with M. Sullivan. 19.05.2021, 14:10 (Wednesday) Luis Diogo (Fluminense Federal University, Brazil) Title: Monotone Lagrangians in cotangent bundles of spheres Location: Zoom session, the link is available upon request by email Abstract: Among all Lagrangian submanifolds of a symplectic manifold, the class of monotone Lagrangians is often very rich and nicely suited to being studied using pseudoholomophic curves. We find a family of monotone Lagrangians in cotangent bundles of spheres with the following property: every compact monotone Lagrangian with non-trivial Floer cohomology cannot be displaced by a Hamiltonian diffeomorphism from at least one element in the family. This follows from the fact that the Lagrangians in the family split-generate the compact monotone Fukaya category. This is joint work with Mohammed Abouzaid. 26.05.2021, 14:10 (Wednesday) Jonathan David Evans (Lancaster University) Title: TBA Location: Zoom session, the link is available upon request by email Abstract: TBA 02.06.2021, 14:10 (Wednesday) Jeff Hicks (University of Cambridge) Title: TBA Location: Zoom session, the link is available upon request by email Abstract: TBA 09.06.2021, 14:10 (Wednesday) Itamar Rosenfeld Rauch (Technion, Haifa) Title: TBA Location: Zoom session, the link is available upon request by email Abstract: TBA 16.06.2021, 14:10 (Wednesday) Marcelo R.R. Alves (University of Antwerp) 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