# Geometry & Dynamics Seminar 2017-18

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 25.10.2017, 14:10 (Wednesday) Orientation meeting for students Location: Schreiber bldg., room 309, Tel-Aviv University 31.10.2017,12:10 (Tuesday) SPECIAL LECTURE - PLEASE NOTE THE DATE,  PLACE & TIME Jean-Michel Bismut, Université Paris-Sud (Orsay) Title: Hypoelliptic Laplacian, index theory and the trace formula Location: Schreiber bldg., room 209, Tel-Aviv University Abstract: The hypoelliptic Laplacian is a family of operators, indexed by $b\in \mathbf{R}_{+}^{*}$, acting on the total space of the tangent  bundle of a Riemannian manifold, that interpolates between the ordinary Laplacian as $b\to 0$ and the generator of the geodesic flow as $b\to + \infty$.  These operators are not elliptic, they are not self-adjoint, they are hypoelliptic. The hypoelliptic deformation preserves subtle invariants of the Laplacian. In the case of locally symmetric spaces, the deformation is essentially isospectral. In a first part of the talk, I will describe the geometric construction of the hypoelliptic Laplacian in the context of de Rham theory. In a second part, I will explain applications to the trace formula. 01.11.2017, 14:10 (Wednesday) Daniel Rosen (TAU) Title: Duality of Caustics in Minkowski Billiards Location: Schreiber bldg., room 309, Tel-Aviv University Abstract: Mathematical billiards are a classical and well-studied class of dynamical systems, "a mathematician’s playground". Convex caustics, which are curves to which billiard trajectories remain forever tangent, play an important role in the study of billiards. In this talk we will discuss convex caustic in Minkowski billiards, which is the generalization of classical billiards no non-Euclidean normed planes. In this case a natural duality arises from, roughly speaking,  interchanging the roles of the billiard table and the unit ball of the (dual) norm. This leads to duality of caustics in Minkowski billiards. Such a pair of caustics is dual in a strong sense, and in particular they have equal perimeters and other classical parameters. We will show that, when the norm is Euclidean, every caustic possesses a dual caustic, but in general this phenomenon fails. Based on joint work with S. Artstein-Avidan, D. Florentin, and Y. Ostrover . 08.11.2017, 14:10 (Wednesday) Mads Bisgaard (ETH) Title: Topology of small Lagrangian cobordisms Location: Schreiber bldg., room 309, Tel-Aviv University Abstract: I will discuss how one can study Lagrangian cobordisms from the point of view of quantitative symplectic topology: It turns out that if a Lagrangian cobordism is sufficiently small (in a sense which can be made precise), then its topology is to a large extend determined by its boundary. I will show how this principle allows one to derive several homological uniqueness results for small Lagrangian cobordisms. In particular (under the smallness assumption) I will prove homological uniqueness of the class of Lagrangian cobordisms which, by Biran-Cornea’s Lagrangian cobordism theory, induces operations on a version of the derived Fukaya category. If time permits it, I will indicate a link from these ideas to Vassilyev’s theory of Lagrange characteristic classes and the classification of caustics. 15.11.2017, 14:10 (Wednesday) Dmitry Novikov (Weizmann Institute) Title: Complex cellular parameterization Location: Schreiber bldg., room 309, Tel-Aviv University Abstract: We introduce the notion of a complex cell, a complex analog of the cell decompositions used in real algebraic and analytic geometry. Complex cells defined using holomorphic data admit a natural notion of analytic continuation called $\delta$-extension, which gives rise to a rich hyperbolic geometric structure absent in the real case. We use this structure to prove that complex cellular decompositions share some interesting features with the classical constructions in the theory of resolution of singularities. Restriction of a complex cellular decomposition to the reals recovers the preparation theorem for subanalytic functions, and can be viewed as an analytic continuation thereof. A key difference in comparison to the classical resolution of singularities is that the cellular decompositions are intrinsically uniform over (sub)analytic families. We deduce a subanalytic version of the Yomdin-Gromov theorem where $C^k$-smooth maps are replaced by mild maps. (joint work with Gal Binyamini) 22.11.2017, 14:10 (Wednesday) Boris Kruglikov (University of Tromsø) Title: Integrability in Grassmann geometry and twistor theory Location: Schreiber bldg., room 309, Tel-Aviv University Abstract: This reviews a series of works joint with E.Ferapontov, D.Calderbank, B.Doubrov and V.Novikov. It will be explained that for several important classes of PDEs the integrability by the method of hydrodynamic reductions is equivalent to a Lax representation. This includes equations of Hirota type and also PDE systems encoded by submanifolds in Grassmannians. For the latter the integrability can be interpreted geometrically. In 3D and 4D the integrability is also shown to be equivalent to Einstein-Weyl and, respectively, self-dual geometry on solutions. This relates dispersionless integrability to the twistor theory. Ref: J.Diff.Geom.97 (2014), arXiv:1503.02274, arXiv:1612.02753, arXiv:1705.06999. 29.11.2017, 14:10 (Wednesday) Igor Uljarevic Title: Floer homology and contact Hamiltonians Location: Schreiber bldg., room 309, Tel-Aviv University Abstract: In the setting of symplectic manifolds which are convex at inﬁnity, we use a version of the Aleksandrov maximum principle to extend the class of Hamiltonians that one can use in the direct limit when constructing symplectic homology. As an application, we detect elements of inﬁnite order in the symplectic mapping class group of a Liouville domain and prove existence results for translated points. The talk is based on joint work with W. Merry. 06.12.2017, 14:10 (Wednesday) Dmitry Faifman (University of Toronto) Title: Contact curvatures and integral geometry of the contact sphere Location: Schreiber bldg., room 309, Tel-Aviv University Abstract: Valuations are finitely additive measures on nice subsets, for example the Euler characteristic, volume and surface area are valuations. During the 20th century, valuations have been studied predominantly on convex bodies and polytopes, in linear spaces and lattices. Valuations on manifolds were introduced about 15 years ago by S. Alesker, with contributions by A. Bernig, J. Fu and others, and immediately brought under one umbrella a range of classical results in Riemannian geometry, notably Weyl's tube formula and the Chern-Gauss-Bonnet theorem. These results circle around the real orthogonal group. In the talk, the real symplectic group will be the central player. Drawing inspiration from the Lipschitz-Killing curvatures in the Riemannian setting, we will construct some natural valuations on contact and almost contact manifolds, which generalized the Gaussian curvature. We will also construct symplectic-invariant distributions on the grassmannian, leading to Crofton-type formulas on the contact sphere and symplectic space. 13.12.2017, 14:10 (Wednesday) Sergey Fomin (University of Michigan) - MINT distinguished lecture Title: Morsifications and mutations Location: Schreiber bldg., room 309, Tel-Aviv University Abstract: I will discuss a surprising connection between singularity theory and cluster algebras, more specifically between (1) the topology of isolated singularities of plane curves and (2) the mutation equivalence of the quivers associated with their morsifications. Joint work with Pavlo Pylyavskyy and Eugenii Shustin. 20.12.2017, 14:10 (Wednesday) NO SEMINAR THIS WEEK 27.12.2017, 14:10 - 15:00 (Wednesday) Kei Irie (Kyoto University and Simons Center for Geometry and Physics) Title: Denseness of minimal hypersurfaces for generic metrics Location: Schreiber bldg., room 309, Tel-Aviv University Abstract: We prove that, on a smooth closed manifold of dimension $3 \le d \le 7$ with a $C^\infty$-generic Riemannian metric, the union of closed embedded minimal hypersurfaces is dense. This is joint work with F.Marques and A.Neves. The proof is based on min-max theory for the volume functional on the space of codimension 1 (flat) cycles,  which was originally developed by Almgren and Pitts. The key ingredient of the proof is the Weyl law''(proved by Liokumovich, Marques and Neves), which says that the asymptotic of min-max values in this theory recovers the volume of a Riemannian manifold. 27.12.2017, 15:10 - 16:00 (Wednesday) Iosif Polterovich (Université de Montréal) Title: Isoperimetric inequalities for Laplace eigenvalues on surfaces: some recent developments Location: Schreiber bldg., room 309, Tel-Aviv University Abstract: Isoperimetric inequalities for Laplace eigenvalues have a long history, going back to the celebrated Rayleigh-Faber-Krahn inequality for the fundamental tone. Still, many basic questions remain unanswered, particularly, for higher eigenvalues. In the talk I will give an overview of some recent developments in the study of isoperimetric inequalities for eigenvalues on compact surfaces with a Riemannian metric. In particular, I will discuss a solution of a conjecture posed by N. Nadirashvili in 2002 regarding the maximization of higher Laplace-Beltrami eigenvalues on the sphere (joint with M. Karpukhin, N. Nadirashvili and A. Penskoi). 03.01.2018, 14:10 (Wednesday) Julian Chaidez (University of California, Berkeley) Title: The Conley-Zehnder Index In Singular Contact Geometry Location: Schreiber bldg., room 309, Tel-Aviv University Abstract: The Conley-Zehnder (CZ) index is an important invariant of closed orbits of smooth Hamiltonian flows and Reeb flows. In this talk, I will discuss a version of the CZ index for various "singular" contact geometry problems, such as Reeb dynamics on polytopes and dynamical billiards. We will show how this CZ index can be applied to convert some results in smooth contact geometry into results about singular contact geometry using a limiting argument. 10.01.2018, 14:00 (Wednesday) Pazit Haim-Kislev (TAU) Title: The EHZ capacity of convex polytopes Location: Schreiber bldg., room 309, Tel-Aviv University Abstract: The EHZ capacity is a well-studied symplectic invariant that measures the "symplectic size" of convex sets, by taking the minimal action of a closed characteristic on the boundary. We introduce a simplification to the problem of finding a closed characteristic with minimal action for the case of convex polytopes. We use this to give a combinatorial formula for the EHZ capacity of convex polytopes, and to prove a certain subadditivity property of the capacity of a general convex body. 17.01.2018, 14:10 (Wednesday) Daniel Alvarez-Gavela (Stanford University) Title: Singularities of fronts and their simplification Location: Schreiber bldg., room 309, Tel-Aviv University Abstract: We will present a full h-principle for the simplification of singularities of Lagrangian and Legendrian fronts. We give several applications to symplectic and contact topology, including relations to pseudo-isotopy theory and to Nadler's program for the arborealization of Lagrangian skeleta. 07.03.2018, 14:10 (Wednesday) Vukasin Stojisavljevic (TAU) Title: Persistence barcodes and Laplace eigenfunctions on surfaces Location: Schreiber bldg., room 309, Tel-Aviv University Abstract: Using barcodes, we will define a family of functionals on the space of continuous functions on a smooth manifold. These functionals behave well with respect to C^0-distance between functions. Furthermore, we will show that on the space of linear combinations of Laplace eigenfunctions on surfaces these functionals have certain upper bounds. As an application of all of these properties we obtain results about C^0-approximations of a function by linear combinations of Laplace eigenfunctions. Based on a joint work with I. Polterovich and L. Polterovich. 14.03.2018, 14:10 (Wednesday) Matthias Meiwes (TAU) Title: Dynamically exotic contact spheres Location: Schreiber bldg., room 309, Tel-Aviv University Abstract: Topological entropy is a dynamical invariant that codifies in a single number the exponential instability of a dynamical system. In this talk I'll discuss some results on the existence of contact structures on manifolds such that every Reeb flow has positive topological entropy. In dimensions greater than 5, even spheres, in contrast to their simple topology, carry such contact structures.  I'll discuss how, in many cases, algebraic growth properties of wrapped Floer homology play a crucial role in detecting positive entropy. Finally, I will describe a more recent approach to the construction of such contact manifolds, using the growth of Rabinowitz Floer homology. This is joint work with Marcelo Alves. 21.03.2018, 14:10 (Wednesday) Peter Ozsvath (Princeton University) - Blumenthal lectures in Geometry Title: An algebraic construction of knot Floer homology Location: Schreiber bldg., room 309, Tel-Aviv University Abstract: Bordered Floer homology is an invariant for three-manifolds with boundary.  I will discuss an algebraic approach to computing knot Floer homology, based on decomposing knot diagrams.  This is joint work with Zoltan Szabo, influenced by earlier joint work with Ciprian Manolescu and Sucharit Sarkar; and Robert Lipshitz and Dylan Thurston. 28.03.2018, 14:10 (Wednesday) Yaniv Ganor (TAU) Title: A homotopical viewpoint at the Poisson bracket invariants for tuples of sets Location: Schreiber bldg., room 309, Tel-Aviv University Abstract: We suggest a homotopical description of the Poisson bracket invariants for tuples of closed sets in symplectic manifolds. It implies that these invariants depend only on the union of the sets along with topological data. 11.04.2018, 14:10 (Wednesday) Jarek Kedra (University of Aberdeen) Title: Boundedness properties of SL(n,Z) Location: Schreiber bldg., room 309, Tel-Aviv University Abstract: A group G is called bounded if any conjugation-invariant norm on G has finite diameter. When looking at conjugation-invariant word norms then the diameter depends on the choice of a generating set. I will discuss the subtleties of this dependence, provide more details on the behaviour of SL(3,Z) and present some applications. Joint work with Assaf Libman and Ben Martin. 25.04.2018, 14:10 (Wednesday) Sheng-Fu Chiu (Academia Sinica Institute of Mathematics, Taiwan) Title: Microlocal Sheaf Theory and Contact Non-squeezing Location: Schreiber bldg., room 309, Tel-Aviv University Abstract: In this talk we will discuss the contact non-squeezing problem of pre-quantized balls in the ambient pre-quantization space. We will provide an approach from the mixed viewpoints of operator valued measurement, categorification of symplectic/contact morphism, and the homological algebra of microlocal sheaf theory. NO SEMINAR THIS WEEK! 09.05.2018, 14:10-15:00 (Wednesday) Michael Brandenbursky (BGU) Title: Entropy, metrics and quasi-morphisms Location: Schreiber bldg., room 309, Tel-Aviv University Abstract: One of the mainstream and modern tools in the study of non abelian groups are quasi-morphisms. These are functions from a group to the reals which satisfy homomorphism condition up to a bounded error. Nowadays they are used in many fields of mathematics. For instance, they are related to bounded cohomology, stable commutator length, metrics on diffeomorphism groups, displacement of sets in symplectic topology, dynamics, knot theory, orderability, and the study of mapping class groups and of concordance group of knots. Let S be a compact oriented surface. In this talk I will discuss several invariant metrics and quasi-morphisms on the identity component Diff_0(S, area) of the group of area preserving diffeomorphisms of S. In particular, I will show that some quasi-morphisms on Diff_0(S, area) are related to the topological entropy. More precisely, I will discuss a construction of infinitely many linearly independent quasi-morphisms on Diff_0(S, area) whose absolute values bound from below the topological entropy. If time permits, I will define a bi-invariant metric on this group, called the entropy metric, show that it is unbounded, and will discuss a relation with Katok's conjecture. Based on a joint work with M. Marcinkowski. 09.05.2018, 15:10-16:00 (Wednesday) Leonid Potyagailo (Université Lille 1) Title: Martin and Floyd boundaries of finitely generated groups Location: Schreiber bldg., room 309, Tel-Aviv University Abstract: ABSTRACT 16.05.2018, 14:10 (Wednesday) Egor Shelukhin (University of Montreal) Title: On the geometry of Lagrangian flux Location: Schreiber bldg., room 309, Tel-Aviv University Abstract: We discuss the space of first cohomology classes of a given Lagrangian submanifold L that can be obtained as the flux of Lagrangian isotopies starting with it. When the flux is required to move along a fixed direction as the isotopy evolves, or when the isotopy stays within a given Weinstein neighborhood of L, we obtain bounds on this space (and restrictions on neighborhoods), and in some cases, its exact calculation. This direction of research was initiated by Benci, Sikorav, and Eliashberg, and was recently studied by Entov-Ganor-Membrez. An important part of our considerations is based on an invariant produced from the Fukaya algebra of the Lagrangian, and its properties. This invariant can be considered to be, roughly, the minimal area of a holomorphic disk on L with a non-trivial algebraic count. This is joint work with Dmitry Tonkonog and Renato Vianna. 23.05.2018, 14:10 (Wednesday) Asaf Kislev (TAU) Title: Sharp bounds on the boundary depth Location: Schreiber bldg., room 309, Tel-Aviv University Abstract: The plan for this talk is to show that the bottleneck distance between Floer barcodes associated to Hamiltonian diffeomorphisms \phi and \psi, is Lipschitz with respect to the spectral norm \gamma(\phi \psi^{-1} ). We use this to show that Usher's boundary depth of RP^n in CP^n with arbitrary Hamiltonian perturbations is bounded above by n / (2n+2), while the absolute boundary depth of any Hamiltonian diffeomorphism of CP^n is bounded above by n/(n+1). We prove that these bounds are sharp, and give some examples and applications. Joint work with Egor Shelukhin. 30.05.2018, 14:10-15:00 (Wednesday) Marie-Claude Arnaud (University of Avignon) Title: Arnol’d-Liouville theorems in low regularity Location: Schreiber bldg., room 309, Tel-Aviv University Abstract: Classical Arnol’d Liouville theorem describes precisely the Dynamics of Hamiltonian systems that have enough independent C2 integrals. For such Hamiltonians, it is known that there is an invariant Lagrangian foliation that is symplectically diffeomorphic to the standard one and that the Dynamics restricted to every invariant leaf is conjugate to a translation flow. Here we focus on what happens when we have lower regularity. The motivation for studying low regularity is that when a Tonelli Hamiltonian has no conjugate points, only the existence of continuous integrals can be proved. More precisely, we will raise the question of  which  continuous Lagrangian foliations are symplectically homeomorphic to the standard one and prove that when the integrals are just C1 and when the Hamiltonian is Tonelli,  we indeed obtain a continuous Lagrangian foliation that is symplectically homeomorphic to the standard one and that Arnol’d-Liouville theorem remains true with a symplectic homeomorphism instead of a C1 change of coordinates. 30.05.2018, 15:10-16:00 (Wednesday) Vinicius Gripp B. Ramos (IMPA) Title: Symplectic embeddings, lagrangian products and integrable systems Location: Schreiber bldg., room 309, Tel-Aviv University Abstract: Symplectic embedding problems are at the core of the study of symplectic topology. There are many well-known results for so-called toric domains, but very little is known about other kinds of domains. In this talk, I will mostly speak about a different kind of domain, namely a lagrangian product. These domains are of a very different nature and are related to billiards, as discovered by Artstein-Avidan and Ostrover. I will explain how to use integrable systems to see that some of these products are secretely toric domains and how to use symplectic capacities to obtain sharp obstructions to many symplectic embedding problems. 06.06.2018, 14:10 (Wednesday) Shira Tanny (TAU) Title: The Poisson bracket conjecture in dimension 2 Location: Schreiber bldg., room 309, Tel-Aviv University Abstract: I will discuss the Poisson bracket invariant of covers and explain how elementary geometric arguments can be used to prove Polterovich's pb-conjecture in dimension 2. This is a joint work with L. Buhovsky and S. Logunov. We express our gratitude to F. Nazarov for his contribution to this work. 13.06.2018, 14:10 (Wednesday) Misha Bialy (TAU) Title: Around Birkhoff conjecture for convex billiards Location: Schreiber bldg., room 309, Tel-Aviv University Abstract: In this talk I will discuss a very old conjecture attributed to G.Birkhoff on convex billiards. It states that the only integrable convex billiards in the plane are ellipses. Here it is very important to specify what is understood by integrability. This talk is about recent important developments toward a positive solution of this conjecture, and on the geometric problems naturally arising on the way. The talk is supposed to be elementary and does not require any prior knowledge.

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