Geometry & Dynamics Seminar 2014-15

The usual place and time are Tel Aviv University's Schreiber Building room 209, on Wednesdays at 14:10.

Please check each announcement since this is sometimes changed.

29.10.2014, 14:10 (Wednesday)	Orientation meeting for students
Location:	Schreiber bldg., room 209, Tel-Aviv University
5.11.2014, 14:10 (Wednesday)	Emmanuel Opshtein (Université de Strasbourg)
Title:	h-principle vs quantitative h-principle in symplectic geometry, with applications.
Location:	Schreiber bldg., room 209, Tel-Aviv University
Abstract:	The main subject of this talk is C^0 symplectic geometry, defined as the geometry of symplectic homeomorphisms : the homeomorphisms which are C^0 limits of symplectic diffeomorphisms. I will review some standard but fundamental "h-principle statements" in symplectic geometry (the easiest of which may be that any two symplectic discs of the same area are symplectic isotopic), and introduce a finer notion, called quantitative h-principle. I will then explain some applications of this notion to C^0 symplectic geometry: classical symplectic invariants associated to submanifolds of low symplectic codimension are also  C^0-invariants, while these invariants are destroyed by symplectic homeomorphisms in high codimension.  This is a joint work with Lev Buhvoski.
12.11.2014, 14:10 (Wednesday)	Cedric Membrez (TAU)
Title:	The Lagrangian cubic equation
Location:	Schreiber bldg., room 209, Tel-Aviv University
Abstract:	This is joint work with Paul Biran. Let M be a closed symplectic manifold and L a Lagrangian submanifold. Denote by [L] the homology class induced by L viewed as a class in the quantum homology of M. This talk is concerned with properties and identities involving the class [L] in the quantum homology ring. We also study the relations between these identities and invariants of L coming from Lagrangian Floer theory. We pay special attention to the case when L is a Lagrangian sphere.
26.11.2014, 14:10 (Wednesday)	Mira Shamis (Weizmann Institute)
Title:	The standard map, and discrete Schroedinger operators
Location:	Schreiber bldg., room 209, Tel-Aviv University
Abstract:	The standard map is a measure-preserving map of the torus; the dynamics generated by it is the subject of numerous conjectures. One of the approaches to the standard map leads to the study of a certain Schroedinger operator. I will start with a brief introduction to discrete Schroedinger operators, and present two results: one pertaining to a general class of discrete Schroedinger operators, and another one -- pertaining to the operator arising from the standard map. Time permitting, I will explain some of the elements of the proof. [Based on joint work with T. Spencer]
3.12.2014, 14:10 (Wednesday)	Henri Berestycki (Ecoles des hautes études en sciences sociales)  MINT Distinguished Lecture
Title:	The effect of domain shape on propagation and blocking for reaction-diffusion equations
Location:	Schreiber bldg., room 209, Tel-Aviv University
Abstract:	I will discuss reaction-diffusion equations motivated by biology and medicine for which the aim is to understand the effect of the shape of the domain on propagation or on blocking of advancing waves. I will first describe the motivations of these questions. I will then discuss various geometric conditions that lead to either blocking, or partial propagation, or complete propagation. These questions involve qualitative results for some non-linear elliptic and parabolic partial differential equations.
10.12.2014, 14:10 (Wednesday)	Yohann Le Floch
Title:	Berezin-Toeplitz operators on surfaces
Location:	Schreiber bldg., room 007, Tel-Aviv University
Abstract:	Berezin-Toeplitz operators appear when studying the semiclassical limit of geometric quantization for compact Kähler manifolds. The aim of this talk will be to explain these terms: I will first define what is meant by "quantization", and I will then introduce geometric quantization and Berezin-Toeplitz operators, emphasizing on the case of surfaces. I will also give several examples. If time permits, I will present some results about the spectral theory of Berezin-Toeplitz operators on surfaces.
17.12.2014, 14:10 (Wednesday)	Iosif Polterovich (Université de Montréal)
Title:	Billiards with a large Weyl remainder
Location:	Schreiber bldg., room 007, Tel-Aviv University
Abstract:	The classical Hardy-Landau lower bound for the error term in the Gauss circle problem can be viewed as an estimate from below for the remainder in Weyl's law for the eigenvalue counting function on a torus. In the talk we  will present  an analogous estimate for certain  planar domains admitting an appropriate one-parameter family of periodic billiard trajectories. Examples include ellipses and smooth domains of constant width. In higher dimensions, lower bounds  on the remainder in Weyl's law are of somewhat different  nature, and they will be discussed as well. The talk  is based on a joint work with S. Eswarathasan and J. Toth.
24.12.2014, 14:10 (Wednesday)	Albert Fathi (École Normale Supérieure de Lyon)
Title:	Mather Duality for non-coercive Hamiltonian convex in the momentum
Location:	Schreiber bldg., room 007, Tel-Aviv University
Abstract:	See here
7.1.2015, 14:10 (Wednesday)	Alexander Caviedes (TAU)
Title:	The Gromov width of coadjoint orbits and curve neighborhoods
Location:	Schreiber bldg., room 007, Tel-Aviv University
Abstract:	In this talk I will show how to estimate upper bounds for the Gromov width of  coadjoint orbits of compact Lie groups by computing certain Gromov-Witten invariants in terms of curve neighborhoods. Curve neighorhoods are unions of rational curves of fixed degree passing through subsets of the coadjoint orbit. If there is some time left, I will explain how lower bounds for the Gromov width can be estimated by constructing embeddings of symplectic balls in the coadjoint orbit by looking at the Gelfand-Zetlin integrable system defined on it
14.1.2015, 14:10 (Wednesday)	Alex Furman (University of Illinois at Chicago) MINT Distinguished Lecture
Title:	Simplicity of the Lyapunov spectrum via boundary theory
Location:	Schreiber bldg., room 007, Tel-Aviv University
Abstract:	Consider products of matrices that are chosen using some ergodic stationary random process on $G=SL(d,R)$, e.g. a random walk on $G$. The Multiplicative Ergodic Theorem (Oseledets) asserts that asymptotically such products behave as $\exp(n\Lambda)$ where $\Lambda$ is a fixed diagonal traceless matrix, called the Lyapunov spectrum of the system. The spectrum $\Lambda$ depends on the system in a mysterious way, and is almost never known explicitly. The best understood case is that of random walks, where by the work of Furstenberg, Guivarch-Raugi, and Gol’dsheid-Margulis we know that the spectrum is simple (i.e. all values are distinct) provided the random walk is not trapped in a proper algebraic subgroup. Recently, Avila and Viana proved a conjecture of Kontsevich-Zorich that asserts simplicity of the Lyapunov spectrum for another system related to Teichmuller flow. In the talk we shall describe an approach to proving simplicity of the spectrum based on ideas from boundary theory that were developed to prove rigidity of lattices. Based on joint work with Uri Bader.
21.1.2015, 14:10 (Wednesday)	Egor Shelukhin (CRM, University of Montreal)
Title:	The contact Hofer norm
Location:	Schreiber bldg., room 007, Tel-Aviv University
Abstract:	It is a well-known theorem of Hofer and Lalonde-McDuff that the uniform norm of normalized Hamiltonians gives a bi-invariant non-degenerate metric on the group of Hamiltonian diffeomorphisms: the Hofer metric. In this talk we show that the analogous construction in contact topology still gives a non-degenerate metric on the (identity component of) the group of contactomorphisms, which, however, is not bi-invariant. We discuss certain properties and applications of this metric, including the construction of two new bi-invariant metrics on the group of Hamiltonian diffeomorphisms of integral symplectic manifolds.
15.3.2015, 14:10 (Sunday) (NOTE SPECIAL DATE & LOCATION)	Franziska Schroeter (TAU)
Title:	The zoo of curve multiplicities in tropical enumerative geometry
Location:	Shenkar - Physics, 104, Tel-Aviv University (NOTE SPECIAL LOCATION)
Abstract:	One advantageous facet of tropical geometry is the ability to tackle enumerative problems from complex and real algebraic geometry by combinatorial means. For example, if we are interested in one particular enumerative problem of curves satisfying point conditions, there is a related tropical enumerative problem. We can assign a combinatorial multiplicity to each tropical curve we consider for it, which encodes how we can transfer precisely results for the tropical enumerative problem to the corresponding algebraic one. But there are also tropical enumerative problems having no known counterpart in algebraic geometry. My talk gives an overview about different enumerative problems at the tropical-algebraic interface studied in the last couple of years and outlines recent research in this direction. No prior knowledge of tropical geometry is required.
18.3.2015, 14:10 (Wednesday)	Patrick Iglesias-Zemmour (CNRS and Hebrew University)
Title:	About Symplectic Diffeology
Location:	Schreiber bldg., room 309, Tel-Aviv University (PLEASE NOTE CHANGE IN LOCATION)
Abstract:	Symplectic Diffeology is an extension of symplectic geometry to the category of diffeological spaces. That includes spaces of infinite dimension, or spaces that are usually regarded as singular; for example: spaces of functions or products of irrational tori. I will discuss what can mean for a parasymplectic form (i.e. a generic a closed 2-form) to be presymplectic or symplectic, in the context of diffeology. I will give a few examples for each case and also, if I have time, an example of singular (para)symplectic reduction in infinite dimension in this framework.
25.3.2015, 14:10 (Wednesday)	Światosław R. Gal (University of Wroclaw / Technion)
Title:	Bounded simplicity of groups acting on one-dimensional spaces.
Location:	Schreiber bldg., room 209, Tel-Aviv University (PLEASE NOTE CHANGE IN LOCATION)
Abstract:	A groups $G$ is simple if, for any elements $f,g\in G$ with $g\neq 1$, $f$ is a product of finitely many conjugates of $g$. If this finite  number is bounded by $N$ (independently on $f$ and $g$) we call $G$ $N$-boundedly simple. We will show that many groups (known to be simple) acting on a line or a circle, such as various Thompson, Higmann-Thompson, or Neretin groups, are $N$-boundedly simple with explicit bound for $N$. The talk would be self-contained including independent proofs of simplicity of those groups. All those groups would be treated dynamically, ie. defined by their action not referring to their presentation. Presented results are a joint work with Kuba Gismatullin from Wrocław/Jerusalem.
15.4.2015, 14:10 (Wednesday)	Grigori Olshanski (IITP and Higher School of Economics, Moscow)
Title:	What are infinite random permutations
Location:	Schreiber bldg., room 309, Tel-Aviv University
Abstract:	TRandom permutations can be viewed as a combinatorial analog of random matrices. In random matrix theory, people study the asymptotic behavior of spectra of large random matrices. Likewise, the literature in combinatorial probability contains many works on the limiting behavior of various characteristics of random permutations of large size. But are there reasonable models of random permutations of actually infinite size? I will describe some positive results in this direction.
29.4.2015, 14:10 (Wednesday)	Michael Entov (Technion)
Title:	Full symplectic packing for tori
Location:	Schreiber bldg., room 309, Tel-Aviv University
Abstract:	The symplectic packing problem is one of the major problems of symplectic topology - it concerns packing symplectic manifolds by symplectically embedded shapes (e.g. balls, polydisks etc.) and, in particular, finding the maximal fraction of the total volume of a symplectic manifold that can be filled out by the shapes. In this talk I will discuss why an even-dimensional torus T equipped with an arbitrary linear symplectic form admits a full symplectic packing by any number of balls - that is, any finite collection of disjoint standard symplectic balls admits a symplectic embedding to T, as long as their total volume is less than the volume of T. The proof uses a number of powerful rigidity results from complex geometry. The full symplectic packing of the torus T by balls can be used to prove the full symplectic packing of T by any number of equal polydisks (or any number of equal cubes), provided the cohomology class of the linear symplectic form on T is not proportional to a rational one. The proof of the latter corollary is based on Ratner's orbit closure theorem. This is a joint work with M.Verbitsky.
06.5.2015, 14:10 (Wednesday)	Misha Bialy (Tel Aviv University)
Title:	Effective bounds in E.Hopf rigidity for billiards and geodesic flows
Location:	Schreiber bldg., room 309, Tel-Aviv University
Abstract:	In this talk I will show that in some cases the E.Hopf rigidity phenomenon allows quantitative interpretation.  More precisely, we estimate from above the measure of the set $\mathcal{M}$ swept by minimal orbits. These estimates are sharp, i.e. if $\mathcal{M}$ occupies the whole phase space we recover the E.Hopf rigidity. We give these estimates in two cases: the first is the case of convex billiards in the plane, sphere or hyperbolic plane. The second is the case of conformally flat Riemannian metrics on a torus. It seems to be a challenging question to understand such a quantitative bound for Burago-Ivanov theorem.
13.5.2015, 14:10 (Wednesday)	NO SEMINAR THIS WEEK
17.5.2015 - 26.5.2015	Computational Symplectic Topology Workshop
27.5.2015, 14:10 (Wednesday)	Alexander Caviedes Castro (Tel Aviv University)
Title:	Calabi Quasimorphisms for monotone coadjoint orbits
Location:	Schreiber bldg., room 309, Tel-Aviv University
Abstract:	In this talk I will explain how the existence of Calabi quasimorphisms on the universal covering of the group of Hamiltonian diffeomorphisms of a monotone coadjoint orbit of a compact Lie group follows from positivity results of Gromov-Witten invariants
2.6.2015, 14:10 (Tuesday) (NOTE SPECIAL DATE & LOCATION)	Frol Zapolsky (Haifa University)
Title:	Spectral invariants for monotone Lagrangian submanifolds
Location:	Schreiber bldg., room 08 Tel-Aviv University
Abstract:	I will outline a generalization of the definition of spectral invariants to the case of monotone Lagrangian submanifolds. As a sample application of the theory I will prove a rigidity result concerning the exotic monotone tori in CP^2 and in CP^1 \times CP^1. The talk is based on joint work with Remi Leclercq.
3.6.2015, 14:10 (Wednesday)	Yurii Neretin (University of Vienna)
Title:	Infinite symmetric group and bordisms of two-dimensional triangulated surfaces.
Location:	Schreiber bldg., room 309, Tel-Aviv University
Abstract:	We start from unitary representations of product $G$ of 3 copies of infinite symmetric group and produce the following constructions in the spirit of 'topological field theory'. There is a category, whose morphisms are triangulated checker-wise colored compact two-dimensional surfaces with boundary. Product of morphisms is a gluing similar to product of bordisms. We show that unitary representations of $G$ produce functors from this category to category of Hilbert spaces and bounded operators.
21.6.2015, 14:10 (Wednesday)	Pat Hooper (City College of NY)
Title:	Classical and non-classical translation surfaces, renormalization, and invariant measures
Location:	Schreiber bldg., room 309, Tel-Aviv University
Abstract:	A translation surface is a surface locally modeled on the plane with transition functions given by translations and with cone singularities. Such a surface admits natural flows called the straight-line flow, which moves each point in a specific direction at unit speed. I will explain that there is an action of SL(2,R) on spaces of translation surfaces which renormalizes the straight-line flow. By work of Masur and others, it is known that the SL(2,R) action can be used to understand the invariant measures of the straight-line flow. This philosophy is unexpectedly robust. After explaining the classical case, I will explain how it works for some infinite genus translation surfaces. In particular, will explain how it works for some spaces of finite covers of infinite genus translation surfaces. This work is joint with Rodrigo Treviño.
17.6.2015, 14:10 (Wednesday) (Blumenthal Lecture in Geometry)	Richard Evan Schwartz (Brown University)
Title:	The plaid model
Location:	Schreiber bldg., room 309, Tel-Aviv University
Abstract:	I will introduce a construction which produces embedded lattice polygons in the plane.  The model is highly structured and has both a combinatorial and number-theoretic feel to it. I call the model the plaid model because one of its descriptions is in terms of grids of parallel lines.  The plaid model exhibits a hierarchical multi-scale kind of structure.  It is closely related to outer billiards on kites, and also seems similar in spirit to Pat Hooper's Truchet tile system.  Mainly I will show off the model with computer demos and explain what theory about it I have worked out so far.
21.6.2015, 14:10 (Sunday) (NOTE SPECIAL DATE & LOCATION)	Netanel Blaier (MIT)
Title:	A symplectic analogue of the Johnson homomorphism coming from quantum Massey products
Location:	Holzblat auditorium 007,  Tel-Aviv University
Abstract:	Mathematicians often try to study an object by considering its group of automorphisms. Therefore, it only seems natural that given a symplectic manifold $(M,\omega)$, we would like to understand $\pi_0 Symp(M,\omega)$. To make the problem nontrivial, we focus on  those isotopy classes which act trivially on cohomology. When $M = \Sigma_g$ is a surface,  the group of such symplectomorphism is well known to low-dimensional topologists: it is the Torelli group, an important but poorly understood subgroup with many interesting  connections to other areas of mathematics. In the early 1980's, Dennis Johnson revolutionized the study of this group by introducing a sequence of homomorphisms $\tau_k$ detecting delicate intersection-theoretic information. We show that the definition of the Johnson homomorphisms can be recast in terms of the Morse $A_\infty$-algebra on the mapping tori $M_\phi$, and then extended to higher dimensional symplectic manifolds using quantum Massey products. As a sample application, we construct an $S^1$-family of embedded surfaces $C \subset \mathbb{P}^3$ whose monodromy is a seperating Dehn twist. Forming a (parametrized, small energy) blowup of the mapping tori, we obtain a six-dimensional symplectic manifold $M = Bl_C \mathbb{P}^3$, and a symplectomorphism $\phi : M \to M$. We then use the quantum Johnson homomorphism to show that $\phi$ is an "exotic" symplectomorphism.