Geometry & Dynamics Seminar 2018-19

The seminar will take place in Schreiber Building room 309, on Wednesdays at 14:10.

Please check each announcement since this is sometimes changed.

17.10.2018, 14:10 (Wednesday)	Orientation meeting for students
Location:	Schreiber bldg., room 309, Tel-Aviv University
24.10.2018, 14:10 (Wednesday)	Chandrika Sadanand (Technion)
Title:	You can hear the shape of a polygonal billiard table
Location:	Schreiber bldg., room 309, Tel-Aviv University
Abstract:	Consider a polygon-shaped billiard table on which a ball can roll along straight lines and reflect off of edges infinitely. In work joint with Moon Duchin, Viveka Erlandsson and Chris Leininger, we have characterized the relationship between the shape of a polygonal billiard table and the set of possible infinite edge itineraries of balls travelling on it. In this talk, we will explore this relationship and the tools used in our characterization (notably a new rigidity result for flat cone metrics).
31.10.2018, 14:10 (Wednesday)	Louis Ioos (Tel Aviv University)
Title:	Geometric Quantization of Hamiltonian flows
Location:	Schreiber bldg., room 309, Tel-Aviv University
Abstract:	Quantum mechanics is a physical theory that describes nature at the scale of atoms, and one of its fundamental aspects is its relation with classical mechanics, which describes nature at human scales. In particular, there are general axioms that a quantum theory must satisfy coming from its classical counterpart. The Geometric Quantization program is an attempt to implement these axioms in the general framework of symplectic geometry. In this talk, I will present how one constructs the quantum evolution operator associated to a given Hamiltonian flow in this context, and I will give estimates on its behavior at the limit of large scales.
07.11.2018, 14:10 (Wednesday)	Michael Khanevsky (Technion)
Title:	Surface quasimorphisms and the Hofer norm
Location:	Schreiber bldg., room 309, Tel-Aviv University
Abstract:	We consider several constructions of quasimorphisms on the Hamiltonian group of surfaces that were proposed by Gambaudo-Ghys, Polterovich and Py. These constructions are based on topological invariants either of individual orbits or of orbits of finite configurations of points and a quasimorphism computes the average value of such invariant in the surface. We show that many quasimorphisms that arise this way are not Hofer continuous.
14.11.2018, 14:10 (Wednesday)	Bo'az Klartag (Weizmann Institute of Science)
Title:	Convex geometry and waist inequalities
Location:	Schreiber bldg., room 309, Tel-Aviv University
Abstract:	We will discuss connections between Gromov's work on isoperimetry of waists and Milman's work on the M-ellipsoid of a convex body. It is proven that any convex body K in an n-dimensional Euclidean space has a linear image K_1 of volume one satisfying the following waist inequality: Any continuous map f from K_1 to R^d has a fiber f^{-1}(t) whose (n-d)-dimensional volume is at least c^{n-d}, where c > 0 is a universal constant. Already in the case where f is linear, this constitutes a slight improvement over known results. In the specific case where K = [0,1]^n, one may take K_1=K and c=1, confirming a conjecture by Guth. We furthermore exhibit relations between waist inequalities and various geometric characteristics of the convex body K.
21.11.2018, 14:10 (Wednesday)	Nicolina Istrati (Tel Aviv University)
Title:	Toric locally conformally Kahler manifolds
Location:	Schreiber bldg., room 309, Tel-Aviv University
Abstract:	A basic problem in geometry is the search of good metrics. On complex manifolds, a most satisfactory class is given by the Kahler metrics, because of the interplay they encode between complex, symplectic and Riemannian geometry. However, there are well-known obstructions to their existence; we are then led to consider new classes which would imitate their good behavior. One way to do this is to consider their conformal analogue, the locally conformally Kahler (LCK) metrics. A Hermitian metric g on a complex manifold is LCK if around every point there exists a local Kahler metric which is conformal to g. The symplectic counterpart of these structures is given by the locally conformally symplectic (LCS) forms. In the first part of this talk, I will give an introduction to this class of manifolds and present some of their main features. Then I will focus on toric LCS manifolds, which can be defined in analogy with toric symplectic geometry. I will present a classification result in terms of moment polytopes for the toric LCS manifolds which admit compatible complex structures.
28.11.2018, 14:10 (Wednesday)	Yaniv Ganor (Tel Aviv University)
Title:	Tropical Approach to Enumerative Algebraic Geometry
Location:	Schreiber bldg., room 309, Tel-Aviv University
Abstract:	Enumerative algebraic geometry is a branch of algebraic geometry that studies problems of counting the number of geometric objects in come class subject to geometric constraints. For example,  we all know that the number of lines in the plane, passing through a pair of distinct points is one. Tropical geometry studies a certain degeneration of complex algebraic geometry, where the complex objects degenerate into some polyhedral complexes. It can be seen as an analogue of algebraic geometry over the tropical semiring, roughly amounting to replacing addition with maximum and multiplication with addition. The tropical approach to enumerative geometry, pioneered by Mikhalkin, is to degenerate complex objects into tropical objects, count the tropical counterparts, which is generally easier, due to their piecewise linear nature, and then lift them back to complex algebraic objects. In this talk we will survey tropical geometry of curves in the plane and its applications to the count of complex curves on complex surfaces, via an algebraic approach due to Shustin. Then we will present some results in the count of singular complex curves, a joint work with Eugenii Shustin. We will assume no previous knowledge of tropical geometry nor of algebraic geometry.
05.12.2018, 14:10 (Wednesday)	Ram Band (Technion)
Title:	Neumann domains on manifolds and graphs
Location:	Schreiber bldg., room 309, Tel-Aviv University
Abstract:	The nodal set of a Laplacian eigenfunction forms a partition of the underlying manifold or graph. Another natural partition is based on the gradient vector field of the eigenfunction (on a manifold) or on the extremal points of the eigenfunction (on a graph).The submanifolds (or subgraphs) of this partition are called Neumann domains. We present some interesting results concerning Neumann domains on manifolds and on graphs. We compare manifolds to graphs in this sense and also relate the Neumann domain results to the nodal domain study. The talk is based on joint works with Lior Alon, Michael Bersudsky, Sebastian Egger, David Fajman and Alexander Taylor.
12.12.2018, 14:10 (Wednesday)	Misha Verbitsky (IMPA)
Title:	Multiplicity of singularities is not a bi-Lipschitz invariant
Location:	Schreiber bldg., room 309, Tel-Aviv University
Abstract:	Zariski has conjectured that multiplicity of a singularity of a hypersurface is a homeomorphism invariant. This conjecture is still unknown. However, a bi-Lipschitz version of Zariski conjecture is true: multiplicity of bi-Lipschitz hypersurface singularities is equal (Comte). The same is also true for surface singuarities (Newmann-Pichon). It was conjectured that the multiplicity is a bi-Lipschitz invariant for singularity of any codimension. Using classification of 5-manifolds, I would explain how to construct bi-Lipschitz equivalent singularities of any multiplicity, disproving this conjecture. This is a joint work with Lev Birbrair, Alexandre Fernandes and J. Edson Sampaio.
19.12.2018, 14:10 (Wednesday)	Oleg Ivrii (Caltech)
Title:	Complex dynamics, dimensions of quasicircles and the Weil-Petersson metric
Location:	Schreiber bldg., room 309, Tel-Aviv University
Abstract:	I will begin this talk by recalling the classical Weil-Petersson  metric on Teichmüller space. McMullen showed that the Weil-Petersson metric is related to the second derivative of the Hausdorff dimensions  of certain quasicircles arising from simultaneous uniformization. A similar construction can be carried out for matings of Blaschke  products, which allows one to define a Weil-Petersson metric on the main cardioid of the Mandelbrot set. In my PhD thesis, I studied the Weil-Petersson metric in degree 2 and  showed that the metric is incomplete and made some progress towards understanding its metric completion (which is an analogue of the Deligne-Mumford compactification). Together with K. Astala, A. Perälä and I. Prause, we used fractal approximation techniques to  understand dimensions of general quasicircles, that is, we used dynamical methods to study extremal problems in conformal mapping. I hope to give a brief overview of these developments. Here are some pictures related to the talk.
26.12.2018, 14:10 (Wednesday)	Albert Fathi (Georgia Tech)
Title:	Singularities of solutions of the Hamilton-Jacobi equation. A toy model: distance to a closed subset.
Location:	Schreiber bldg., room 309, Tel-Aviv University
Abstract:	ABSTRACT
02.01.2019, 14:10-15:00 (Wednesday)	Egor Shelukhin (University of Montreal)
Title:	Upper bounds on the Lagrangian spectral norm
Location:	Schreiber bldg., room 309, Tel-Aviv University
Abstract:	We discuss recent developments in establishing uniform bounds on the spectral norm and related invariants of Lagrangian submanifolds of open and closed symplectic manifolds. Furthermore, we outline a few applications. This talk is partially based on joint work with Asaf Kislev.
02.01.2019, 15:10-16:00 (Wednesday)	Alexander Bobenko (Technical University of Berlin)
Title:	On a discretization of confocal quadrics: Geometric parametrizations, integrable systems and incircular nets
Location:	Schreiber bldg., room 309, Tel-Aviv University
Abstract:	We propose a discretization of classical confocal coordinates. It is based on a novel characterization thereof as factorizable orthogonal coordinate systems. Our geometric discretization leads to factorizable discrete nets with a novel discrete analog of the orthogonality property. A discrete confocal coordinate system may be constructed geometrically via polarity with respect to a sequence of classical confocal quadrics. The coordinate functions of discrete confocal quadrics are computed explicitly. The theory is illustrated with a variety of examples in two and three dimensions. These include confocal coordinate systems parametrized in terms of Jacobi elliptic functions. Connections with incircular nets and elliptic billiards are established.
09.01.2019, 14:10 (Wednesday)	Daniel Tsodikovich (Tel Aviv University)
Title:	Hamiltonian Dynamics on Simplexes
Location:	Schreiber bldg., room 309, Tel-Aviv University
Abstract:	In this talk I will discuss the Hamiltonian dynamics on the four-dimensional standard simplex, in particular the classical dynamical properties, such as transitivity, integrability, existence of periodic points, and entropy (some of the results are generalized to weighted standard simplexes in any dimension). I will explain how to find the periodic points and the integrals of motion, and will prove that the entropy is zero. The talk will show that the standard simplex exhibits a mixture of chaotic and non-chaotic properties.
27.02.2019, 14:10 (Wednesday)	Vukasin Stojisavljevic (Tel Aviv University)
Title:	Symplectic Banach-Mazur distance and persistence modules
Location:	Schreiber bldg., room 309, Tel-Aviv University
Abstract:	We will define a distance, called symplectic Banach-Mazur distance, on the space 𝒞M of all "nice" star-shaped domains inside the cotangent bundle of a smooth manifold M. Our main goal is to study large-scale geometry of the space 𝒞M equiped with this distance when M is a closed surface of positive genus. The key technical ingredient which we use is stability of persistence modules coming from filtered symplectic homology with respect to this distance. We will also show certain results related to the study of closed geodesics. The talk is based on a joint work with Jun Zhang.
06.03.2019, 14:10 (Wednesday)	Louis Ioos (Tel Aviv University)
Title:	Canonical Kähler metrics and quantization
Location:	Schreiber bldg., room 309, Tel-Aviv University
Abstract:	The search for canonical Kähler metrics on projective manifolds is one of the fundamental problems in algebraic geometry. In the beginning of the century, Donaldson discovered a link between this problem and a geometric point of view on quantum mechanics proposed by Berezin in the seventies. In this talk, I will explain this link via the point of view of Positive Operator Valued Measures, and show how these can be applied to Donaldson's program. This talk is based on a joint work with Victoria Kaminker, Leonid Polterovich and Dor Shmoish.
13.03.2019, 14:10 (Wednesday)	Marcelo Alves (ULB Brussels)
Title:	Periodic motions and forcing of positive entropy for Reeb flows
Location:	Schreiber bldg., room309, Tel-Aviv University
Abstract:	A celebrated theorem of Li-Yorke states that a continuous map of the interval possessing a period three periodic point admits a subset where the dynamics is chaotic. I will present a related result for Reeb flows on contact 3-manifolds: the existence of certain periodic orbits implies positivity of topological entropy. This generalizes a beautiful result of Denvir and Mackay which asserts that a geodesic flow on the 2-torus with a simple contractible closed geodesic has positive topological entropy, and is a joint work with Pedro Salomão and Umberto Hryniewicz. I will also explain some ongoing work with Abror Pirnapasov aimed at finding infinitely many distinct links in the three torus which force positive topological entropy. If time allows I will discuss some (to the best of my knowledge) open problems on the forcing of slow entropy of Reeb flows and symplectomorphisms, which are motivated by works of Polterovich and Frauenfelder-Schlenk, and explain an approach to investigate them based on work of Matthias Meiwes.
20.03.2019, 14:10 (Wednesday)	Jordan Payette (University of Montreal)
Title:	The representation problem of symplectic submersions
Location:	Schreiber bldg., room 309, Tel-Aviv University
Abstract:	Symplectic submersions are functions between symplectic manifolds which generalize symplectomorphisms. One motivation for their study resides in the fact that Gromov's non-squeezing theorem for symplectic embeddings of balls in R^{2n} admits a simple and somewhat more intrinsic formulation in terms of such functions. From this perspective, it seems natural to ask which symplectic submersions are subject to a non-squeezing phenomenon. A starting point is to characterize those symplectic submersions which appear in Gromov's theorem; this is what we call the "representation problem". The main goals of this talk is to introduce the notion of symplectic submersions and to provide partial solutions to the representation problem; for instance, the symplectic submersions appearing in Gromov's theorem are in a sense the simplest ones.
27.03.2019, 14:10 (Wednesday)	Michael Brandenbursky (Ben-Gurion University)
Title:	Fragmentation norm and relative quasimorphisms
Location:	Schreiber bldg., room 309, Tel-Aviv University
Abstract:	We prove that manifolds with complicated enough fundamental group admit measure-preserving homeomorphisms which have positive stable fragmentation norm with respect to balls of bounded measure. This is a joint work with Jarek Kedra.
03.04.2019, 14:10 (Wednesday)	Misha Bialy (Tel Aviv University)
Title:	Geometry of Gutkin billiard tables
Location:	Schreiber bldg., room 309, Tel-Aviv University
Abstract:	Eugene Gutkin invented a class of planar Birkhoff billiards with remarkable invariant curves. It is natural question to ask if there exist Gutkin billiard tables in higher dimensions. It turns out that the only higher-dimensional convex billiards with Gutkin property are round spheres. In the first part of the talk I am going to explain this result. In the second part (joint with Andrey Mironov and Lior Shalom), I will discuss the so called Outer billiards and will show an example of outer billiard with Gutkin property. The dynamics of this example seems to be very interesting.
10.04.2019, 14:10-15:00 (Wednesday)	Amit Wolecki (Tel Aviv University)
Title:	Illumination and blocking in rational billiards
Location:	Schreiber bldg., room 309, Tel-Aviv University
Abstract:	Does a light source in a polygonal mirror room illuminate the whole region? Are there pairs of points in a mirror room that do not illuminate each other? Examples for a polygonal room that affirms the latter question were found by Tokarsky in 1995 and raised the question of possible cardinality of pairs of points that do not illuminate each other. Breakthrough results by Eskin, Mirzakhani and Mohammadi paved the way for solving classical illumination problems within the general framework of flat surfaces and dynamics on the their moduli spaces. We shall see some of the general results that stems from Eskin Mirzakhani and Mokhammadi's works and applied for solving problems in illumination, including a recent result about the cardinality of non-illuminating pairs or points.
10.04.2019, 15:10-16:00 (Wednesday)	Jarek Kedra (University of Aberdeen)
Title:	The geometry of the fundamental group
Location:	Schreiber bldg., room 309, Tel-Aviv University
Abstract:	It is a classical observation due to John Milnor and Albert Schwarz that the word metric on the fundamental group of a closed manifold carries an information about the Riemannian metric of the universal cover (the metrics are quasi-isometric).  In the above approach the word metric on the fundamental group is associated with a finite generating set. In the talk I will explore the word metrics on the fundamental group associated with geometrically meaningful generating sets. Specifically, I will focus on the elements of the fundamental group represented by closed geodesics and on examples of Riemannian manifolds where such elements generate the fundamental group. I will then ask the most basic question whether the diameter of such a word metric is finite or infinite. The first answer is interpreted as abundance of closed geodesics while the second as their scarcity.  I will then present examples for both cases (they are locally symmetric spaces). This is a join work with Bastien Karlhofer, Michał Marcinkowski and Alexander Trost.
17.04.2019, 14:10-15:00 (Wednesday)	Brian Tervil (University of Haifa)
Title:	Translated points for prequantization spaces over monotone toric manifolds
Location:	Schreiber bldg., room 309, Tel-Aviv University
Abstract:	I will present a version of Sandon's conjecture on the number of translated points of contactomorphisms in the context of prequantization bundles over monotone toric manifolds. I will outline the construction of a natural prequantization space for which the conjecture holds, as well as of a cohomology group which is the main character in the proof of the theorem. The cohomological construction is based on the theory of generating functions and equivariant cohomology as developed by Givental for toric manifolds.
17.04.2019, 15:10-16:00 (Wednesday)	Yohann Le Floch (University of Strasbourg)
Title:	Semitoric families
Location:	Schreiber bldg., room 309, Tel-Aviv University
Abstract:	Semitoric systems form a class of integrable systems generalizing toric systems in dimension four, for which one of the integrals generates acircle action. Their classification was obtained by Pelayo and Vu Ngoc (2009-2011) in terms of five symplectic invariants. However, some problems remain: firstly, the construction of a semitoric system with prescribed invariants is quite involved, and secondly, few explicit examples of such systems are known. In the first part of the talk, I will review toric and semitoric integrable systems. In the second part, I will discuss some progress towards the explicit construction of a semitoric system given part of its symplectic invariants, and describe some new explicit examples. This is based on joint work with Joseph Palmer (Rutgers).
01.05.2019, 14:10 (Wednesday)	Weiwei Wu (University of Georgia)
Title:	Semi-toric spherical systems and symplectomorphism groups
Location:	Schreiber bldg., room 309, Tel-Aviv University
Abstract:	We will explain a generalization of semi-toric systems. In dimension four, such systems can be easily obtained by generalizing the notion of "toric blow-up". As it turns out, this construction gains new understandings of the symplectic mapping class groups.  We will explain its relation to a long-standing question between Lagrangian Dehn twists and symplectic mapping class groups of rational manifolds, and potential construction of exotic finite group actions. This is a combination of several on-going joint works with Liat Kessler, Jun Li and Tian-Jun Li.
Conference: C^0 aspects of symplectic geometry and Hamiltonian dynamics, Technion (Haifa), May 12-16. NO SEMINAR THIS WEEK!
22.05.2019, 14:10 (Wednesday)	David Fisher (Indiana University, Bloomington, USA) - TIDY Distinguished Lecture
Title:	Zimmer's conjecture: subexponential growth, measure rigidity and strong property (T)
Location:	Schreiber bldg., room 309, Tel-Aviv University
Abstract:	This talk is a sequel to the colloquium of Monday.  I will try to make it logically independent and self-contained, but most of the history and motivation will occur in the colloquium talk and this talk will emphasize ideas of proofs of the following theorem.   Let G be a cocompact lattice in SL(n,R) where n>3, M a compact manifold and a: G---> Diff(M) a homomorphism.  If dim(M)<n-1, a(G) is finite. Furthermore if dim(M)=n-1 and a(G) preserves a volume form, a(G) is finite.   The proof  has many surprising features, including that it uses hyperbolic dynamics to prove an essentially elliptic result and that it uses results on homogeneous dynamics,  including Ratner's measure classification theorem, to prove results about inhomogeneous  system. If time permits I will say a few words about the difficulties that arise when G is not cocompact. This is joint work with Aaron Brown and Sebastian Hurtado.
SPECIAL ANNOUNCEMENT - COLLOQUIUM TALK:
20.05.2019, 12:15 (Monday)	David Fisher (Indiana University, Bloomington, USA) - TIDY Distinguished Lecture
Title:	Recent Progress on Zimmer's Conjecture
Location:	Schreiber bldg., room 6, Tel-Aviv University
Abstract:	Lattices in higher rank simple Lie groups are known to be extremely rigid.  Examples of this are Margulis' superrigidity theorem, which shows they have very few linear represenations, and Margulis' arithmeticity theorem, which shows they are all constructed via number theory.   Motivated by these and other results, in 1983 Zimmer made a number of conjectures about actions of these groups on compact manifolds.  After  providing some history and motivation, I will discuss a very recent result, proving many cases of the main conjecture. While avoiding  technical matters, I will try to describe some of the novel flavor of the proof. The proof has many surprising features bringing together  ideas from homogeneous dynamics, hyperbolic dynamics and functional analyis. This is joint work with Aaron Brown and Sebastian Hurtado.
28.05.2019, 13:10 (Tuesday)	Gang Tian  (Peking University, Princeton University) - Blumenthal Lecture in Geometry
SPECIAL LECTURE - PLEASE NOTE THE DATE, PLACE & TIME
Title:	Recent progress on Kahler-Ricci flow
Location:	Orenstein bldg., room 110, Tel-Aviv University
Abstract:	In this talk, I will first recall some basic results on Kahler-Ricci flow. Then I will discuss some recent progress on long-time behavior of the flow. The progress is based on a generalization of Perelman's famous non-collapsing result.
SPECIAL ANNOUNCEMENT - COLLOQUIUM TALK:
27.05.2019, 12:15 (Monday)	Gang Tian  (Peking University, Princeton University) - Blumenthal Lecture in Geometry
Title:	Analytic Minimal Model Program
Location:	Schreiber bldg., room 6, Tel-Aviv University
Abstract:	The analytic Minimal Model Program was introduced a decade ago and provided a method to classifying projective manifolds birationally through Ricci flow. If successful , the method can be also used to classify Kahler manifolds which are more general than projective manifolds. In this expository talk, I will give an overview of the Analytic Minimal Model Program and discuss some fundamental results. I will also discuss some open problems.
05.06.2019, 14:10 (Wednesday)	Gerhard Knieper (Ruhr-Universitat Bochum) - TIDY Distinguished Lecture
Title:	Coarse hyperbolicity, measures of maximal entropy and growth rate of closed geodesics
Location:	Schreiber bldg., room 309, Tel-Aviv University
Abstract:	The class of systems which we consider are geodesic flows on closed manifolds without conjugate points. Furthermore, we assume some hyperbolic background structure, given by a metric of negative curvature. Under this assumptions we show the existence of a measure of maximal entropy (MME). For surfaces we able to prove mixing, uniqueness of the MME and the uniform distribution of closed orbits w.r.t. this measure. Using the mixing properties of the MME we are able to get precise asymptotics of the growth rate of closed geodesics known before (due to the work of Margulis and Bowen) only in the case of strictly negative curvature. We also formulate conditions for which results carry over to higher dimensions. This talk is partially based on joint work with Vaughn Climenhaga and Khadim War.
12.06.2019, 14:10 (Wednesday)	Henri Berestycki (EHESS, PSL University, Paris) - Sackler Distinguished Lecture
Title:	Reaction-diffusion equations in general domains
Location:	Schreiber bldg., room 309, Tel-Aviv University
Abstract:	Reaction-diffusion equations are ubiquitous in modelling in the life sciences as an approach to spatial propagation and diffusion. They also arise in physics, and, more recently, in social sciences. After describing the mechanism of reaction and diffusion, I will review some classical properties. I will then mention more recent works dealing with non-homogeneous media. I will emphasize here the role played by the domain of propagation.