Geometry & Dynamics Seminar 2012-13
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.
| 24.10.2012, 14:10 (Wednesday) | Orientation meeting for students | |
| Location: | Schreiber bldg., room 209, Tel-Aviv University | |
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| 30.10.2012, 11:10 (Tuesday) | Valentin Ovsienko (CNRS, Institut Camille Jordan, Universite Claude Bernard Lyon) | |
| Title: | Pentagram map and frieze patterns | |
| Location: | Schreiber bldg., room 209, Tel-Aviv University | |
| Abstract: | The pentagram map is a dynamical system that has amazing relations to geometry, algebra and combinatorics. It was invented by Richard Schwartz 20 years ago and it has recently attracted much interest. I will give a survey of recent activity around the pentagram map, explain the role of Coxeter's frieze patterns and formulate open questions. The talk will be elementary. |
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| 31.10.2012, 14:10 (Wednesday) | Boris Dubrovin (SISSA, Trieste, Italy & Steklov Math. Institute) | |
| Title: | On topological recursion for integrable systems | |
| Location: | Schreiber bldg., room 209, Tel-Aviv University | |
| Abstract: | We consider the class of hierarchies of integrable PDEs satisfying topological recursion coming from Deligne-Mumford moduli spaces of stable algebraic curves. Many classical examples like Korteweg - de Vries, nonlinear Schroedinger, Toda lattice equations belong to this class but there are many new hierarchies depending on continuous parameters. Construction of such "hierarchies of topological type" will be explained in the talk. |
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| 7.11.2012, 14:10 (Wednesday) | Yaron Ostrover (Tel Aviv University) | |
| Title: | Mathematical billiards in the eye of symplectic geometry | |
| Location: | Schreiber bldg., room 209, Tel-Aviv University | |
| Abstract: | Abstract: Mathematical billiards describe the motion of a massless particle in a domain with elastic reflections from the boundary. Despite the simplicity of the model, mathematical billiards present a broad variety of dynamical behaviors, from complete integrability to chaotic motion. Furthermore, the study of billiard dynamics has vast applications to different fields such as mathematical physics, number theory, acoustics, optics, etc. In this talk we will highlight some of the geometric aspects of billiard dynamics. In particular, we will explain how a certain symplectic invariant on the classical phase space can be used to obtain bounds and inequalities on the length of the shortest periodic billiard trajectory. The talk is based on a joint work with Shiri Artstein-Avidan. |
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| 14.11.2012, 14:10 (Wednesday) | Jake Solomon (Hebrew University) | |
| Title: | The Calabi homomorphism, Lagrangian paths and special Lagrangians | |
| Location: | Schreiber bldg., room 209, Tel-Aviv University | |
| Abstract: | The talk will have two parts. First, I'll discuss a generalization of the Calabi homomorphism to a functional on Lagrangian paths and its relationship with special Lagrangian geometry. Then, I'll explain how these ideas fit in the framework of mirror symmetry. |
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| 21.11.2012, 14:10 (Wednesday) | Francois Charette (Tel Aviv University) | |
| Title: | A categorification of the relative Seidel morphism | |
| Location: | Schreiber bldg., room 209, Tel-Aviv University | |
| Abstract: | The Lagrangian cobordism category recently introduced by Biran and Cornea provides a way to encode Lagrangian submanifolds algebraically, via a functor to the Fukaya category. We will discuss the effect of a simple version of this functor on the Lagrangian suspension given by a Hamiltonian isotopy. The talk is based on joint work with Paul Biran and Octav Cornea. |
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| 28.11.2012, 14:10 (Wednesday) | Yael Karshon (University of Toronto) | |
| Title: | Morse-Bott-Kirwan theory | |
| Location: | Schreiber bldg., room 210, Tel-Aviv University | |
| Abstract: | One of the central players in equivariant symplectic geometry is the momentum map, a map to Euclidean space that encodes symmetries of the system. One of the central techniques in equivariant symplectic geometry is to apply Morse Theory to components of the momentum map or to the norm-square of the momentum map. Unfortunately, the norm-square of the momentum map is not quite a Morse function. Fortunately, the norm-square satisfies a condition, identified by Kirwan, under which the usual techniques of Morse-Bott theory still work. Together with Tara Holm, we prove that if a function satisfies Kirwan's condition locally then it also satisfies this condition globally. As an application, we use the local normal form theorem to recover Kirwan's result that the norm square of a momentum map satisfies Kirwan's condition. |
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| 28.11.2012, 15:10 (Wednesday) | Maia Fraser (University of Chicago) | |
| Title: | Equivariant generating function homology and general contact non-squeezing in $\R^{2n} \times S^1$ | |
| Location: | Schreiber bldg., room 210, Tel-Aviv University | |
| Abstract: | I will give an overview of generating function homology for contact manifolds and describe a variant of equivariant generating function homology which makes it possible to mimic a ${\mathbb Z}$-action in the time-direction on loops. As an application I will show how this may be used to prove general non-squeezing in the contact manifold $R^{2n}\times S^1$. (first stated and partially proved by Eliashberg, Kim and Polterovich [EKP2006]): given any $R_1 > 1$ and $R_2 < R_1$ it is impossible to map $B(R_1)\times S^1$ into $B(R_2) \times S^1$ by a contactomorphism isotopic to the identity through compactly supported contactomorphisms, where $B(R)$ is the ball of symplectic area $R$. |
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| 5.12.2012, 14:10 (Wednesday) | Lev Buhovski (Tel Aviv University) | |
| Title: | Unboundedness of the first eigenvalue of the Laplacian in the symplectic category |
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| Location: | Schreiber bldg., room 209, Tel-Aviv University | |
| Abstract: | The main subject of this talk is the question of Leonid Polterovich, about the symplectic flexibility of the first eigenvalue of the Laplacian. At the beginning I will describe the history of this problem, and the first partial solutions of it. In the second half of the talk, I will give a sketch of my solution of the question. |
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Dmitry Faifman (Tel Aviv University) | |
| Title: | Applying symplectic techniques to metric geometry | |
| Location: | Schreiber bldg., room 209, Tel-Aviv University | |
| Abstract: | One natural approach of constructing invariants of a normed space is to attach to it some compact Finsler manifold, and then study its geometric invariants such as volume and closed geodesics. The girth of a normed space is such an invariant. We will discuss several such constructions, and survey some classical theorems and conjectures describing them. Then we will see how techniques of symplectic geometry and Hamiltonian group actions can be applied to study such invariants. |
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| 19.12.2012, 14:10 (Wednesday) | Michael Brandenbursky (Vanderbilt University) | |
| Title: | Bi-invariant metrics on diffeomorphism groups | |
| Location: | Schreiber bldg., room 209, Tel-Aviv University | |
| Abstract: | In this talk I will discuss various metrics on groups of diffeomorphisms of smooth manifolds, which do or do not preserve some additional structure (usually volume or symplectic form). Then I will restrict my talk to the case of the group G of area-preserving diffeomorphisms of the 2-disc. This group admits a natural bi-invariant Autonomous metric. I will show that any finitely generated free abelian group embeds bi-Lipschitz into the group G. If time permits I will discuss some results concerning groups of Hamiltonian diffeomorphisms of surfaces. This is a joint work with J. Kedra. |
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| 26.12.2012, 14:10 (Wednesday) | Egor Shelukhin (Université de Montréal) | |
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| Location: | Schreiber bldg., room 209, Tel-Aviv University | |
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| 2.1.2013, 14:10 (Wednesday) | Iosif Polterovich (Universite de Montreal) | |
| Title: | Seeing sounds, hearing shapes and beyond | |
| Location: | Schreiber bldg., room 210, Tel-Aviv University | |
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Anatoly Libgober (University of Illinois at Chicago) | |
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| Location: | Schreiber bldg., room 210, Tel-Aviv University | |
| Abstract: | I'll discuss relation between the Alexander polynomials of fundamental groups of the complements to plane singular curve and the Mordell Weil rank of isotrivial abelian varieties, in particular elliptic curves, over field of complex rational functions in two variables. Study of this relation leads to new invariants plane curves singularities, called local Albanese varieties, encoding Hodge theoretical information about a singular point. One of the corollaries yields a family of simple Jacobians for which the Mordell Weil rank can be arbitrary large. |
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| 9.1.2013, 14:10 (Wednesday) | Stéphane Nonnenmacher (Institut de Physique Théorique, CEA-Saclay) | |
| Title: | Quantum scattering in presence of classical hyperbolicity | |
| Location: | Schreiber bldg., room 209, Tel-Aviv University | |
| Abstract: | A quantum (or wave) scattering system consists of waves coming from infinity, interacting with a "scatterer", and then escaping to infinity or to a measuring device. At certain wave frequencies the scattering system "resonates", leading to peaks in the scattering intensity. Mathematically, resonances can be obtained as the complex poles of the resolvent of the quantum Hamiltonian. In such an "open" setting, the resonances replace the discrete spectrum of compact geometries. They are associated with metastable modes, instead of eigenmode; the imaginary part of the resonance is proportional to the (quantum) decay rate of the associated mode. Interesting geometric examples are given by the case of the Laplace-Beltrami operator on a Riemannian manifold of infinite volume ("nice" near infinity). We are interested in the high-frequency distribution of those resonances, in particular the "long-living" ones, close to the real axis: how numerous are those resonances? Is there a "resonance gap", meaning a uniform lower bound for the quantum decay rates? Those questions naturally lead to consider the corresponding classical Hamiltonian dynamics, in particular the set of trapped trajectories ("trapped set"). We will review a few situations in which the dynamics on this trapped set presents some hyperbolicity. In particular, when the trapped set is a chaotic hyperbolic repeller, we prove show a criterium (in terms of the classical dynamics) ensuring a resonance gap, and prove that the density of resonances is bounded above by a "fractal Weyl law". We will also consider the case where the trapped set is a normally hyperbolic symplectic submanifold. These results use various tools from semiclassical analysis, which allow to "microlocalize" the quantum Hamiltonian in the close vicinity of the trapped set. At some point we use a form of hyperbolic dispersion estimate, which combines classical hyperbolicity with the quantum uncertainty principle. |
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| 16.1.2013, 14:10 (Wednesday) | Tobias Hartnick (Technion) | |
| Title: | An introduction to the random geometry of quasimorphisms | |
| Location: | Schreiber bldg., room 209, Tel-Aviv University | |
| Abstract: | The notion of a quasimorphism goes back all the way to work of Eudoxus (~400 BC), and has been frediscovered many times over history, most notably by Poincare in the late 1880s. A first culmination point in its study appeared in the 1980s in the groundbreaking work of Traubel, Gromov, Brooks, Ivanov, Grigorchuk and others, which places quasimorphisms in the context of bounded cohomology. Since then, a huge amount of new examples has been constructed using tools from geometric group theory, differential geometry, morse theory and dynamics. On the other hand, the general theory of quasimorphisms has not developed at the same speed as these examples, so that many basic structural questions on quasimorphisms remain unanswered today. In this talk we will focus on the random geometry of quasimorphisms, which is the general structure theory of quasimorphisms on groups in the presence of a probability measure. In this context, we will discuss harmonicity properties of quasimorphisms ("bounded Hodge theory") as well as their behavior along trajectories of i.i.d. random walks (central limit theorems, laws of iterated logarithm), following work of Burger-Monod, Calegari-Fujiwara and Björklund and the speaker. Time permitting we will discuss various applications due to Malyutin and Calegari-Maher. No previous exposure to quasimorphisms is required. |
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| 23.1.2013, 14:10 (Wednesday) | Danny Calegari (University of Chicago) | |
| Title: | Surfaces From Linear Programming | |
| Location: | Schreiber bldg., room 209, Tel-Aviv University | |
| Abstract: | A famous question of Gromov asks whether every hyperbolic group contains a subgroup which is isomorphic to the fundamental group of a closed surface. Surface subgroups play a very important role in many areas of low-dimensional topology, for example in Agol's recent proof that every hyperbolic 3-manifold has a finite cover which fibers over the circle. I would like to describe several ways to build surface subgroups in certain hyperbolic groups. The role of hyperbolicity is twofold here: first, hyperbolic geometry allows one to certify injectivity by *local* data; second, hyperbolic dynamics allows one to use ergodic theory to produce the pieces out of which an injective surface can be built. I would like to sketch a proof of the fact that the extension of a free group associated to a ''random'' endomorphism contains a surface subgroup with probability one. This is joint work with Alden Walker. |
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| 30.1.2013, 14:10 (Wednesday) | Florent Balacheff (Laboratoire Paul Painleve, Universite Lille 1, Lille, France) | |
| Title: | From systolic geometry to Mahler's conjecture : an uncertainty principle in the geometry of numbers | |
| Location: | Schreiber bldg., room 209, Tel-Aviv University | |
| Abstract: | In this talk, we will explain how to derive a generalization of first Minkowski's theorem for asymmetric convex bodies by carefully analyzing the systolic geometry of asymmetric Finsler tori. This result can be viewed as an uncertainty principle, and is related to Mahler's conjecture. This is joined work with Juan-Carlos Álvarez Paiva and Kroum Tzanev. |
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| 6.3.2013, 14:10 (Wednesday) | Gabriel Paternain (University of Cambridge) | |
| Title: | Spectral rigidity and invariant distributions on Anosov surfaces | |
| Location: | Schreiber bldg., room 209, Tel-Aviv University | |
| Abstract: | I will discuss inverse problems on a closed Riemannian surface whose geodesic flow is Anosov. More specifically, I will try to establish spectral rigidity (a problem that has been open for some time) and surjectivity results for the adjoint of the geodesic ray transform. These surjectivity results imply the existence of many geometric distributions in H^{-1} invariant under the geodesic flow and play a key role to prove spectral rigidity. This is joint work with Mikko Salo and Gunther Uhlmann. |
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| 6.3.2013, 15:10 (Wednesday) | Alexander Plakhov (University of Aveiro Portugal and Institute for Information Transmission Problems, Russia) |
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| Title: | Invisibility and retro-reflection in billiards | |
| Location: | Schreiber bldg., room 209, Tel-Aviv University | |
| Abstract: | I will discuss inverse problems on a closed Riemannian surface whose geodesic flow is Anosov. More specifically, I will try to establish spectral rigidity (a problem that has been open for some time) and surjectivity results for the adjoint of the geodesic ray transform. These surjectivity results imply the existence of many geometric distributions in H^{-1} invariant under the geodesic flow and play a key role to prove spectral rigidity. This is joint work with Mikko Salo and Gunther Uhlmann. |
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| 12.3.2013, 11:10 (Tuesday) | Sergei Tabachnikov (Pennsylvania State University) | |
| Title: | Pentagram Map, twenty years after. (Lecture #1) |
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| Location: | Schreiber bldg., room 210, Tel-Aviv University | |
| Abstract: | Introduced by R. Schwartz about 20 years ago, the pentagram map acts on plane n-gons, considered up to projective equivalence, by drawing the diagonals that connect second-nearest vertices and taking the new n-gon formed by their intersections. The pentagram map is a discrete completely integrable system whose continuous limit is the Boussinesq equation, a completely integrable PDE of soliton type. I shall survey recent work on generalizations of the pentagram map, emphasizing its close ties with the theory of cluster algebras, a new and rapidly developing area with numerous connections to diverse fields of mathematics. In particular, I shall describe a higher-dimensional version of the pentagram map and, rather counter-intuitively, its 1-dimensional version. |
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| 13.3.2013, 14:10 (Wednesday) | Stefan Nemirovski (Steklov Mathematical Institute) | |
| Title: | Spacetimes and Contact Geometry | |
| Location: | Schreiber bldg., room 209, Tel-Aviv University | |
| Abstract: | Penrose and Low observed that the space of light rays of a "nice" spacetime carries a natural contact structure. This leads to a correspondence between notions from the Lorentz geometry of the spacetime and the contact geometry of its space of light rays. I will explain this correspondence, sketch some known results, and pose a few open problems. |
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| 13.3.2013, 15:10 (Wednesday) | Sergei Tabachnikov (Pennsylvania State University) | |
| Title: | Pentagram Map, twenty years after, Lectures #2. | |
| Location: | Schreiber bldg., room 209, Tel-Aviv University | |
| Abstract: | Introduced by R. Schwartz about 20 years ago, the pentagram map acts on plane n-gons, considered up to projective equivalence, by drawing the diagonals that connect second-nearest vertices and taking the new n-gon formed by their intersections. The pentagram map is a discrete completely integrable system whose continuous limit is the Boussinesq equation, a completely integrable PDE of soliton type. I shall survey recent work on generalizations of the pentagram map, emphasizing its close ties with the theory of cluster algebras, a new and rapidly developing area with numerous connections to diverse fields of mathematics. In particular, I shall describe a higher-dimensional version of the pentagram map and, rather counter-intuitively, its 1-dimensional version. |
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| 20.3.2013, 14:10 (Wednesday) | Peter Ozsváth (Princeton University) | |
| Title: | Bordered Heegaard Floer homology | |
| Location: | Schreiber bldg., room 209, Tel-Aviv University | |
| Abstract: | Bordered Floer homology is an invariant for three-manifolds with boundary, developed in collaboration with Robert Lipshitz and Dylan Thurston. I will outline the construction, and describe recent work, some in progress. I will try to describe how this construction works in the special case where the three-manifold has torus boundary. |
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| 10.4.2013, 14:10 (Wednesday) | Paul Busch (University of York) | |
| Title: | On the "Zoo" of Heisenberg Uncertainties, and How to Measure Incompatible Quantum Observables |
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| Location: | Schreiber bldg., room 209, Tel-Aviv University | |
| Abstract: | Developing the theme of the introductory lecture "Quantum Uncertainty - in All Its Guises", I will give a more detailed review of the mathematical modeling of joint measurements of incompatible quantum observables, using the theory of operator measures on Hilbert spaces. I will show how the noncommutativity of observables can be overcome as an obstruction to their joint measurability by allowing for appropriate trade-offs for the relevant measurement accuracies and the degrees of necessary disturbance caused by the measurements. |
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| 17.4.2013, 14:10 (Wednesday) | Boris Botvinnik (University of Oregon) | |
| Title: | Surgery and the moduli spaces of metrics of positive scalar curvature |
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| Location: | Schreiber bldg., room 209, Tel-Aviv University | |
| Abstract: | We will review major ideas to study manifolds with positive scalar curvature metrics by means of surgery, cobordism theory and conformal geometry. Then we discuss several results describing the topology of spaces (and moduli spaces) of positive scalar curvature metrics on simply connected manifolds. |
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| 24.4.2013, 14:10 (Wednesday) | Sobhan (Ecole Normale Supérieure) | |
| Title: | Rigidity of coisotropic submanifolds and their characteristic foliations | |
| Location: | Schreiber bldg., room 209, Tel-Aviv University | |
| Abstract: | The Gromov-Eliashberg theorem states that a diffeomorphism which can be written as a C^0 limit of symplectomorphisms is itself a symplectomorphism. In this talk, I will show that coisotropic submanifolds and their characteristic foliations exhibit similar rigidity properties. This is joint work with Vincent Humiliere and Remi Leclercq. |
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| 19.5.2013, 11:10 (Sunday) | Dmitry Burago (Penn State University) | |
| Title: | A Math Mosaic ("Tales from both pockets") | |
| Location: | Schreiber bldg., room 210, Tel-Aviv University | |
| Abstract: | This won't be a typical seminar lecture. Instead I'll give a number of mini-talks on very different topics. The only thing linking the topics together is that they have all been of interest to me in the past several years. A very important part of the lecture will be the presentation of open problems. These will be formulated using only basic material at a level accessible to graduate student. |
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| 28.5.2013, 11:10 (Tuesday) | Vadim Kaloshin (University of Maryland) | |
| Title: | Kirkwood gaps and instability for three body problems | |
| Location: | Schreiber bldg., room 210, Tel-Aviv University | |
| Abstract: | It is well known that, in the Asteroid Belt, located between the orbits of Mars and Jupiter, the distribution of asteroids has the so-called Kirkwood gaps exactly at mean motion resonances of low order. We study the dynamics of the Newtonian Sun--Jupiter--Asteroid problem near such resonances. We construct a variety of diffusing orbits which show a drastic change of the osculating eccentricity of the asteroid, while the osculating semimajor axis is kept almost constant. We shall also discuss stochastic aspects of dynamics in near mean motion dynamics. This might be an explanation of presence of Kirkwood gaps. This is a joint work with J. Fejoz, M. Guardia, P. Roldan |
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| 29.5.2013, 14:10 |
Vadim Kaloshin (University of Maryland) | |
| Title: | Quasiergodic hypothesis and Arnold diffusion in dimension 3 | |
| Location: | Schreiber bldg., room 209, Tel-Aviv University | |
| Abstract: | The famous ergodic hypothesis claims that a typical Hamiltonian dynamics on a typical energy surface is ergodic. However, KAM theory disproves this. It establishes a persistent set of positive measure of invariant KAM tori. The (weaker) quasi-ergodic hypothesis, proposed by Ehrenfest and Birkho ff, says that a typical Hamiltonian dynamics on a typical energy surface has a dense orbit. This question is wide open. In early 60th Arnold constructed an example of instabilities for a nearly integrable Hamiltonian of dimension n >2 and conjectured that this is a generic phenomenon, nowadays, called Arnold di usion. In the last two decades a variety of powerful techniques to attack this problem were developed. In particular, Mather discovered a large class of invariant sets and a delicate variational technique to shadow them. In a series of preprints: one joint with P. Bernard, K. Zhang and two with K. Zhang we prove Arnold's conjecture in dimension n= 3. |
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| 4.6.2013, 11:10 (Tuesday) | Maurice de Gosson (University of Vienna) | |
| Title: | Symplectic Eggs, Quantum Blobs, and Uncertainty | |
| Location: | Schreiber bldg., room 210, Tel-Aviv University | |
| Abstract: | A symplectic egg is the image of a phase space ball by a linear symplectic transformation. When the radius of the ball is $\sqrt[\bar h]$ then the symplectic ball is a "quantum blob", the smallest phase space unit allowed by the quantum uncertainty principle. The latter is one of the hallmarks of quantum me- chanics, and is abundantly discussed in both theoretical and experimental physics. We show that a strong version of the uncertainty principle can be stated in terms of quantum blobs, using the related notion of symplectic capacity, which is closed related to Gromov’s symplectic non-squeezing the- orem. Our approach in principle only requires a knowledge of elementary linear algebra and of the most basics concepts of quantum mechanics. |
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| 5.6.2013, 14:10 |
Maurice de Gosson (University of Vienna) | |
| Title: | The Wigner Transform and the Uncertainty Principle | |
| Location: | Schreiber bldg., room 209, Tel-Aviv University | |
| Abstract: | We pursue the discussion of the uncertainty principle of quantum me- chanics initiated in the first talk. A fundamental mathematical object is the Wigner transform; it plays a central role in the theory of Weyl pseudo- differntial operators, to which it is closely related. In this talk we put an emphasis on its properties of symplectic covariance, which allow us to prove an analogue of Hardy’s uncertainty principle for Wigner transforms. This result, which says that a Wigner transform cannot be "too concentrated" in phase space will expressed in terms of the notion of symplectic capacity of a certain ellipsoid. We will also show that the notion of quantum blob introduced in the first talk can be interpreted in terms of the Wigner trans- forms of Gaussian functions (the "squeezed coherent states" introduced by Schrödinger, and familiar from quantum optics) |
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| 19.6.2013, 14:10 (Wednesday) | Frédéric Bourgeois (Université Libre de Bruxelles) | |
| Title: | S^1-equivariant symplectic homology and linearized contact homology | |
| Location: | Schreiber bldg., room 209, Tel-Aviv University | |
| Abstract: | We define an S^1-equivariant version of symplectic homology via various equivalent approaches. We show that, over rational coefficients, S^1-equivariant symplectic homology is isomorphic to linearized contact homology. This is a joint work with Alexandru Oancea. |
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Organized by Misha Bialy, Lev Buhovsky, Yaron Ostrover, and Leonid Polterovich