Geometry & Dynamics Seminar

Fall 2007 & Spring 2008

14.07.2008 Regina Rotman (PennState)
Geodesic loops on Riemannian manifolds

7.07.2008 Michael Khanevsky (TAU)
Hofer's geometry on the space of curves

30.06.2008 Yuliy Baryshnikov (Bell Labs)
Spherical billiards with many 3-periodic orbits
Abstract: It is known that the Lebesgue measure of 3-periodic trajectories in
planar (Birkhoff) billiards is zero (and a well-known conjecture states
that the same is true for any period). On the sphere, however, it is
easy to construct a billiard domain with 2-dimensional family of
3-periodic orbits (take the intersection of the sphere with the positive
octant). In this talk I will explain - using non-holonomic distributions
- why this is essentially the only possible construction.

23.06.2008 Patrick Iglesias-Zemmour (Hebrew University)
Moment maps

17.06.2008 (Tuesday) Michael Bialy (TAU)
Periodic solutions for reduction of Benney chain
Abstract:We study periodic solutions for a quasi-linear system of PDEs, which
is the so called dispersionless Lax reduction of the
Benney moments chain. It is a very well known problem for
a quasi-linear systems to find smooth solutions or to establish
their finite time lifespan. I will show how the quasi linear
system is related to a question of integrability of a classical
Hamiltonian system of the form $ H=p^2/2+u(q,t) $ and will explain
how Lax analysis can be performed for it. The main result states
that the only periodic solutions of the 3 by 3 reduction of the
Benney are of the form of traveling
waves. I will relate this result to classical conjecture by
Birkhoff on integrable billiards.

2.06.2008 Andrej Mironov (Sobolev Institute, Novosibirsk)
On Hamiltonian-minimal Lagrangian submanifolds in C^n and CP^n
Abstract: In the talk I will explaine a method of construction of
Hamiltonian-minimal Lagrangian immersions of some manifolds in C^n and
CP^n. By this method one can construct, in particular, immersiones of such
manifolds as the generalized Klein's bottle K^n, K^{n-1}\times S^1,
S^{n-1}\times S^1, and others.

27.05.2008 Pierre Py (ENS,Lyon)
Mixed Action-Maslov invariants
Abstract: we consider Hamiltonian diffeomorphisms of a monotone
symplectic manifold. In this situation, Polterovich associated to
each contractible fixed point of the diffeomorphism, a quantity which
combines the action and the Maslov index of the fixed point, but
which does not depend on the choice of a "filling disc". We will
explain how to define such an invariant for any invariant measure of
the diffeomorphism (assuming the measure has zero asymptotic cycle)
and the relation of this construction with Entov and Polterovich's
quasi-morphism.

07.04.2008 Pierre Py (ENS,Lyon)
                    Some dynamical cocycles for groups of Hamiltonian
                    diffeomorphisms of surfaces

Abstract: I will explain how various classical (and less classical)
invariants associated to Hamiltonian diffeomorphisms of surfaces give
rise to some cohomology classes on the group of Hamiltonian
diffeomorphisms, and explain why these classes might be useful for
the study of group actions on surfaces. I will review the necessary
definitions of group cohomology.

31.03.08 Dan Mangoubi (IHES)
               Tubular Neighborhoods of the Nodal Set.

Abstract: Let M be a closed compact smooth Riemannian Manifold of dimension n.
Let f be an eigenfunction of the Laplacian on M with eigenvalue \mu^2.
{f=0} is called the (\mu)-nodal set.
Yau's conjecture says: The (n-1) dimensional volume of {f=0} is comparable to C\mu.
This conjecture was proved in the case where M is real analytic by Donnelly and Fefferman.
We consider a tubular neighborhood T_r of the nodal set of radius r.
We show that on real analytic manifold   C_1 \mu r < Vol(T_r) < C_2 \mu r.
This shows further regularity of the nodal set and could lead to curvature estimates of the nodal set.
This is joint work with Dima Jakobson.

17.03.2008 Albert Fathi (ENS,Lyon)
                 Hamilton-Jacobi and Denjoy-Schwartz
Abstract:Why dynamics matter in the regularity of smooth subsolutions
In this follow-up lecture of the colloquium, we will explain that, in dimension n=2, the
classical work of Denjoy on C2 diffeomorphisms of the circle (or rather its generalization by
Schwartz to flows on surfaces) puts strong restrictions on the existence of smoother critical
subsolutions of the Hamilton-Jacobi equation. For this we will have to introduce Mather's
\alpha function (called the homogenized Hamiltonian in PDE) and Schwartzman asymptotic
cycles. We will also develop the results obtained by Patrick Bernard (and even some
generalization) for existence of smoother critical subsolutions.

24.03.08 Felix Schlenk (Université Libre de Bruxelles)
               Exotic monotone Lagrangian tori

Abstract: We explain a construction of a family of monotone Lagrangian tori
in $R^{2n}$, $CP^n$ and products of spheres.
All these tori are Lagrangian isotopic,
but there are about $4^n$ Hamiltonian isotopy classes.
This is work joint with Yuri Chekanov.

10.03.2008 Leva Buhovsky (TAU)
                 "The 2/3 convergence rate for the Poisson bracket"

03.03.2008 Roman Muraviev
                  Growth Gap vs. Smoothness for Diffeomorphisms of The Interval

28.01.2008 Lev Birbrair (Universidade Federal do Ceara, Brazil)
   Metric Geometry of Singularities of Complex Surfaces

21.01.2008 Michael Khanevsky
                 A Gysin-Floer exact sequence for Lagrangian Floer homology

14.01.2008 Jake Solomon (IAS, Princeton)
                      "A differential equation for the open Gromov-Witten potential"

Abstract:

I will describe a system of differential equations for the genus 0 open
Gromov-Witten potential of a Lagrangian submanifold of a symplectic
4-manifold fixed by an anti-symplectic involution. These equations involve
both the open Gromov-Witten potential and the closed Gromov-Witten
potential. They are sufficiently restrictive that in significant examples
they completely determine both the open and closed Gromov-Witten
potentials up to a finite number of constants. This provides a general and
very efficient approach to calculating real enumerative invariants. The
proof relies on an open-closed generalization of the topological conformal
field theory behind the WDVV equation.

7.01.2008 Oren Ben Bassat, Hebrew University
                "Introduction to Generalized Complex Geometry"
                Abstract
                    I will give an introduction to Generalized Complex Geometry.
                This is a kind of geometry that includes both complex and symplectic
                geometry. I will explain how it naturally incoporates T-duality in
                its absolute and relative forms.

31.12.2007 Yaron Ostrover, MIT
                  "On the semi-simplicity of the quantum homology"
                   (Joint work with Ilya Tyomkin)

24.12.2007 Shiri Artstein-Avidan, TAU
                  "A Brunn-Minkowski-type inequality for a symplectic capacity, and applications"
                        (joint work with Yaron Ostrover)

17.12.2007 Lev Buhovsky, TAU
                  "One explicit construction of a relative packing"

10.12.2007 Dmitry Novikov, Weizmann Institute
                   "Non-oscillation of PseudoAbelain integrals"

3.12.2007   Michael Brandenbursky, Technion
                  "Quasi-morphisms and knot theory"

26.11.2007   Dmitry Turaev, Ben Gurion University
                   "Unbounded energy growth in slowly-perturbed Hamiltonian systems"

19.11.2007    Mikhail Katz, Bar Ilan
                    "Quaternion and Lie algebra applications in systolic geometry
                    and topology"

12.11.2007    Gershon Wolansky, Technion
                    "Dirichlet functionals and the weak KAM Theory"

5.11.2007     Egor Shelukhin, TAU
                    "A connection between Hamiltonian and Kahler invariants"

29.10.2007     Par Kurlberg, KTH Stokholm
                    "Quantum ergodicity - unique or not?"