Schedule for: 16w5017 - Integrability and Near-Integrability in Mechanics and Geometry

A welcome drink will be served at the hotel.

The ideas of projective geometry have a certain « universality » and appear in many different fields of mathematics and physics. As mathematics itself, the destiny of this old area is to remain forever young… The goal of this talk is to illustrate this universality on contemporary examples; our main characters are the Schwarzian derivative, S, and the pentagram map, T. I will explain what this S and T have in common.

Recall Rutherford scattering. A 2-body problem was solved with the key ingredient being scattering: incoming and outgoing lines which were asymptotes to associated hyperbolae. In the N-body problem at positive energies, when a certain limit is taken, families of solutions converge to a system of N incoming and N outgoing rays with a finite number of elastic collisions in between. We abstract this situation to a ``point billiard process'' whose data consist of a Euclidean vector space E endowed with a finite collection of codimension d linear subspaces called ``collision subspaces''. The solutions to the `process'' are unit speed continuous piecewise linear trajectories corresponding to billiards played on the table E minus the union of the collision subspaces. These solutions, or ``billiard trajectories'' move in straight lines away from the collision subspaces. Upon hitting a subspace the reflect off according to the standard law of reflection. The ``dynamics'' associated to the process is not deterministic since for a given ray incoming to a collision subspace there is a $d-1$ dimensional sphere's worth of allowable outgoing rays. The ``itinerary'' of such a trajectory is the list of subspaces it hits, in the order hit. Two basic questions are: (A) Are itineraries finite? (B) What is the structure of the space of all trajectories having a fixed itinerary? In two beautiful papers from 1998 Burago-Ferleger-Kononenko [BFK] use non-smooth metric geometry ideas (CAT(0) and Hadamard spaces) to answer (A) affirmatively. We answer (B): this space of trajectories is a Lagrangian relation on the space of oriented lines in E. Our proof relies on two techniques, (1) generating families for Lagrangian relations, and (2) the metric geometry introduced by BFK and relying crucially on a theorem of Reshetynak. We will focus on (2). This is joint work with Andreas Knauf and Jacques Fejoz.

The geometry of planar bicycles paths has been shown, by work of Sergei Tabachnikov and his collaborators, to be a beautiful and rich subject. I will show, mainly via computer-generated animations, a natural extension of some of the results to 3-dimensional bike paths. This is work in collaboration with Mark Levi, Ron Perline and Sergei Tabachnikov.

I'll explain a combinatorial model, which I call the plaid model. For each rational parameter, the construction produces a finite union of embedded polyhedral surfaces in a cube. When the surfaces are sliced in one direction, the resulting curves encode the dynamics of outer billiards on kites. When the surfaces are sliced in other directions, they give the same families of curves (up to isotopy) as those produced by Pat Hooper's Truchet tile system.

(Joint work with Phil Bowers, Florida State) The famous "Penrose" tiling is perhaps the most well known hierarchical, aperiodic tiling of the plane. We consider this and other infinite tilings generated by subdivision rules. However, we put conformal structure rather than euclidean structure on the tiles, giving so-called "conformal tilings". Conformal tiling is determined by combinatorics alone, and is not limited by the rigid geometric constraints of classical tilings, thus it brings up new issues in tiling theory. This talk will rely heavily on tiling images. Among other things, those images suggest that aggregates of conformal tiles may converge in shape to their classical euclidean counterparts. This raises an interesting issue: how can rigid euclidean shapes be encoded in abstract combinatorics? This is a new side of tiling theory with many open questions.

I will describe counterintuitive behavior of particles in rotating potentials, as well as some related effects such as the curious Coriolis-like force acting on binaries in an ambient gravitational field. I will also mention the unexpected role of the Gaussian curvature in this connection.

Complexity theory deals with determining when there does or does not exist a faster algorithm than the standard for some basic task such as multiplying matrices. In the case of matrix multiplication, Strassen shocked the world in 1969 by finding an algorithm faster than the standard one and computer scientists now make the astonishing conjecture that as the size of the matrices increase, it becomes almost as easy to multiply two nxn matrices as it is to add them. The famous P v. NP problem essentially conjectures that there is no fast solution to the traveling salesperson problem. Recently algebraic geometry and representation theory have led to advances regarding these conjectures. I will give an overview of these questions and the recent advances.

I shall report on a recent progress inThe problem of integrability of Birkhoff and other models of billiards. The approach is motivated by a paper of Sergei Tabachnikov on Outer billiards. This talk is based on joint works with A.E.Mironov.

We describe the dynamics of a simple piecewise linear map of a strip, which can be thought as a toy model for various economical problems. It turns out that even such a simple model possess non-trivial dynamics. We classify the attractors for the different values of the parameters of this model.

The classical Birkhoff conjecture states that the only integrable convex planar domains are circles and ellipses. In a joint work with A. Avila and J. De Simoi we show that this conjecture is true for perturbations of ellipses of small eccentricity. It turns out that the method of proof gives an insight into deformational spectral rigidity of planar axis symmetric domains and gives a partial answer to a question of P. Sarnak. The latter is a joint work with J. De Simoi and Q. Wei.

We present a class of nonconvex billiards with a boundary composed of arcs of confocal conics which contain reflex (nonconvex) angles. We present their basic dynamical, topological, and arithmetic properties. Such systems are not integrable, but carry strong traces of integrability. We study their periodic orbits and establish a local Poncelet porism. A connection with interval exchange transformation is established together with the Keane-type conditions for minimality. A transformation from pseudo-integrable billiards to rectangular billiards is constructed. This research is done jointly with M. Radnovic.

There have already been numerous studies and interpretations of the famous separation of variables in the integrable top of S. Kovalevskaya. In this talk we show how the original Kovalevskaya curve of separation can be obtained, by a simple one-step transformation, from the spectral curve of the Lax representation found by Bobenko, Reyman, and Semenov-Tian-Shansky. The algorithm works for the general constants of motion of the top and is based on W. Barth's description of Prym varieties via pencils of genus 3 curves. This also allows us to derive existing and new curves of separation for the Kovalevskaya gyrostat in one and two force fields.

We present a new class of integrable surfaces associated with Bertrand curves. These surfaces are foliated by constant-torsion curves evolving according to a novel integrable geometric flow. Curves transverse to the constant-torsion curves (orbit curves) are Bertrand curves on the surface. The surfaces discussed interpolate two known integrable systems and we establish the connection. We also use tools from soliton theory to generate surface solutions using B\"{a}cklund transformations.

A circle pattern with the combinatorics of the square grid is an embedding of the square grid such that every face admits a circumcircle. Using Miquel's six circles theorem, we define a dynamical system on the space of such circle patterns on a flat torus. Simulations seem to indicate that this is an integrable system. I will discuss work in progress which provides some results in this direction.

In 1966, V.Arnold suggested a group-theoretic framework for ideal hydrodynamics. In this approach, the motion of an incompressible fluid on a Riemannian manifold is described as the geodesic flow of a right-invariant metric on the group of volume-preserving diffeomorphisms. In my talk, I will review Arnold's picture and show how it can be extended to incorporate certain discontionus fluid motions, known as vortex sheets. This is done by replacing groups and algebras in Arnold's approach by certain groupoids and algebroids. This is joint work with B.Khesin.

The evolution, through spatially periodic linear dispersion, of rough initial data leads to surprising quantized structures at rational times, and fractal, non-differentiable profiles at irrational times. The Talbot effect, named after an optical experiment by one of the founders of photography, was first observed in optics and quantum mechanics, and leads to intriguing connections with exponential sums arising in number theory. Ramifications of these phenomena and recent progress on the analysis, numerics, and extensions to nonlinear wave models, both integrable and non-integrable, will be discussed.

In d=2 with variables (x,t), the superalgebraic trick of adding a supplementary odd variable allows the construction of a "square root of time", an operator D satisfying D^2=∂/∂t in superspace of dimension (2|1). We used it to obtain a Miura transform in dimension (2|1) for non stationary Schrödinger type operators. We shall discuss here the construction of an algebra of pseudodifferential symbols in dimension (2|1); that algebra generalizes the one for d=1, used in construction of hierarchies from isospectral deformations of stationary Schrödinger type operators.

Decades ago Semenov-Tian-Shansky defined the so-called twisted Poisson structure on $G^N$, where $G$ is a Poisson Lie group. We will review the definition and show that in the case $G = PSL(n+1)$, it can be reduced to the moduli space of projective polygons (under the projective action), as defined by the discrete projective curvatures. We will also show that any reduced Hamiltonian evolution is induced on the curvatures by a simple polygon evolution that can be defined directly from the variation of the Hamiltonian function. We will also define a second structure reducing a right invariant tensor, which is proven to be compatible with the previous reduction for dimensions 2 and 3, and conjectured to be for any dimension. The pair are Hamiltonian structures for integrable discretizations of W_n algebras in any dimension. Thus, one can write a very simple realization of this integrable system as an evolution of projective polygons. This is joint work with Jing Ping Wang.

Take a solid shell bounded by two homothetic ellipsoids. Ivory's theorem says that the gravity inside the shell is zero; besides, if the shell is infinitely thin, the equipotential surfaces outside of it are confocal ellipsoids. Arnold's theorem generalizes the first part of the Ivory theorem (zero gravity) to certain algebraic surfaces. In this talk we present analogs of both theorems in the spherical and the hyperbolic space. This is a part of a joint work with Sergei Tabachnikov.

Quiver mutations play important role in definition of cluster algebra and also appeared independently as Seiberg duality in mathematical physics. In this talk I will discuss quivers with finite mutation class, more exactly, classification result and its application. ​This is a joint work with A.Felikson and P.Tumarkin.

We discuss integrable structures on spaces of polygonal tilings. Special cases include the pentagram maps and their generalizations, dimer integrable systems of Goncharov and Kenyon, and resistor network integrable systems.

Two particle systems are said to be “dual” if the maps that linearize their dynamics are inverses. “Bispectrality", on the other hand, describes the situation in which a function satisfies two eigenvalue equations with the roles of spacial and spectral variables exchanged. Although bispectrality does not initially appear to be dynamical in nature, it turns out that duality manifests itself as bispectrality on both the classical and quantum levels. (For example, the Calogero-Moser system is self-dual since its action-angle map is an involution, and this can be seen either in a symmetry of the eigenfunction of its quantum Hamiltonian or through the appearance of this particle system in the pole dynamics of bispectral solutions of the KP Hierarchy.) This talk will review both old and recent results on this subject and then conclude with a list of associated open problems.

We show how a Hamiltonian diffeomorphism can be lifted to the symplectic one-point blow up. Then we consider loops of Hamiltonians diffeomorphimsm and show that the rank of fundamental group of the group of Hamiltonian diffeomorphisms of the symplectic one-point blow up is positive.