Place: Konferenzraum, 5th floor - Mathematikon INF 205
Hofer–Zehnder capacity of disk tangent bundles via billiards
The Hofer–Zehnder capacity is a numerical symplectic invariant defined using Hamiltonian dynamics. Its finiteness implies the Weinstein conjecture for any compact contact hypersurface, demonstrating that finiteness alone is a fairly strong statement. In this talk, I will survey known (and missing) results for subsets of (co-)tangent bundles, with a focus on explicit computations. Many of these rely on Riemannian billiards to obtain lower bounds. So far, all known results concern highly degenerate metrics, such as constant curvature or more generally symmetric spaces. I will present work in progress that aims to extend these results to more generic (bumpy) metrics.
Rigidity in Frenkel Kontorova model.
The Frenkel-Kontorova is an important model for nearest neighbor interactions in physics. An example of a non-trivial integrable potential for this model was found by Suris in the 80's. This is somewhat analogous to billiards in an ellipse. We will explain how the action angle coordinates look like in this system, and how we can use them to show local rigidity of integrable potentials near the Suris potential, and that the Suris potential is spectrally rigid.
Symplectic billiards: integrability and spectral rigidity
Symplectic billiards were introduced by P. Albers and S. Tabachnikov as a dynamical system where, unlike Birkhoff billiards, the generating function is the area instead of the length. After briefly recalling the definition and main properties of symplectic billiards, this presentation aims to provide an overview of some recent results concerning integrability and (area-)spectral rigidity for this class of billiards.
Based on joint work with L. Baracco and O. Bernardi.
Existence and nonexistence of invariant curves of coin billiards. (Joint work with A. Clarke)
In this paper we consider the coin billiard introduced by M. Bialy. It is a modification of the classical billiard, obtained as the return map of a nonsmooth geodesic flow on a cylinder that has homeomorphic copies of a classical billiard on the top and on the bottom (a coin). The return dynamics is described by a map of the annulus. We prove the following three main theorems: in two different scenarios (when the height of the coin is small, or when the coin is near-circular) there is a family of KAM curves close to, but not accumulating on, the boundary; for any noncircular coin, if the height of the coin is sufficiently large, there is a neighbourhood of the boundary through which there passes no invariant essential curve; and the only coin billiard for which the phase space is foliated by essential invariant curves is the circular one. These results provide partial answers to questions of Bialy. Finally, we describe the results of some numerical experiments on the elliptical coin billiard.
Multidimensional Birkhoff Theorem for Recurrent Lagrangian Submanifolds
A theorem by Birkhoff, established in 1922, states that an essential invariant curve under a twist map of the cylinder is a Lipschitz graph over the circle. Several generalizations of this theorem to higher dimensions have followed. One notable result, due to Marie-Claude Arnaud (2010) concerns the cotangent bundle of a closed manifold and establishes that any exact Lagrangian submanifold (Hamiltonianly isotopic to the zero section) and invariant under the flow of a Hamiltonian that is convex on the fibers, is a Lipschitz graph over the zero section. More precisely, it is the graph of a weak KAM solution to the Hamilton-Jacobi equation.
We propose an extension of this result to recurrent Lagrangian submanifolds under the action of the Hamiltonian flow, using a topology of convergence that controls the dilations of these submanifolds. In this case, such submanifolds are Lipschitz graphs over the zero section and, more precisely, graphs of differentials of recurrent viscosity solutions.
Infinitely many chords, homoclinic orbits, and nice Birkhoff sections for Reeb flows in 3D
The main result of this talk is the following statement: If a Reeb vector field on a closed 3-manifold admits a boundary strong Birkhoff section then every Legendrian knot has infinitely many geometrically distinct Reeb chords, except possibly when the ambient manifold is a lens space or the sphere and the Reeb flow has exactly two periodic orbits. This is joint work with Vincent Colin and Ana Rechtman.
Quantitative almost-existence in dimension four
In 1987, Hofer and Zehnder showed that for any smooth function H on R^2n, almost every compact and regular level set contains at least one closed characteristic. I'll show that, when n=2, almost every compact and regular level set contains at least two closed characteristics.
Bounds on capacities in cotangent bundles using Floer homology with differential graded coefficients
Liouville domains with vanishing symplectic homology have finite spectral capacities. In contrast, unit cotangent bundles typically have non-vanishing symplectic homology. Over the years, several methods that circumvent this feature were used successfully in order to bound spectral capacities: S1-equivariant symplectic homology (Viterbo, Frauenfelder, Pajitnov), symplectic homology with local coefficients (Ritter, Albers, Frauenfelder, and also the speaker), Gromov-Witten invariants (Bimmermann, Maier). I will focus in the talk on a recent method, developed in joint work with Barraud, Damian, and Humilière, which uses a new type of Floer homology with differential graded coefficients.
On the number of closed geodesics on S^2
Given an arbitrary reversible Finsler metric on S^2, the question of how many closed geodesics this metric has has a long history. In this talk, we will present a result stating that the number of closed geodesics grows at least quadratically with respect to length. The proof is based on methods from cylindrical contact homology, dynamics of annulus maps, and spheres of revolution.
Braid Stability and the spectral distance
In this talk, we establish a result known as braid stability on symplectic surfaces, for the spectral distance. This leads to two persistence-type results for topological entropy: effective robustness and lower semicontinuity. Here, effective robustness means that the topological entropy does not drop by more than epsilon under perturbations supported in a disk whose area is bounded by a constant depending on epsilon.This talk is based on an ongoing joint work with Marcelo Alves and Matthias Meiwes.
A Poincaré-Birkhoff-type theorem for crossed, possibly non-convex billiard tables
In this talk I will summarize a proof that on every smooth, possibly non-convex billiard table carrying a pair of intersecting 2-bounce orbits A and B with a mild curvature condition at their endpoints one can find - for every coprime (p,q) with p/q in an interval specified by the endpoint curvatures- a periodic orbit that links p times with A and q times with B. Hence such tables, though possibly non-convex, carry infinitely many geometrically distinct periodic orbits. The proof relies, among other things, on a simple quantum mechanical interpretation of the Morse index.
The Symplectic Billiard Table
In this informal talk, I will show software demos of billiards and related systems. I will give examples of how computer experimentation can be used in both forming conjectures and validating proof strategies for these systems.
Symbolic dynamics of triangle tiling billiards and interval exchange transformations with flips
A tiling billiard is a model for the light propagation in heterogeneous media : a ray of light moves through a tiling of a plane, and refracts each time it crosses a boundary of a tile. The refractive index being -1 brings the study of such billiards to the world of non-orientable flat surfaces and interval exchange transformations with flips.
For a family of such billiards in a periodic triangular tiling made by a single triangle, I will discuss several questions about their symbolic dynamics, shading a light on their connection to Rauzy fractals, interval exchange transformations with flips and linear foliations on surfaces.
Convex billiards and rigidity of Mather $\beta$ function.
In this talk, I shall discuss convex planar billiards, including Birkhoff, Symplectic, and Outer billiards.
In the first part, I will explain how integrability results can be stated in terms of Mather $\beta$-function.
In the second part, I discuss new rigidity results for Mather $\beta$-function in the spirit of isoperimetric inequalities.
Surprisingly, the Outer billiards case turns out to be somewhat exceptional.
Based on joint work with A.Mironov, S. Baranzini, and A.Sorrentino.
Periodic Orbits of Rotating Kepler Problem
In this talk, I will discuss the natural parametrization of the moduli space of periodic orbits in the rotating Kepler problem and present computations of their Conley–Zehnder indices. The analysis is based on the use of the Morse–Bott spectral sequence. If time permits, I will also touch on possible variations of the rotating Kepler problem.
Embedding dynamics in cosymplectic geometry: Turing complete solutions to the Navier-Stokes' equations
Cosymplectic manifolds admit a notion of compatible metrics for which in particular the Reeb vector field solves the Navier-Stokes' equations.
A construction by Moore [1] realizes Turing machines as certain piecewise linear maps of the square, and this was used by Cardona, Miranda, Peralta-Salas, and Presas [2] to embed Turing machines as (return maps of) solutions to the (stationary) Euler equations, obtained by embedding them into contact manifolds and using a similar notion of compatible metrics.
In this talk I will discuss these constructions and our work (joint with Ángel González-Prieto, Eva Miranda, and Daniel Peralta-Salas) using cosymplectic manifolds to embed Turing machines as solutions to the Navier-Stokes equations.
[1] Moore, C. “Generalized Shifts: Unpredictability and Undecidability in Dynamical Systems.” Nonlinearity 4, no. 2 (May 1991): 199. https://doi.org/10.1088/0951-7715/4/2/002.
[2] Cardona, Robert, Eva Miranda, Daniel Peralta-Salas, and Francisco Presas. “Constructing Turing Complete Euler Flows in Dimension $3$.” Proceedings of the National Academy of Sciences 118, no. 19 (May 11, 2021): e2026818118. https://doi.org/10.1073/pnas.2026818118
On the rigidity of Mather's beta function and the stable norm for Riemannian metrics
In this talk, I will discuss rigidity phenomena associated with Mather's beta function and the stable norm for Riemannian metrics.
Mather's beta function arises in the study of Lagrangian systems: it assigns to each real homology class the minimal average action of invariant probability measures with that homology class (or Schwartzman asymptotic cycle). The stable norm, a related concept in Riemannian geometry, is defined via the homogenized limit of the minimal lengths of cycles in multiples of a given homology class, yielding a norm on homology that encodes the asymptotic geometry of geodesics.
Both functions serve as fundamental tools for understanding the large-scale dynamics of geodesic flows and the interplay between geometry and dynamics. A central question is whether these objects remember the underlying metric, either globally or locally. For instance, given two metrics, can specific features of their associated beta functions or stable norms reveal whether the metrics are related by a simple transformation, such as a homothety?
In this talk, I will present some answers to this question, based on a joint work with Anna Florio and Martin Leguil.
A counterexample to Viterbo's conjecture
Symplectic capacities are invariants emerging from Hamiltonian dynamics and symplectic topology which provide a way to measure the "size" of symplectic manifolds. Some, like the Hofer-Zehnder capacity when restricted to convex domains, have a compelling dynamical interpretation: they can sometimes be viewed as the minimal length of a closed billiard trajectory within a convex body, measured with a specific Minkowski distance.
A central isoperimetric-type problem in the field has been Viterbo's conjecture (2000), which posited that the ball should have the largest capacity among all convex domains of the same volume. This elegantly stated question has captured the relationship between convex and symplectic geometry.
In this talk, I will present a recent counterexample to Viterbo's conjecture, based on joint work with Yaron Ostrover. I will also discuss some open questions that follow, including the formulation of modified versions of the conjecture and a continued exploration of the precise role that convexity plays in symplectic geometry.
Outer symplectic billiards
Outer billiards in the plane is, by now, a well-understood and thoroughly studied dynamical system akin to the conventional, inner billiards. Outer symplectic billiards is a multidimensional relative of this system, defined in the standard symplectic space. The role of the "billiard table" is played by a submanifold, not necessarily a hypersurface, and the respective billiard map is symplectic but, in general, multivalued. I shall survey some recent (and not so recent) results about this system with a focus on open problems.
Pensive and vortex billiards
We define a class of plane billiards in which, while preserving the billiard rule of equality of the angles of incidence and reflection, before reflecting the billiard ball travels along the boundary for some distance depending on the incidence angle. This generalizes "puck billiards" proposed by Misha Bialy, as well as the motion of vortex dipoles in 2D hydrodynamics near the domain boundary. We describe the variational origin and invariance of a symplectic structure for such billiards. We also observe the appearance of both the golden and silver ratios in the corresponding vortex setting. This is a joint work with D.Glukhovskiy and T. Drivas.
On the topological invariance of helicity
Helicity is an invariant of divergence free vector fields on a three-manifold. One of its fundamental properties is invariance under volume preserving diffeomorphisms. Arnold, having derived an ergodic interpretation of helicity as an asymptotic Hopf invariant, asked whether helicity remains invariant under volume preserving homeomorphisms, and more generally, whether it admits an extension to topological volume preserving flows. In this talk, I will present an affirmative answer to both questions for non-singular flows. The proof draws on recent advances in C^0 symplectic geometry, in particular regarding the algebraic structure of the group of area preserving homeomorphisms, which I will also survey. This is based on joint work with Sobhan Seyfaddini.
Outer Billiards of Symplectically Self-polar Convex Bodies.
We have shown that the class of symplectically self-polar convex bodies is a natural generalization of the class of Radon curves for the higher even dimensions. Precisely, it is a generalization that yields invariant hypersurfaces with 4-periodic centrally symmetric orbits the same way the Radon curves do. In contrast to the classical result of Berger and Gruber regarding the triviality of conventional high-dimensional Birkhoff billiards with caustics, the aforementioned class is non-trivial in every dimension. This talk is based on our joint work with Prof. Misha Bialy.