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Open problems in billiards and quantitative symplectic geometry

Bernhard Albach, Jean-François Barraud, Misha Bialy, Johanna Bimmermann, Ana Chávez Cáliz, Mihai Damian, Lina Deschamps, Umberto Hryniewicz, Vincent Humilière, Boris Khesin, Levin Maier, Agustin Moreno, Alexandru Oancea, Olga Paris-Romaskevich, Alfonso Sorrentino, Serge Tabachnikov

TL;DR

This Open Problems collection gathers foundational questions at the intersection of billiards and quantitative symplectic geometry arising from the 2025 Heidelberg workshop. It develops a framework for conformally and locally conformal symplectic dynamics, including CS vector fields with a constant $c$ and the Lee differential $d_\eta$, and then catalogs a broad set of open problems spanning projective and outer billiards, inner/outer periodic orbits on immersed submanifolds, complexity of polygonal symplectic billiards, and optical transformations of oriented lines. Key themes include Ivrii-type conjectures for outer symplectic billiards, growth and density questions for periodic trajectories and wavefronts, and finiteness questions for cotangent bundle capacities such as $D^*N$, connected to symplectic and Floer-theoretic techniques. The collection aims to bridge methods from symplectic geometry, Hamiltonian dynamics, and geometric optics, with potential implications for translation surfaces, pseudoholomorphic curve techniques, and the broader study of dynamical systems in non-conservative settings. Overall, it serves as a roadmap for rigorous results and cross-pollination of ideas across the geometry of billiards, conformal dynamics, and symplectic invariants.

Abstract

This document collects contributions to the Open Problem List in Billiards and Quantitative Symplectic Geometry, compiled following discussions during the workshop ``Billiards and quantitative symplectic geometry'' that took place at the University of Heidelberg on July 14--18, 2025.

Open problems in billiards and quantitative symplectic geometry

TL;DR

This Open Problems collection gathers foundational questions at the intersection of billiards and quantitative symplectic geometry arising from the 2025 Heidelberg workshop. It develops a framework for conformally and locally conformal symplectic dynamics, including CS vector fields with a constant and the Lee differential , and then catalogs a broad set of open problems spanning projective and outer billiards, inner/outer periodic orbits on immersed submanifolds, complexity of polygonal symplectic billiards, and optical transformations of oriented lines. Key themes include Ivrii-type conjectures for outer symplectic billiards, growth and density questions for periodic trajectories and wavefronts, and finiteness questions for cotangent bundle capacities such as , connected to symplectic and Floer-theoretic techniques. The collection aims to bridge methods from symplectic geometry, Hamiltonian dynamics, and geometric optics, with potential implications for translation surfaces, pseudoholomorphic curve techniques, and the broader study of dynamical systems in non-conservative settings. Overall, it serves as a roadmap for rigorous results and cross-pollination of ideas across the geometry of billiards, conformal dynamics, and symplectic invariants.

Abstract

This document collects contributions to the Open Problem List in Billiards and Quantitative Symplectic Geometry, compiled following discussions during the workshop ``Billiards and quantitative symplectic geometry'' that took place at the University of Heidelberg on July 14--18, 2025.
Paper Structure (19 sections, 5 theorems, 38 equations, 3 figures)

This paper contains 19 sections, 5 theorems, 38 equations, 3 figures.

Table of Contents

  1. Question:
  2. Question:
  3. Conformal symplectic dynamics on a symplectic manifold
  4. Setting
  5. Properties of the flow
  6. Locally conformal symplectic systems
  7. Setting
  8. Conformally Symplectic Cotangent Bundles
  9. Projective billiards.
  10. Inner and outer periodic orbits on immersed submanifolds.
  11. Complexity of polygonal symplectic billiards.
  12. Chaotic symplectic billiards?
  13. Planar characteristic curves.
  14. Density of reflected wave fronts.
  15. Ivrii's conjecture for outer symplectic billiards.
  16. ...and 4 more sections

Key Result

Proposition 3

[Théo Marty, Olga P.-R.] The universal cover $\widetilde{X}$ of a parallelogram-tiled plane deprived of the vertices of the parallelograms with angles $\alpha$ is well defined if and only if the following two conditions hold:

Figures (3)

  • Figure 1: Pensive Billiard.
  • Figure 2: On the left, a periodic trajectory of a triangle tiling billiard. On the right, a trajectory of a refractive wind-tree model confined to a band.
  • Figure 3: On the left, a trajectory in a parallelogram tiling with parameters $\alpha=\frac{\pi}{4}$ and $a=b=1$ (a trajectory is a leaf of a non-orientable foliation of the plane) after $138$ refractions. On the right, a trajectory with $a=1, b=1.05189074$ and $\alpha=47.8695014$ after $3000$ refractions (a trajectory re-intersects itself).

Theorems & Definitions (17)

  • Proposition 3
  • Definition 1
  • Remark 2
  • Remark 3
  • Definition 4
  • Lemma 5
  • proof
  • Lemma 6: Gauge invariance
  • proof
  • Lemma 7: Modified Cartan's formula
  • ...and 7 more