Mathematical Billiards

June 28 - August 13

TIME: MWF 10:10 am - 12:05 pm

INSTRUCTOR: SERGEI TABACHNIKOV

MAIN TOPICS include:

  1. Mechanical systems, configuration and phase spaces. Optics—mechanics analogy. Variational principles. Huygens principle. Example: elastic particles on the line and on the circle. Finsler billiards, magnetic billiards.
  2. Billiards inside the circle and the square. Dynamics of circle rotation, equidistribution. Multi-dimensional versions. Application: distribution of first digits in sequences. Symbolic description: continued fractions, Sturm sequences. Complexity of sequences.
  3. Optical properties of conics. Integrability, various proofs. Applications: whispering galleries, trap for a parallel beam, illumination problem, Urquhart's theorem, beating second law of thermodynamics. Poncelet porism, various proofs.
  4. Evolutes and involutes. Caustics of billiards and string construction. Evolute as the locus of centers of curvature. Four vertex theorem. Sturm-Hurwitz theorem and its four proofs. Topology of wave fronts and Fabricius-Bjerre theorem. Projective and spherical duality.
  5. Phase space of a billiard. Symplectic structure and symplectic properties of the billiard transformation. Integral geometry, Crofton formula. Applications: isoperimetric inequality, Fary's theorem (DNA inequality). Hilbert's fourth problem.
  6. Poincare recurrence theorem. Applications: periodic trajectories in rational polygons and in right triangles. Unfolding billiard trajectories in polygons.
  7. Periodic trajectories: variational approach. Birkhoff's theorem. Introduction to Morse theory. Non-convex and polygonal billiards: open problems. Two-periodic billiard trajectories and binormals.
  8. Mirror equation. Non-existence of caustics, Mather's theorem. Existence of caustics and KAM theory. Birkhoff's conjecture and Bialy's theorem.
  9. Outer (or dual) billiards. Motivation via projective duality. Area preserving property and area construction. Behavior at infinity. Rational and quasi-rational polygons. Example: regular pentagon. Outer billiard in the hyperbolic plane; Poncelet porism revisited.
  10. Complete integrability of the billiard map inside the ellipsoid and the geodesic flow on the ellipsoid.