In Summer 2004 the Mathematics Department of the Pennsylvania State University hosted a Research Experiences for Undergraduates site. Eleven participants were selected from qualified applicants.

The program ran for 7 weeks from **June 28 to August 13, 2004**. It combined learning with research and included:

- one
**six-week course**,*Mathematical Billiards*

Instructor: Sergei Tabachnikov, Professor of Mathematics, Director of MASS Program

(Billiards by S. Tabachnikov in PDF, about 700K) **Weekly seminar**run by the program coordinator**Research projects**- Chris Culter (Submitted for publication),

Dual billiards for polygons always have periodic orbits - Van Cyr and Mit'ka Vaintrob,

Bicycle curves part 1 (by Van Cyr)

Bicycle curves part 2 (by Mitka Vaintrob) - Chris Biermann and Stas Sheynkop,

The Benford's Law and random iterations of 2x² and 3x² (by Chris Biermann)

The Benford's Law and areas of countries (by Stas Sheynkop) - Rob King, Haijian Lin, Sarah Mall, and Greg McNulty

Lower bounds on mean curvature of closed curves contained in convex boundaries - Sarah Mall,

Lower bounds on mean curvature of closed curves contained inL

shape - James Krysiak and Zachary McCoy,

Alexandrov's conjecture: On the intrinsic diameter and surface area of Convex surfaces - Casey Bush and Toan Phan,

Bending fields, rigidity and bellows conjecture

- Chris Culter (Submitted for publication),

## Courses

#### MATHEMATICAL BILLIARDS

June 28 - August 13

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

INSTRUCTOR: SERGEI TABACHNIKOV

MAIN TOPICS include:

- 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.
- 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.
- 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.
- 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.
- 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.
- Poincare recurrence theorem. Applications: periodic trajectories in rational polygons and in right triangles. Unfolding billiard trajectories in polygons.
- Periodic trajectories: variational approach. Birkhoff's theorem. Introduction to Morse theory. Non-convex and polygonal billiards: open problems. Two-periodic billiard trajectories and binormals.
- Mirror equation. Non-existence of caustics, Mather's theorem. Existence of caustics and KAM theory. Birkhoff's conjecture and Bialy's theorem.
- 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.
- Complete integrability of the billiard map inside the ellipsoid and the geodesic flow on the ellipsoid.