This book results from a unique and innovative program at Pennsylvania State University. Under the program, the ""best of the best"" students nationwide are chosen to study challenging mathematical areas under the guidance of experienced mathematicians. This program, Mathematics Advanced Study Semesters (MASS), offers an unparalleled opportunity for talented undergraduate students who are serious in the pursuit of mathematical knowledge. This volume represents various aspects of the MASS program over its six-year existence, including core courses, summer courses, students' research, and colloquium talks. The book is most appropriate for college professors of mathematics who work with bright and eager undergraduate and beginning graduate students, for such students who want to expand their mathematical horizons, and for everyone who loves mathematics and wants to learn more interesting and unusual material. The first half of the book contains lecture notes of nonstandard courses. A text for a semester-long course on $p$-adic analysis is centered around contrasts and similarities with its real counterpart.A shorter text focuses on a classical area of interplay between geometry, algebra and number theory (continued fractions, hyperbolic geometry and quadratic forms). Also provided are detailed descriptions of two innovative courses, one on geometry and the other on classical mechanics. These notes constitute what one may call the skeleton of a course, leaving the instructor ample room for innovation and improvisation. The second half of the book contains a large collection of essays on a broad spectrum of exciting topics from Hilbert's Fourth Problem to geometric inequalities and minimal surfaces, from mathematical billiards to fractals and tilings, from unprovable theorems to the classification of finite simple groups and lexicographic codes.
A brief description of MASS program by S. Katok and S. Tabachnikov Teaching in the MASS program by G. E. Andrews Lecture notes: $p$-adic analysis in comparison with real by S. Katok Geometrical methods of mechanics by M. Levi Geometric structures, symmetry and elements of Lie groups by A. Katok Continued fractions, hyperbolic geometry and quadratic forms by S. Katok MASS colloquium: MASS colloquium by S. Tabachnikov Hilbert's fourth problem in two dimensions by J. C. Alvarez Paiva Integral lexicographic codes by J. Conway The classification of finite simple groups by E. Formanek Billiard balls count $\pi$ by G. Galperin Rep-tiles revisited by V. Nitica Fractals and dynamics by Y. Pesin Unprovable theorems and fast-growing functions by S. G. Simpson Minimal surfaces and random walks by A. B. Sossinsky The tale of a geometric inequality by S. Tabachnikov Student research papers: Summer REU program at Penn State by M. Guysinsky Partitions of $n$ and connected triangles by S. Chuba Triangles gone wild by J. Kantor and M. Maydanskiy Determinacy of games by A. Medvedev On the nonexistence of odd perfect numbers by J. Voight Appendices: MASS courses and instructors by S. Katok, A. Sossinsky, and S. Tabachnikov MASS colloquia by S. Katok, A. Sossinsky, and S. Tabachnikov MASS participants by S. Katok, A. Sossinsky, and S. Tabachnikov.