2003; 313 pp; hardcover ISBN10: 0821833634 ISBN13: 9780821833636 List Price: US$69 Member Price: US$55.20 Order Code: MASS
 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 sixyear 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 semesterlong 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. Readership Professors of mathematics; general mathematical audience. Table of Contents  S. Katok and S. Tabachnikov  A brief description of MASS program
 G. E. Andrews  Teaching in the MASS program
Lecture notes  S. Katok  \(p\)adic analysis in comparison with real
 M. Levi  Geometrical methods of mechanics
 A. Katok  Geometric structures, symmetry and elements of Lie groups
 S. Katok  Continued fractions, hyperbolic geometry and quadratic forms
MASS colloquium  S. Tabachnikov  MASS colloquium
 J. C. Álvarez Paiva  Hilbert's fourth problem in two dimensions
 J. Conway  Integral lexicographic codes
 E. Formanek  The classification of finite simple groups
 G. Galperin  Billiard balls count \(\pi\)
 V. Niţică  Reptiles revisited
 Y. Pesin  Fractals and dynamics
 S. G. Simpson  Unprovable theorems and fastgrowing functions
 A. B. Sossinsky  Minimal surfaces and random walks
 S. Tabachnikov  The tale of a geometric inequality
Student research papers  M. Guysinsky  Summer REU program at Penn State
 S. Chuba  Partitions of \(n\) and connected triangles
 J. Kantor and M. Maydanskiy  Triangles gone wild
 A. Medvedev  Determinacy of games
 J. Voight  On the nonexistence of odd perfect numbers
Appendices  S. Katok, A. Sossinsky, and S. Tabachnikov  MASS courses and instructors
 S. Katok, A. Sossinsky, and S. Tabachnikov  MASS colloquia
 S. Katok, A. Sossinsky, and S. Tabachnikov  MASS participants
