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MASS Selecta: Teaching and Learning Advanced Mathematics

Svetlana Katok, Alexei Sossinsky, and Serge Tabachnikov, editors
American Mathematical Society
Publication Date: 
Number of Pages: 
[Reviewed by
Mihaela Poplicher
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This is an unique book that describes and includes materials from an unique program in the United States: the MASS (Mathematics Advanced Study Semesters) program at Penn State. MASS was started seven years ago, and has been very successful with bright and eager undergraduate and beginning graduate students.

The book includes a detailed description of MASS and the related REU (Research Experience for Undergraduates) program: core courses, lecture notes for a few courses, colloquium talks, summer courses, students' research. The REU program is, of course, not unique in the US, but its connection with the MASS program enriches it greatly and makes it very successful.

The MASS program started in 1996 and is funded by Penn State and NSF (through a VIGRE grant.) The number of MASS participants varies from year to year, with an average of 15 students each year. These students are selected from applicants currently enrolled in US colleges and universities, mostly juniors or seniors, who "have demonstrated a sustained interest in mathematics and a high level of mathematical ability."

Each year the MASS program has as its main elements three core courses on topics from Analysis, Algebra/Number Theory, and Geometry/Topology, a seminar, a lecture series, and student research projects. The final exams are oral and have a format described as a "creative development of certain European traditions." The weekly 2-hour interdisciplinary seminar is run by the director of the MASS program; its goal is to unify all the activities in the program. A weekly lecture series includes talks either by distinguished mathematicians from Penn State or by visitors. Most of the student research projects are related to the core courses.

The book includes lecture notes from several MASS courses: "p-adic Analysis in Comparison with Real" (by Svetlana Katok); "Geometric Methods of Mechanics" (by Mark Levi); "Geometric Structures, Symmetry and Elements of Lie Groups" (by Anatole Katok); "Continued Fractions, Hyperbolic Geometry and Quadratic Forms" (by Svetlana Katok). All these are very interesting and very different from regular college courses.

Also included are a list of all the colloquium titles (Appendix 2) and a chapter with many of the talks. The book includes five of the student research projects in a special chapter.

The MASS program is very intense and encourages interactions among students, teaching assistants and mathematicians. "MASS participants are literally immersed into mathematical studies." As a practical matter it should be noted that all the courses, the seminar and the colloquium from the MASS program (16 credit hours) are recognized as Honors classes and are transferable to MASS participants' home universities.

Although REU programs exist at other schools, at Penn State Summer REU program is viewed as an extension of the MASS program. The REU program has the following components: Individual/small group projects supervised by faculty, two short courses, a weekly seminar, and MASS Fest, a 3-day conference at the end of REU at which the participants present their research projects. . Appendix 1 of the book contains a list with the REU courses.

This book describes the MASS program in detail, and as such could be used by any department/university interested in developing a similar program. The lecture notes and the colloquium talks are a great source for any college instructor interested in developing honors courses or in challenging bright students. The undergraduate students eager to study interesting and challenging mathematical (research) problems, as well as the beginning graduate students who want to expand their mathematical horizons will also find a wonderful resource in this book.

Mihaela Poplicher is assistant professor of mathematics at the University of Cincinnati. Her research interests include functional analysis, harmonic analysis, and complex analysis. She is also interested in the teaching of mathematics. Her email address is

  • 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. Nitica -- Rep-tiles revisited
  • Y. Pesin -- Fractals and dynamics
  • S. G. Simpson -- Unprovable theorems and fast-growing 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


  • 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