Department of Mathematics of the Pennsylvania State University runs a yearly semester-long intensive program for undergraduate students seriously interested in pursuing a career in mathematical sciences. The *Mathematics Advanced Study Semesters* (MASS) program started in the Fall of 1996 and is held during the Fall semester of each year.

The principal part of the program consists of *three core courses* chosen from major areas in *Algebra/Number Theory*, *Analysis*, and *Geometry/Topology* respectively, specially designed and offered exclusively to MASS participants, and a weekly *MASS seminar*.

Additional features include *colloquium-type lectures* by visiting and resident mathematicians and mathematical research projects.

The following courses will be offered in the Fall of 2005:

## Schedule

*Integer Partitions*

Instructor: George Andrews, Evan Pugh Professor of Mathematics

Teaching Assistant: Uuye Otgonbayar

113 MCALLISTER BUILDING, MWRF 10:10-11:00*P-adic analysis in comparison with real*

Instructor: Svetlana Katok, Professor of Mathematics

Teaching Assistant: Bryce Weaver

113 MCALLISTER BUILDING, MWRF 11:15-12:05*Geometry and Billiards*

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

Teaching Assistant: Gordana Stojanovic

113 MCALLISTER BUILDING, MWRF 1:25-2:15*MASS Seminar*

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

113 MCALLISTER BUILDING, Tuesday 10:10-11:00*MASS Colloquium*

Instructor: Multiple invited speakers

113 MCALLISTER BUILDING, Thursday 2:30-3:30

## Course Outline

Math 497A - Honors MASS Algebra

Integer Partitions

Instructor: George E. Andrews, Professor of Mathematics

MWRF - 10:10-11:00am

Course objectives and Honors Objectives: This course will be an introduction to the theory of paritions. This venerable field (its foundation trace back to Euler in the 17th century) has become of interest to computer scientist and physicists as well as to mathematicians interested in number theory and combinatorics. We shall provide an introduction to both the combinatorial and analytic aspects of the subject.

Mode of Instruction: Lectures and recitation periods with homework, written mid-term examination, oral final examination, and project.

Typical Readings: The central text for the course will by Integer Partitions (by Andrews and Eriksson) which has just been published by Cambridge University Press. In addition original papers in the literature will be assigned.

Work Requirements/Evaluation Criteria: Homework, written mid-term examination, oral final examination, and project.

Math 497B - Honors MASS Analysis

P-adic Analysis in Comparison with Real

Instructor: Svetlana Katok, Professor of Mathematics

MWRF - 11:15-12:05 pm

Suggested Prerequisites: Acceptance into MASS program (see above).

Course Objectives: Both real and p-adic numbers are obtained from the rationals by a procedure called completion, which can be applied to any metric space, by using different distances on the rationals: the usual Euclidean distance for the reals and a new p-adic distance for each prime p, for the p-adics. The p-adic distance satisfies the strong triangle inequality

that causes surprising properties of p-adic numbers and leads to interesting deviations from the classical real analysis much like the renunciation of the fifth postulate of Euclid's Elements

, the axiom of parallels, leads to non-Euclidean geometry. Similarities, on the other hand, arise when the fact does not depend on the strong triangle inequality

, and in these cases the same proof works in the real and p-adic cases. Analysis of the differences and similarities will help the students to better understand the proofs in both contexts.

I included several topics from real analysis and elementary topology which are not usually covered in undergraduate courses (totally disconnected spaces and Cantor sets, points of discontinuity of maps and the Baire category theorem, surjectiveness of isometries of compact metric spaces). They will enhance the students' understanding of real analysis and intertwine the real and p-adic contexts of the course.

Typical Readings: p-adic analysis in comparison with real

by Svetlana Katok, in MASS Selecta: Teaching and Learning Advanced Undergraduate Mathematics}, 11-87, AMS, Providence, 2003 (based on a MASS 2000 course)

Mode of Instruction: Lectures, problem solving, projects

Work Requirements/Evaluation Criteria: Homework, written mid-term examination, oral final examination and project.

Math 497C - Honors MASS Geometry

Geometry and billiards

Instructor: Sergei Tabachnikov, MASS Director and Professor of Mathematics

MWRF - 1:25-2:15pm

The course is an introduction to geometry and dynamics of billiards. As a motivation, configuration and phase spaces of mechanical systems with elastic collisions, that can be described as billiards, will be discussed. Topics include: billiard in a rectangle and the dynamics of a circle rotation, optical properties of conics and Poncelet theorem, evolutes and involutes of plane curves and the four-vertex theorem, integral geometry and Hilbert's fourth problem, periodic billiard trajectories and variational principles, billiards in polygons, chaotic billiard dynamics and others.

Prerequisites: acceptance into MASS program

Typical Readings: Billiards

by S. Tabachnikov and lecture notes that will be available prior to the lectures

Work Requirements/Evaluation Criteria: weekly homework, written mid-term examination, oral final examination, and research project.

Mode of Instruction: three weekly lectures and one weekly recitation session with a TA

Math 497D - MASS Interdisciplinary seminar

Instructor: Sergei Tabachnikov

T - 10:10-12:05pm

This seminar is designed to focus on selected interdisciplinary topics in algebra, geometry and analysis to coordinate core courses and to prepare students to MASS Colloquium. Seminar sessions may include presentations from student research projects.

Typical Readings: N/A

Math 497E - MASS Colloquim

Instructor: Multiple visiting speakers

R 2:30-3:30 pm

Covers selected topics in mathematics.

Typical Readings: N/A

## Calendar of Events

Arrival Day | August 27 |

MASS Welcome Party & Orientation | August 30 |

Classes Begin | August 30 |

Labor Day Holiday | September 5 |

Study Day (No classes) | October 14 |

Midterm Exams | October 10, 11, 12 |

Thanksgiving Holiday | November 23-25 |

Classes End | December 2 |

Study Period | December 5-8 |

Final Exams | December 9, 12, 14 |

MASS Graduation Ceremony | December 15 |

## Enrollment

Participants are selected from applicants who will be juniors or seniors in the following academic year (sophomores may be admitted in some cases). All participants are expected to have demonstrated a sustained interest in mathematics and a high level of mathematical ability and to have mastered basic techniques of mathematical proof. The expected background includes a full calculus sequence, basic linear algebra, a transition course with proofs (such as discrete mathematics) and advanced calculus or basic real analysis. The search for participants is nationwide. International applications are invited as well. Each participant is selected based on academic record, two recommendation letters from faculty, and an essay (international applicants should demonstrate their mastery of English).

Candidates should submit:

- Application Form
- Transcript
- Record of Mathematics Courses
- A short essay describing their interest in mathematics
- Two letters of recommendation
- Financial disclosure form
- Transfer Protocol form

Application materials may be retrieved off the web, or requested by mail, fax, or e-mail.

Applications should be submitted through MathPrograms.org, ID: PSUMASS or sent by mail, fax, or e-mail to

107 McAllister Building

Department of Mathematics

Penn State University

University Park, PA 16802

(814) 863-8730 / Fax:(814) 865-3735

E-mail: mass@math.psu.edu

## Financial Arrangements

Successful applicants currently enrolled in U.S. colleges and universities will be awarded the Penn State MASS Fellowship which reduces the tuition to the in-state level. Best efforts will be made not to increase their out of pocked expenses. See the Financial Information for more details.

## Housing

All participants not enrolled at Penn State will be provided an opportunity to live in one of the residence halls on campus.

## Credits

The program elements total 16 credits, all of which are recognized by Penn State as honors credits and are transferable to participants' home universities. Students will also receive a certificate from the MASS Program at Penn State. Additional recognition may be provided through prizes for outstanding performance and for best projects.

## Administration

The overall supervision of the MASS program is provided by the Scientific Advisory Board which includes senior members of Penn State's mathematics faculty, and several outstanding mathematicians from other institutions.

The program is managed by the Director Sergei Tabachnikov.

Stephanie Zerby is the Administrative Assistant for the MASS program.

Participants are chosen by the *Selection Committee* headed by a member of the Scientific Advisory Board.