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 2006:

## Schedule

*Finite Fields and their Applications*

Instructor: Gary Mullen, Professor of Mathematics

Teaching Assistant: Van Cyr

113 MCALLISTER BUILDING, MWRF 10:10-11:00*Aspects of Symmetry: from Representations to Quantum Field Theory*

Instructor: Adrian Ocneanu, Professor of Mathematics

Teaching Assistant: Daniele Signori

113 MCALLISTER BUILDING, MWRF 11:15-12:05*Mathematical Theory of Waves*

Instructor: Alberto Bressan, Professor of Mathematics

Teaching Assistant: Andriy Gogolev

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

Finite Fields and their Applications

Instructor: Gary Mullen, Professor of Mathematics

MWRF - 10:10-11:00am

This course will consist of an introduction to the theory of finite fields. We will also discuss some of the many practical applications of finite fields. These applications will include algebraic coding theory for the error-free transmission of information, and cryptology for the secure transmission of information. Finite fields are also of great use in the construction of various kinds of combinatorial designs.

Typical Reading: lecture notes provided by instructor

Math 497C - Honors MASS Geometry

Aspects of symmetry: from representations to Quantum Field Theory

Instructor: Adrian Ocneanu, Professor of Mathematics

MWRF - 11:15-12:05pm

We shall describe the structure of group, illustrated by the finite rotation groups in 3 dimensions. We shall construct concretely the representations of these groups and their tensor product decompositions. With the coefficients obtained this way we shall construct topological quantum field theories and obtain knot and manifold invariants. The structure and representations of the 3 dimensional rotation group SO(3) and of its cover SU(2) will also be discussed. We shall describe these groups in several concrete ways, several of which use the departmental sculpture.

Prerequisites: We start from scratch, but a working knowledge of linear algebra is recommended.

Math 497B - Honors MASS Analysis

Mathematical theory of waves

Instructor: Alberto Bressan, Professor of Mathematics

MWRF - 1:25-2:15 pm

The first part of the course will be concerned with linear waves: sound waves and vibrations in elastic strings. We shall derive a partial differential equation describing wave motion, and study various properties of solutions; in particular, the superposition principle and the speed of propagation of disturbances. The second part of the course will be devoted to non-linear wave motion. In this case, the wave profiles can change in time, eventually leading to the formation of shocks. Although shock waves are described by discontinuous functions, they can still be described in terms of a P.D.E, taking the form of a conservation law. Various models and applications to gas dynamics and to traffic flow will be discussed.

Basic reference: Roger Knobel, An Introduction to the Mathematical Theory of Waves. AMS 2000

Prerequisites: basic linear algebra, calculus in several variables.

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 | |

MASS Welcome Party & Orientation | September 5, 10:10-12:05 |

Classes Begin | September 5 |

Study Day (No classes) | October 6 |

Midterm Exams | October 16-18 |

Thanksgiving Holiday | November 22-24 |

Classes End | December 8 |

Study Period | December 11-14 |

Final Exams | December 15-20 |

MASS Graduation Ceremony | December 21 |

## 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.