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 2008:

## Schedule

*Elliptic Curves and Applications to Cryptography*

Instructor: Kirsten Eisentraeger, Assistant Professor of Mathematics

Teaching Assistant: Van Cyr

113 McAllister Building, MWRF 1:25-2:15pm*Elements of Fractal Geometry and Dynamics*

Instructor: Yakov Pesin, Distinguished Professor of Mathematics

Teaching Assistant: Vaughan Climenhaga

113 McAllister Building, MWRF 10:10-11:00am*Introduction to Symplectic Geometry*

Instructor: Kris Wysocki, Associate Professor of Mathematics

Teaching Assistant: Vivek Srikrishnan

113 McAllister Building, MWRF 11:15-12:05pm*MASS Seminar*

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

113 McALLISTER BUILDING, Tuesday 10:10-12:05*MASS Colloquium*

Instructor: Multiple invited speakers

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

## Course Outline

Math 497A - Honors MASS Algebra

Elliptic Curves and Applications to Cryptography

Instructor: Kirsten Eisentraeger, Assistant Professor of Mathematics

MWRF - 1:25-2:15pm

**Description**: The study of diophantine equations is an area of number theory that deals with finding solutions to polynomial equations. Looking for solutions of equations in integers or rationalnumbers has a long history that goes back to ancient Greece. In this class we will focus on elliptic curves, a special class of diophantine equations given by certain cubic equations in two variables. We will study these equations through a combination of techniques from number theory and algebraic geometry.

We will cover the group law for elliptic curves, both in terms of the geometry of the curve and in terms of explicit equations. Then we will discuss points of finite order and isogenies. We will also study elliptic curves over the rationals and prove the Mordell-Weil theorem, which says that the group of rational points on an elliptic curve is finitely generated.

After that, we will talk about elliptic curves defined over finite fields. Elliptic curves over finite fields have many applications to cryptography, and we will discuss their use in discrete-log based cryptosystems and in applications of the Weil and Tate pairings.

**Readings**: Silverman-Tate, *Rational points on elliptic curves* or Silverman, *The Arithmetic of elliptic curves*

Math 497B - Honors MASS Analysis

Elements of Fractal Geometry and Dynamics

Instructor: Yakov Pesin, Distinguished Professor of Mathematics

MWRF – 10:10-11:00am

**Description**: Fractals are strange but beautiful objects that appear in nature and arts as results of self-organization and self-similarity. In dynamics they are responsible

for the presence of highly-irregular, chaotic motions. The course is an introduction to a circle of topics in fractal geometry and chaotic dynamics.

**Readings**: K. Falconer, *Fractal Geometry, Mathematical Foundations and Applications*, John Wiley & Sons, 1990

M. Schroeder, *Fractals, Chaos, Power Laws*, W. H. Freeman & Company, 1991

Math 497C - Honors MASS Geometry

Introduction to Symplectic Geometry

Instructor: Kris Wysocki

MWRF – 11:15-12:05pm

**Description**: This course is intended to give an introduction to the field of symplectic topology. Symplectic topology has its roots in the study of classical mechanics and these days plays an important role in many areas of modern mathematics. The course will begin with linear symplectic geometry and Hamiltonian flows. After that we will define symplectic invariants and discuss some examples of symplectic invariants in details, then will proceed to study the existence of periodic orbits of Hamiltonian vector fields.

**Readings**: Chapters from the book: *Hamiltonian Dynamics and Symplectic Invariants* by H. Hofer and E. Zehnder

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 23-24 |

MASS Welcome Party & Orientation | August 26 |

Classes Begin | August 25 |

Labor Day — No Classes | September 1 |

Midterm Exams | October 6 - 8 |

Thanksgiving Holiday — No Classes | November 24-28 |

Classes End | December 5 |

Final Exams | December 12, 15, 17 |

MASS Graduation Ceremony | December 18 |

## 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.