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 2004:

## Schedule

*Finite Fields and Applications*

Instructor: Gary Mullen, Professor of Mathematics

Teaching Assistant: Carl Mummert

124 POND BUILDING, MWRF 10:10-11:00*Introduction to Dynamical Systems*

Instructor: Anatole Katok, Raymond N. Shibley Professor of Mathematics

Teaching Assistant: Travis Fisher

124 POND BUILDING, MWRF 11:15-12:05*Differential Geometry and Topology of Curves and Surfaces*

Instructor: Mohammad Ghomi, Visiting Associate Professor of Mathematics

Teaching Assistant: Robert Yuncken

124 POND BUILDING, MWRF 1:25-2:15*MASS Seminar*

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

124 POND BUILDING, Tuesday 10:10-11:00*MASS Colloquium*

Instructor: Multiple invited speakers

122 POND BUILDING, Thursday 3:35-4:25

## Course Outline

Math 497A - Honors MASS Algebra

Finite Fields and 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: material will come from readings and books such as Introduction to Finite Fields and Their Applications

by R. Lidl and H. Niederreiter, Revised ed., Cambridge, 1994.

Math 497B - Honors MASS Analysis

Introduction to Dynamical Systems

INSTRUCTOR: Anatole Katok, Raymond N. Shibley Professor of Mathematics

MWRF 11:15-12:05 pm

The theory of dynamical systems is a major discipline closely intertwined with most major areas of modern mathematics. It has greatly stimulated research in many areas of science and given rise to a vast new interdisciplinary area known as applied dynamics, nonlinear science or chaos theory. In this course we will combine rigorous mathematical development of the main concepts of modern theory of dynamical systems with motivations coming from both inside and outside mathematics and with copious examples of important applications. The only prerequisites beyond calculus and elementary linear algebra is a basic undergraduate real analysis course. We will discuss as a motivation a variety of scientific and mathematical questions that dynamics can address. Then we will build a number of key examples and present concepts and tools for describing asymptotic behavior in dynamical systems gradually increasing a level of complexity. A variety of more advanced topics including logistic maps, strange attractors, twist maps, closed geodesics and applications to number theory will be available as topics for projects and research.

Typical Readings: A first course in dynamics with a panorama of recent developments by B. Hasselblatt and A. Katok, Cambridge University Press, 2003. ISBN: 0521587506 (in part, based on a MASS 1996 course, the first year of the program's existence)

Math 497C - Honors MASS Geometry

Differential Geometry and topology of curves and surfaces

INSTRUCTOR: Mohammad Ghomi, Visiting associate Professor of Mathematics

MWRF 1:25-2:15pm

This class will be a motivated introduction to modern differential geometry and topology which focuses on the classical objects in this area: curves and surfaces in Euclidean space. Many specific examples and a host of open problems will be discussed at each stage. There will also be some exercises and experiments using computers. The readings will consist of lecture notes written by the instructor, as well as excerpts from a variety of texts including those by M. do Carmo (Differential Geometry of Curves and Surfaces) and A. Gray (Modern Differential Geometry of Curves and surfaces with Mathematica). While curves and surfaces will be treated as the central objects of study, we will also lay the foundations for the study of abstract Riemannian manifolds.

Typical Readings: Lecture notes by Instructor, excerpts from a variety of texts including those by M. do Carmo Differential Geometry of Curves and Surfaces

and A. Gray Modern Differential Geometry of Curves and Surfaces with Mathematica

.

Math 497D - MASS Interdisciplinary seminar

INSTRUCTOR: Sergei Tabachnikov, MASS Director and professor of mathematics

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: Sergei Tabachnikov, MASS Director and Professor of Mathematics

R 3:00-4:00 pm

Covers selected topics in mathematics.

Typical Readings: N/A

## Calendar of Events

Arrival Day | August 28 |

MASS Welcome Party & Orientation | August 30 |

Classes Begin | August 31 |

Labor Day Holiday | September 6 |

Study Day (No classes) | October 15 |

Midterm Exams | October 11, 12, 13 |

Thanksgiving Holiday | November 24-26 |

Classes End | December 3 |

Study Period | December 4-9 |

Final Exams | December 10, 13, 15 |

MASS Graduation Ceremony | December 16 |