In Summer 2002 the Mathematics Department of the Pennsylvania State University hosted a Research Experiences for Undergraduates site. Support for the site was provided by the National Science Foundation VIGRE grant.

The program ran for 7 weeks from June 17 to August 2, 2002, combined learning with research and included:

- Two
**two-week mini-courses***Partitions and the Omega Package*

Instructor: George Andrews, Evan Pugh Professor of Mathematics*Differential Equations As Dynamical Systems*

Instructor: Yakov Pesin, Professor of Mathematics

**Weekly seminar**was run by the program coordinator**Research projects**

Each REU 2002 participant recieved a stipend of $2,100, reimbursement for room and board, and travel reimbursement up to $500.

### ACADEMIC CALENDAR

Arrival Day | June 15 |

REU Welcome Party & Orientation 4:30 p.m., 212 McAllister Bldg. |
June 16 |

Problem Solving and Weekly Seminars: Misha Guysinski, Director for REU 11:00 a.m., 222 Thomas Bldg. |
June 17 - July 18 |

Mini Course I: Andrews - Partitions and the Omega Package M,W,F, 10:10 a.m. - 12:05 p.m., 223 Thomas Bldg. |
June 24 - July 5 |

Mini Course II: Pesin - Differential Equations as Dynamical Systems T,R, 2:30 p.m. - 3:35 p.m., 222 Thomas Bldg. |
July 9 - July 18 |

Completion of projects and preparation for conference presentations | July 19 - 26 |

MASS Fest | July 27 - 29 |

Dorm Check-out 12:00p.m. |
August 4 |

## Courses

PARTITIONS AND THE OMEGA PACKAGE

June 24 - July 5

TIME: MWF 10:10 am - 12:05 pm

#### INSTRUCTOR: GEORGE ANDREWS

The subject of the course was the elementary theory of partitions. The first part was devoted to the basic use of generating functions in the theory. Euler's discoveries including the Pentagonal Number Theorem were the main focus. The last half was devoted to the Omega package, a software program implemented in Mathematica and designed by Paule, Riese (University of Linz) and me. This package enables the study of rather complicated partition problems thus making possible interesting research projects very soon after learning the basics of the subject.

#### DIFFERENTIAL EQUATIONS AS DYNAMICAL SYSTEMS

July 9 - July 18

TIME: TR 2:30 p.m. - 3:35 p.m.

INSTRUCTOR: YAKOV PESIN

During the course I will present a systematic geometric approach to the theory of non-linear ordinary differential equations. Various types of solutions (fixed points, periodic cycles, etc.) will be considered and the stability theory of these solutions will be discussed. The application to the population biology (Lotka-Voltera models) will be given.

MAIN TOPICS include:

- Geometric description of ordinary differential equations: phase portrait, phase flow, trajectories. Vector fields, rectification theorem.
- Phase flow in the one-dimensional case.
- Vector fields on the plain. Fixed points and periodic cycles.
- Stability of solutions.
- Lotka-Volterra models.
- Lorenz system of differential equations.

TEXTBOOKS:

The primary text-book — K. Alligood, T. Sauer, J. Yorke, Chaos: An Introduction to Dynamical Systems, Springer-Verlag, 1996

The secondary text-book — V.I. Arnold, Ordinary Differential Equations, The MIT Press, 1991