#### 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