**21-624 Selected topics: Differential equations, groups and Galois theory**

Rami Grossberg (rami@cmu.edu)

MWF 12:30

HBH 1510

12 Units

The fundamental question: ``Why the integral $\int e^{-x^2} dx$ cannot be expressed in terms of elementary functions?'' lead Picard and Vessiot (in the 19th century) to develop an algebraic theory of ordinary differential equations that is analogous to classical Galois theory. This theory was continued in this century by Ritt and Kolchin. In the sixties the theory was connected to the theory of algebraic groups and number-theoretic questions. My interest in the field is motivated by some recent applications of model theory to arithmetical geometry (by Hrushovski) and the importance of the field to model theorists. The course will be an introduction to the basic classical results.

*Prerequisite:* No knowledge of logic is assumed, it is desirable that the
students have had an introductory course in algebra covering Galois theory
21-610 or 21-374 (field theory).