M 390 C Differential Algebra

CLASS HOURS: Tue, Thu 11:00 -- 12:30 RLM 11.176

OFFICE HOURS: Tue, 9:00 -- 11:00 or by appointment.

TEXTBOOK: A. Magid, Lectures on Differential Galois Theory, AMS.

SYLLABUS: Ever wondered why is not an "elementary function"? This is a theorem of Liouville and we will see a proof of this theorem halfway through this course. In general, linear ordinary differential equations (such as y"+2xy'=0) have a "differential Galois group" and the solvability of this group, in the usual group-theoretic sense, is equivalent to the "solvability" of the equation in terms of nice functions in a sense that will be made precise in the course. The course will consist of the development of this Differential Galois Theory with some applications. Time permitting, we will talk about related topics such as monodromy groups and the Riemann-Hilbert problem, Buium's approach to Diophantine Geometry via Differential Algebra or Computer Algebra methods of integration in finite terms. Contrary to what we teach in Calculus, indefinite integration can be approached algorithmically and does not require tricks.

Some links:

• A survey by Magid in the Notices of the AMS.
• Review of Magid's book by D. Bertrand.

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