Math 711 — The Arithmetic of Dynamical Systems — Winter 2010

Instructor: Michael Zieve    email:
Office: 3835 East Hall Office Hours: 2:30–4 (T, Th)  or by appointment
Phone: 615-3650 Class: T, Th 1–2:30 in 3088 East Hall

This course provides an introduction to a recently developed area of mathematics combining dynamical systems and number theory. It addresses number-theoretic aspects of orbits of polynomials (such as integrality and rationality questions), dynamics over local and global fields, dynamical generalizations of number theoretic results (such as the Mordell conjecture and the uniform boundedness conjecture), and dynamics of endomorphisms of algebraic groups and associated maps.

Much of this material is in the book "The Arithmetic of Dynamical Systems" by J.H.Silverman (Springer, 2007), which is recommended but not required. This course will also include more recent (published and unpublished) results.

The main prerequisite is fluency in the language of groups, rings, fields, and Galois theory, at the level of Math 593/594. We will also use basic concepts from algebraic number theory (rings of integers, units, valuations, completions, ramification), but students can learn these during the course.

Related activities January–June 2010:

Special session on arithmetic dynamics at the AMS/MAA Annual Meeting in San Francisco in January.

This year's Arizona Winter School in Number Theory (March 13–17) focuses on Number Theory and Dynamics. This annual conference series includes lectures aimed at graduate students, but more importantly includes projects which groups of students work on with the speakers. I highly recommend them to students interested in number theory.

In this year's Michigan Lectures in Number Theory (April 19–21),  Matt Baker explains applications of potential theory on the Berkovich projective line to problems in arithmetic dynamics.

The Bellairs Workshop in Number Theory (May 2–9) in Barbados addresses moduli spaces in arithmetic dynamics.

Laura DeMarco is organizing a conference on the arithmetic of complex dynamics at the University of Illinois at Chicago, May 17–21.

There is an algebraic dynamics workshop at the CUNY Graduate Center, June 7–11.

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