Sample Abstract
This talk centers around the deceptively innocent problem
of determining what happens when we repeatedly apply a polynomial
function, such as f(x) = x2 + c, to the rational numbers. This
question can be answered in part by the classical theory of
dynamical systems, in which we consider repeated application of
a polynomial function to the complex numbers and represent the
dynamics through pictures of the complex plane. However, to
unravel more deeply the problem we must also resort to number-
theoretic techniques which make use of the special properties of
rational numbers and integers, such as factorization into primes.
To appreciate the talk, you need only an undergraduate background: arithmetic in the complex plane, properties of logarithms, prime numbers, and so forth. If you know something about the p-adic numbers, you will fathom more of the rich structure underlying heights. If you don't know something about the p-adic numbers, after this talk you might.
The above abstract was for a graduate-level talk
Description
Among my expository lectures, this talk is exceptional
in that it concerns an area of my own
mathematical research. As a consequence, it is much more variable
in title and content than the talks above.
Suppose f is a polynomial with rational coefficients. Repeated application of f to the rational numbers produces a discrete dynamical system. To study the properties of this dynamical system, we bring to bear both classical complex dynamics and the number-theoretic properties of rational numbers.
For a sufficiently sophisticated audience, I describe the use of p-adic dynamics in the study of dynamical systems over Q. I can also formulate this talk as a motivation for constructing the p-adic numbers and studying p-adic analysis.
Needless to say, I also give research talks on this subject for more advanced audiences.
Level
Undergraduate or Graduate.
For elementary versions of this talk, the listener needs an understanding of the complex plane, of the arithmetic properties of the natural logarithm, and of prime factorization in the integers and rational numbers. I will tailor the talk to suit the level of the audience.
Mechanics
The complex dynamics portion of the talk is usually on transparencies,
and the number theory is on a blackboard.
Appearances
Printed Flier for Radical Pi Talk
