Interested undergraduates may enroll in MATH 499, section C, reference number 16780. This 1 credit course will count toward the 15 hours of upper level math required for the B.S. in the College of Arts and Sciences with major in mathematics.

Concerning the background required, Professor Edward Burger writes: "Any background at all in abstract algebra and real analysis would be terrific but less background is fine too if the student has taken at least one upper level course and/or is mathematically enthusiastic. If someone is interested and enthusiastic, I would love to have them, even if their background is modest. We'll be starting from scratch anyway."

Course Description: Almost your entire mathematical life has been spent on the real line and in real space working with real numbers. Some have dipped into complex numbers, which are just the real numbers after you throw in i. Are these the only numbers that can be built from the rationals? The answer is no. There are entire parallel universes of number that are totally unrelated to the real and complex numbers. Welcome to the world of p-adic analysis--where arithmetic replaces the tape measure and numbers take on a whole new look. Here we will explore this new notion of number and discover its impact on arithmetic, geometry, and calculus. It turns out that p-adic analysis not only dramatically simplifies many mathematical areas but also provides a powerful tool for analyzing number theoretic issues.

The text will be Exploring the Number Jungle: A Journey into Diophantine Analysis by Edward B. Burger, American Mathematical Society, 2000.

Homework 0: (From Exploring the Number Jungle). Read pp. 1-5. Read Modules 1 and 2 and see how many of the results stated there you can prove. Write up any proofs you find.

First Lecture Title: "How to Always Win at Limbo" or "You can sum some of the series some of the time, and some of the series none of the time... but can you sum some of the series ALL of the time?"

First Lecture Abstract: Have you ever gone out with someone for a while and asked yourself: "How close are we?" This presentation will answer that question by answering: What does it mean for two things to be close to one another? We'll take a strange look at infinite series, dare to mention a calculus student's fantasy, and do some brief transcendental meditation. In fact, we'll even attempt to build some very unusual and exotic series that can be used if you ever have to flee the country in a hurry: we'll either succeed or fail... you'll have to come to the talk to find out which. Will you be at the edge of your seats? Perhaps; but if not, then you'll probably fall asleep and either way, after the talk, you'll feel refreshed and great. No matter what, you'll learn a sneaky way to always win at Limbo.