COURSE DESCRIPTION: Gauss
said "Mathematics is the queen of the sciences and number theory the queen of
mathematics"; in this class we shall meet some of her subjects. We will discuss
many of the most important questions in analytic and additive number theory,
with an emphasis on techniques and open problems. Topics will range from
Goldbach's Problem and the Circle Method to the Riemann Zeta Function and Random
Matrix Theory. Other topics will be chosen by student interest, coming from sum
and difference sets, Poissonian behavior, Benford's law, the dynamics of the
3x+1 map as well as suggestions from the class. We will occasionally assume some
advanced results for our investigations, though we will always try to supply
heuristics and motivate the material. No number theory background is assumed,
and we will discuss whatever material we need from probability, statistics or
Fourier analysis.
NOTE: this course
can be taken either as a 300 level or as a 400 level tutorial. All students will
be required to give a presentation to the rest of the class, review and
summarize a paper on the math arxiv, and write a short note on something from
the course.
Format:
Tutorial. Evaluation will be based primarily on homework, presentations and
write-ups, and scholarship.
Prerequisites:
For those taking 308T: at least one of 301/305/312/315/317; for those taking
406T: one of 301/305 AND one of 12/315/317
CONTACTING ME: You can reach me in Bronfman 202 (if I'm there it's office hours), email sjm1@williams.edu, or anonymously through mathephs@gmail.com (password 11235813, the first seven Fibonacci numbers).
OBJECTIVES: The goal is to introduce students to advanced concepts and problems in analytic number theory, and to practice skills that are useful for graduate students and professional mathematicians. This includes reading high level mathematics and summarizing it, checking it and presenting it. For a fuller statement as to the objectives of this course, please click here. This includes some fascinating videos with some thought provoking comments about what you should get out of your education.
TEXTBOOK: The textbook will be Miller and Takloo-Bighash’s `An Invitation to Modern Number Theory’ (errata for the book is here).
COURSE NOTES: click here for course notes from when I taught the class as a senior seminar in 2009.
COURSE DISCLAIMER: I may occasionally say things such as `Probability is one of the most useful courses you can take' or 'If you know probability, stats and a programming language then you'll always be able to find employment'. I really should write `you should always be able to find employment', as nothing is certain. Thus, please consider yourself warned and while you may savor the thought of suing me and/or Williams College, be advised against this! I'm saying this because of the recent lawsuit of a graduate who was upset that she didn't have a job, and sued her school!