MATH 406: Analysis and Number Theory: MWF 9 - 9:50am
Professor Steven Miller (Steven.J.Miller AT williams.edu), 202 Bronfman Science Center (413-597-3293)
Office hours: TBD and by appointment (click here for my schedule)
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.
Format:
lecture/discussion and almost surely presentations. Evaluation will be based on
scholarship, discussions, homework and examinations (and if there is student
interest, papers and presentations in place of some of the exams).
Prerequisites: Multivariable calculus, linear algebra, Math 301 or 305, Math 312
or 315. No
enrollment limit (expected: 21).
SYLLABUS / GENERAL: The textbook will be Miller and Takloo-Bighash’s `An Invitation to Modern Number Theory’. On the first day of class we will describe many of the topics (including Goldbach's Problem and the Circle Method, the Riemann Zeta Function and Random Matrix Theory, Benford's Law, Poissonian behavior, the 3x+1 map and sum and difference sets), and then determine which topics to explore in detail. Please feel free to swing by my office or mention before, in or after class any questions or concerns you have about the course. If you have any suggestions for improvements, ranging from method of presentation to choice of examples, just let me know. If you would prefer to make these suggestions anonymously, you can send email from mathephs@gmail.com (the password is the first seven Fibonacci numbers, 11235813).
OBJECTIVES: There are two main goals to this course: to explore modern number theory, and to learn problem solving skills. We will constantly emphasize the techniques we use to solve problems, as these techniques are applicable to a wide range of problems in the sciences.
Below we provide links for some of the possible topics. The textbook for the course is An Invitation to Modern Number Theory (by S. J. Miller and R. Takloo-Bighash). From there you can click on links to the chapters on Poissonian behavior and on the connections between Number Theory and Random Matrix Theory. The homepage for the book also contains numerous links to papers and survey articles on the subject (see, for example, the paper by Lagarias for more on the 3x+1 map). For sum and difference sets, see the paper by Nathanson (if we choose to discuss this subject, we will almost surely examine the papers by Martin and O'Bryant and Hegarty and Miller); standard references for Benford's law are Hill and Raimi. Common advice to all students is to 'read the masters'; thus, see Hardy and Littlewood for more on the (Hardy-Littlewood) Circle Method. If you have any questions or comments, email me at Steven.J.Miller AT williams.edu
This is my first year at Williams; click here for some personal information about me, my family and my research interests.
For extra credit, find the flaw (or flaws) in any of the following papers (or, to be fair, convince me that they're correct):