Course
title:
Math 21-499: Introduction to Undergraduate Research
Instructor:
Steven J. Miller (sjm1@williams.edu)
General Reading on L-functions: I am not
sure we will do a project here, as these require significantly higher
backgrounds, but will include in case some member of the class are
exceptionally well-prepared. These projects require analytic number theory, a
little complex analysis and some combinatorics. Some of the projects include
the Ratios Conjecture for various families, determining lower order terms in
n-level densities, and determining optimal test functions for various
symmetric spaces and n-level densities.
- General Theory
- Hayes: The
Spectrum of Riemannium: a light description of the connection between
random matrix theory and number theory (there are a few minor errors in the
presentation, basically to simplify the story). This is a quick read, and
gives some of the history.
- Conrey: L-functions and
Random Matrix Theory: This is a high level description of the
similarities between number theory and random matrix theory.
- Katz-Sarnak:
Zeros of Zeta Functions and Symmetry: Another high level article similar
to the others.
- Diaconis:
Patterns in Eigenvalues: this is a bit more readable than the others,
and is based on a distinguished lecture he delivered.
- Miller and Takloo-Bighash:
An Invitation to
Modern Number Theory: This is the textbook I and a colleague wrote,
based on years of supervising undergraduate research classes. I know several
of you already have a copy -- it will be a good resource for the summer, as
a lot of the background material we need is readily available here.
Particularly important chapters for us are:
-
chapter15
(which discusses the connections between random matrix theory and number
theory, and is available online);
- chapter 18 (which does the explicit formula for Dirichlet
characters);
-
chapter 3
(which reviews L-functions and is also online).
- Firk and Miller:
Nuclei, primes and the Random Matrix connection: a survey paper on the
history of the subject, including both the nuclear physics experiments and
the theoretical calculations. Here is another survey with
Owen Barrett, Frank W. K. Firk and Caroline Turnage-Butterbaugh.
- L-function Papers
- Iwaniec-Luo-Sarnak:
Low lying zeros of families of L-functions: This is a must read.
This is the first major paper calculating the 1-level density for families
of L-functions. We will not need much of the paper, but you must read
section 1, skim the beginning of section 2 (section 2 is devoted to
developing good averaging formulas for specific families; as we will be
looking at different families, we won't need this), skim section 3 (same
comments apply), section 4 (very important: here they prove the
explicit formula relating sums over zeros to sums over primes, and while we
will use the explicit formula for different families, the calculation will
follow similarly), section 5 (just skim, same caveats as before).
- Conrey-Snaith:
Applications of
the L-functions Ratios Conjecture: This is a very recent conjecture
which is enjoying remarkable success in predicting answers. There is a lot
I'll say about this during the summer. I somewhat jokingly call it the
conjecture of the four lies, as there are five steps and four of the steps
are provably wrong (ie, the assumptions in those steps fail); however,
miraculously, all the errors seem to cancel to phenomenal level! I've become
very interested in testing this conjecture as much as possible, and have
written several papers in this (and have ideas for a few more which will be
very accessible to us). Read through page 17.
- Rubinstein: Low
lying zeros of L-functions and Random Matrix Theory: this is his
dissertation, and in it he analyzes the 1-level density of the family of
quadratic Dirichlet characters, and shows agreement with Random Matrix
Theory. This is one of the easiest families to look at, and a great testing
ground.
- Hughes - Miller: Low-lying
zeros of L-functions with orthogonal symmetry: this paper finds a more
tractable version of the Katz-Sarnak determinantal expansion for the n-level
density, but for restricted support.
- Miller:
A
symplectic test of the L-functions Ratios Conjecture: This paper builds
on those by Conrey-Snaith and Rubinstein and uses the Ratios Conjecture to
predict the lower order terms up to square-root cancellation, and then shows
(for suitable test functions) that this is the correct answer. An obvious
project is to generalize this test for other families or to enlarge the
support.
- L-function Talks