Title: Quantum unique ergodicity and number theory.
Abstract: A fundamental problem in the area of quantum chaos is to understand the distribution of high eigenvalue eigenfunctions of the Laplacian on certain Riemannian manifolds. A particular case which is of interest to number theorists concerns hyperbolic surfaces rising as a quotient of the upper half-plane by a discrete "arithmetic" subgroup of SL_2(R) (for example, SL_2(Z), and in this case the corresponding eigenfunctions are called Maass cusp forms). In this case, Rudnick and Sarnak have conjectured that the high energy eigenfunctions become equi-distributed. I will discuss some recent progress which has led to a resolution of this conjecture, and also on a holomorphic analog for classical modular forms.
About The Speaker: Kannan Soundararajan (Sound) is currently a professor at Stanford University. He obtained his doctoral degree at Princeton University. Sound was the 2003 Salem Prize winner. His other honors show a
propensity toward being the original winner for importnat prizes. As an undergraduate, Sound won the first Morgan prize awarded annually by the AMS, MAA and SIAM for outstanding research by an undergraduate. In 2005, Sound was awarded the first SASTRA Ramanujan Prize, a $10,000 prize awarded annually for a young mathematician doing outstanding work related to Ramanujan's interests.
Fall 2009:
The seminars are held on Tuesdays
from 3:30 to 4:30
in room LeConte 303B.
If you would like to speak, please contact Samuel Gross at ssgross@math.sc.edu. Open dates are highlighted in the schedule below.
Title: Analytic properties of Shintani zeta functions. Abstract: The Shintani zeta function is a Dirichlet series which counts cubic rings, or equivalently, orbits of SL_2(Z) on the lattice of integral binary cubic forms.
Shintani proved that this zeta function has an analytic continuation with a functional equation, without any Euler product or simple representation in terms of Euler products. In this talk, we will examine the Shintani zeta function from an analytic point of view. In particular we will ask questions about the distribution of its zeroes. The talk will concentrate more on open questions and possible approaches than proved results. Where applicable, we will also consider these questions in a more general setting.
Title: Arithmetic of the 13-regular partition function modulo 3. Abstract: A k-regular partition is a partition where none of the parts are divisible by k. In this talk, we will present some new results regarding congruences relating values of the 13-regular partition function modulo 3. In particular, we identify an infinite family of non-nested arithmetic progressions modulo arbitrary powers of 3 such that all values are divisible by 3, confirming a conjecture from a 2008
seven-author paper. The proof relies on modular forms, so much of the
talk will be an introduction to properties of modular forms and certain standard operators. The talk should be accessible to everyone.
Title: Part II of "From Gauss to Hooley, a look at the class number of indefinite binary quadratic forms." Abstract: Hudson will continue his talk from last week, October 27.
Title: From Gauss to Hooley, a look at the class number of indefinite binary quadratic forms. Abstract: In this talk, I will introduce the basic theory of the class number of
binary quadratic forms of discriminant D. Then, I will discuss the basic
principles behind Hooley's conjectures. We will see how modern calcula-
tions and other heuristics such as the Cohen-Lenstra Heuristic support
Hooley. Finally, we will discuss new applications of Hooley's ideas to
indefnite forms of modern discriminant d.
Title: On the intersections of Fibonacci, Pell, and Lucas numbers. Abstract: We show how to compute the intersection of two Lucas sequences of the forms { U_n(P,+/-1) } or { V_n(P,+/-1) } with P in Z, including Fibonacci, Pell,
Lucas, and Lucas-Pell numbers.
We describe how to compute such an intersection and prove that it may be
infinite only for two $V$-sequences when the product of their
discriminants is a perfect square. Our approach relies on solving systems
of homogeneous quadratic Diophantine equations as well as Thue equations.
In particular, we prove that 0, 1, 2, and 5 are the only numbers that are
both Fibonacci and Pell.
Title: Life after Newton Abstract: Imagine yourself in a situation where you have an infinite class of polynomials and you have been able to use Newton polygons to establish that they are all irreducible ... but there are some exceptional cases. Then what? What happens after Newton polygons seemingly should establish irreducibility but they don't? The likely obstacle is that the polynomials in question are reducible, but as we will see showing that this is the case is not necessarily an easy task but one that can often be managed by computations or some combinatorial arguments.
Title: Successive square values of quadratic polynomials. Abstract: If a quadratic polynomial in x assumes square values for a sequence of successive integers x, then those squares will have constant second difference (equal to twice the lead coefficient). Nontrivial sequences of four successive squares are known for quadratics x^2 + bx + c, but it is not known if sequences of five or more do/can exist. Infinitely many sequences of eight successive squares have been shown to exist for differences > 2. Recently, Browkin and Brzezinski have shown that only finitely many "symmetric" sequences exist with even length >= 10. We will discuss these problems and relate them to simultaneous Pell equations, elliptic curves, and real quadratic number fields.
Title: The arithmetic-geometric mean and p-adic limits of modular forms. Abstract: The arithmetic-geometric mean of Gauss is the coincident limit of two sequences of complex numbers which arise naturally from systematically taking arithmetic and geometric means (choosing the appropriate square roots). Gauss proved that these sequences and their limit (the AGM) are parametrizable by values of modular forms.
In this talk, we will exhibit a sequence of modular forms whose p-adic limit parametrizes values of the AGM. To this end, we will review the basic properties of the arithmetic-geometric mean. We will also discuss the p-adic valuation of power series with integer coefficients, and we will define the notion of a modular form and a harmonic weak Maass form.
Title: Connections of the arithmetic-geometric mean of Gauss to number theory.
Abstract: We will give an overview of the arithmetic-geometric mean (AGM) of Gauss and describe classical connections to elliptic integrals, hypergeometric functions, modular forms, and elliptic curves. It is intended that this talk serve as an introduction to these objects.
It turns out that the AGM arises as a limit which parametrizable by theta functions, a certain class of modular forms. Next week's talk will describe recent work (joint with Sharon Garthwaite) on a paramtrization of the AGM as a p-adic limit of modular forms. The p-adic limit arises via the the interplay between classical modular forms and harmonic weak Maass forms. The recent successes connecting harmonic Maass forms to partitions, Ramanujan's mock theta functions, Lie algebras, probability, mathematical physics, and topological invariants motivates independent interest in their study.
Title:You can't do THAT in a seminar. Abstract: We will begin by talking about an iPhone/iPod app that was recently listed as the number one free app on iTunes. It is a version of
a puzzle called "Lights Out". The "THAT" in the title is referring
to related $n$ by $n$ puzzles which literally cannot be done in (as well as not in) a seminar. After a picturesque introduction, the talk will focus on the existence of such impossible puzzles for varying $n$. The talk will be of a recreational nature using some simple algebra, combinatorics and number theory.
Title: Zeros of Eisenstein series Abstract: Eisenstein series are basic building blocks in the theory of modular forms; they also appear in a particular differential equation for the Weierstrass p-function in elliptic curve theory. Here we will focus on topics related to the vanishing of these functions. In particular, we will work through a simple but elegant argument of F.
Rankin and Swinnerton-Dyer about the location of zeros by viewing the Eisenstein series as sums over integer lattices. We will also briefly look at the limitations in extending this argument to more generalized Eisenstein series.