Algebra and Number Theory Seminar, Fall 2002

Thursdays, 11:15-12:05, Room 371, Willard.

Link to Spring 2003

September 13       Andrei Zelevinsky (Northeastern University)

Cluster algebras of finite type

I will discuss work in progress with Sergey Fomin.  We study a class of commutative rings called cluster algebras and introduced two years ago as an attempt to design an algebraic framework for the dual canonical bases in quantum groups and their representations.  There is an appropriate notion of cluster algebras of finite type. Their classification turns out to be one more instance of the famous Cartan-Killing classification. I will present the main definitions, give a precise formulation of the classification result, and review the main steps in its proof.

Note that the first seminar has a non-standard venue, 1:25 on Friday 13th September, in Room 102 McAllister.  Andrei Zelevinsky is also speaking in the MASS Colloquium on the 12th.

September 19       Ravi Rao (Tata Institute)

A Witt Group Structure on Orbit Spaces of Unimodular Rows

L.N. Vaserstein initiated  the algebraic study of a group structure on orbit spaces of unimodular rows. Generalized Mennicke  $n$-symbols were studied by Fossum-Foxby-Iversen, and A. Suslin related them with his completion of the ``factorial powered'' unimodular row  $(a_0, a_1, a_2^2, \cdots, a_{n-1}^{n-1})$. W. van der Kallen combined these themes, with existing topological intuition, to get a universal weak Mennicke symbols interepretation of the group structure on orbit spaces of unimodular rows of size bigger than half the Krull dimension. We show that there is also a Witt group structure interpretation, as was shown by L.N. Vaserstein in dimension two. Our key lemma enriches the possibility of the orbit spaces having interesting combinatorial properties.

September 26       Dale Brownawell (Penn State)

Independence of Function Field Gamma Values

October 3              George Andrews (Penn State)

25 Years With Ramanujan's Lost Notebook, Some New Perspectives

Ramanujan's Lost Notebook (rediscovered in 1976) has been a recurrent focus of my research for 25 years.  In it there are approximately 600 formulas stated without proof and each is often unrelated to the one that follows it.  In the past, I have spoken about specific collections of formulas that have been quite challenging.  In this talk I hope to step back a little from the specifics and discuss the impact of the study of the Lost Notebook within number theory.  I shall conclude with some insights concerning the most bizarre formula in the entire notebook (along with a proof that this assertion is indisputable).

October 10             Kannan Soundararajan (University of Michigan)

Real Zeros of L-functions

I will discuss recent work with Conrey which provides infinite families of L-functions having no non-trivial real zeros.  The examples are drawn from quadratic Dirichlet L-functions and L-functions attached to Hecke eigenforms of full level.

October 17              James Sellers (Penn State)

On the Number of Graphical Forest Partitions

October 24              Robert Vaughan (Penn State)

A mean value theorem for cubic fields

Let $r(n)$ denote the number of integral ideals of norm $n$ in a degree $k$ extension $K$ of  the rationals, and define $S(x)=\sum_{n\le x}r(n)$ and $\Delta(x)=S(x)-\alpha x$ where $\alpha$ is the residue of the Dedekind zeta function $\zeta(s,K)$ at $1$.  Ayoub [1958] showed that in the quadratic case the abscissa of convergence of
$$ \int_0^{\infty} \Delta(e^y)^2e^{-2y\sigma} dy$$
is $\frac14$.  We will show that in the cubic case the abscissa of convergence in $1/3$ and that in general the abscissa of convergence is at least $\frac{k-1}{2k}$.

October 31             Josh Lansky (Bucknell University)

New forms for SL(2) and U(1,1)

The theory of new forms, originally developed by Atkin and Lehner in the classical context of cusp forms on the upper half-plane, was reinterpreted in terms of the representation theory of GL(2).  The theory was later extended to GL(n) by Jacquet, Piatetski-Shapiro, and Shalika.  The significance of new forms to the theory of automorphic forms will be discussed in this setting.  We will then present recent extensions of this theory to SL(2) and the quasi-split unramified unitary group U(1,1).

November 7           Emre Alkan (University of Wisconsin)

Estimates on the sizes of gaps in the Fourier expansion of modular forms

I will present short interval results and estimates for the gaps in the Fourier expansion of certain modular forms. In particular we will sharpen earlier results which were obtained by Serre, by the Rankin-Selberg method and by Balog and Ono.

November 14          Hiren Maharaj (Clemson University)

On the construction of asymptotically good towers of function fields over finite fields

Much work has been devoted to the construction of asymptotically good sequences of function fields over finite fields, that is, sequences of function fields over a finite field with  asymptotically many rational places relative to the genus.  The main motivation for such constructions is their usefulness in the construction of sequences of arbitrarily long codes with parameters exceeding or close to  the Gilbert-Varshamov bound.  There are essentially two approaches to construct such sequences of function fields: non-explicit (using class field theory, for example) and explicit (where the function fields are explicitly presented with generators and relations). For applications to coding theory, one requires an explicit presentation. Explicit constructions began in 1995 in a breakthrough paper by Garcia and Stichtenoth.  In this talk, we will give a survey of explicit constructions of function fields. Several open problems will be presented.

SPECIAL SEMINAR Wednesday, 111 Boucke, 3:35
November 20           Roger Heath-Brown (AIM and University of Oxford)

Pairs of quadratic forms

The Local to Global Principle (Hasse Principle) for the existence of rational zeros of quadratic forms can fail if one asks for zeros of pairs of quadratic forms.  In such cases the Hardy-Littlewood asymptotic formula for the density of rational zeros also fails, a fortiori.

The talk will describe a particular case when one can none the less obtain a modified aymptotic formula.

November 21           Eric Mortenson (University of Wisconsin)

Supercongruences Between Truncated $_2F_1$ Hypergeometric Functions and Their Gaussian Analogs

Fernando Rodriguez-Villegas conjectured a number of supercongruences for hypergeometric Calabi-Yau manifolds of dimension $d\le3$.  For manifolds of dimension $d=1$, he observed four potential supercongruences.  Here we prove a general result on supercongruences between values of truncated $_2F_1$ hypergeometric functions and Gaussian hypergeometric functions. As a corollary to our main result, we prove the four supercongruences for dimension $d=1$.

December 5             Scott Parsell (Penn State)

A generalization of Vinogradov's mean value theorem

Let $J_{s,k}(P)$ denote the number of integral solutions of the system of equations $\sum_{j=1}^s (x_j^i-y_j^i)=0 \ (1 \leq i \leq k)$ with ${\mathbf x}, {\mathbf y} \in [1,P]^s$.  Bounds of the form $J_{s,k}(P) \ll P^{2s-\frac{1}{2}k(k+1)+\eta(s,k)}$, collectively known as Vinogradov's mean value theorem, have been applied to establish the asymptotic formula in Waring's problem and to study the Riemann zeta function.  When $d \geq 1$, we write ${\mathbf x}^{\mathbf i} = x_1^{i_1} \cdots x_d^{i_d}$ and consider the generalized system $$\sum_{j=1}^s ({\mathbf x}_j^{\mathbf i}-{\mathbf y}_j^{\mathbf i}) = 0 \quad (1 \leq i_1 + \dots + i_d \leq k).$$  We describe new bounds for the number of integral solutions of this system lying in a given box and discuss applications, via the circle method, to counting rational linear spaces of projective dimension $d$ on the  hypersurface defined by an additive equation of degree $k$ in $s$ variables.

December 12            Ling Long (Institute for Advanced Study, Princeton)

Isogenous elliptic curves

A one parameter family of elliptic curves with non-constant $j$-invariant satisfies an order 2 ordinary linear Fuchsian equation, called the Picard-Fucsh equation of the family. Relation between isogenous classes of the family of elliptic curves and their Picard-Fuchs equations will be studied in this talk. We will also discuss some applications.