**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.