*Vertex and conformal algebras*

I will begin by introducing vertex algebras and briefly discuss different
approaches to these objects and the role they played in various mathematical
fields. The rest of the talk will be focussed on conformal algebras
-- the purely algebraic structures that allow for a good description of
vertex algebras. I will also discuss some recent results from the
representation theory of conformal algebras.

**Janaury 31 Ulrike
Vorhauer (Kent State University)**

*Greedy sums of distinct squares*

**NOTE: Hugh Montgomery (University of Michigan) will speak in
the colloquium.**

**February 7 Alexander
Borisov (PSU)**

*Periodic orbits of algebraic morphisms*

I will discuss some examples, results and conjectures regarding periodic
points of algebraic maps. Particular emphasis of the talk will be on Zariski
closure of periodic points of maps of affine spaces over the algebraic
closure of a finite field. My interest in this topic stems from its applications
to group theory research of Mark Sapir. The talk will be based on
our ongoing joint work with him.

**February 15 Ling Long (PSU)**
**NOTE, this week the seminar will be at 2:30pm on Friday, in room
103 Osmond.**

*Modularity of elliptic surfaces*

**February 21 Mike Dancs (PSU)**

*On a variance arising in the Gauss circle problem*

**February 28 Nathan Ng (Institute
for Advanced Study)**

*The summatory function of the Mobius function*

We present some conditional results on the summatory function of the
Mobius function, denoted by M(x). These results depend on a
conjecture of Gonek and Hejhal concerning negative discrete moments of
the Riemann zeta function. The object of this talk is to show how
this conjecture leads to much better results than were previously known.
For example, we construct a limiting distribution that encodes informating
regarding M(x).

**March 14
Bill Hoffman (Louisiana State University)**

*Topology of Siegel Modular Varieties*

Let $\Gamma $ be an arithmetic subgroup of the symplectic group $Sp(2g,
\bf{R})$. $\Gamma $ acts on the Siegel half space $S_g$
and the quotient $S_g /\Gamma$ is a moduli space parametrizing abelian
varieties with extra structures. This talk discusses the general problem
of determining the topological properties, principally the cohomology,
of these spaces. We begin with a survey of known general results,
especially the connection to automorphic forms and zeta functions. The
classical case $g=1$ is recalled. Next we discuss the case $g=2$,
and results obtained in collaboration with S. Weintraub for this. Finally
we close with indications of future directions for this research.

**March 21
Joszef Beck (Rutgers): COLLOQUIUM**

*Lattice point problems, quadratic fields, Yokoi's conjecture*

In this survay type talk we begin with problems like counting lattice
point in tilted hyperbola-segments (i.e. inhomogeneous Pell inequality)
and in right triangles (first investigated by Hardy, Littlewood, and Ostrowski)
where the slope is a quadratic irrational number. There is
a surprising difference between the cases of (say) square-root-2 and square-root-3,
which leads us to problems in real quadratic fields. Finally, we discuss
a well-known "real class number one problem" called the Yokoi's conjecture,
which is a perfect real analogue of the famous Gauss' problem of finding
all imaginary quadratic fields of class number one (solved by Heegner,
Stark, and Baker in 1950-60's). In 1997 we published a heuristic argument
of how to prove the Yokoi's conjecture. This heuristic argument was
very recently developed into a precise proof by a young Hungarian number-theorist
A. Biro.

**March 28
Kiran Kedlaya (Berkeley)**

*Monsky-Washnitzer Cohomology and Computing Zeta Functions*

Monsky-Washnitzer cohomology is a p-adic cohomology theory for algebraic
varieties over finite fields, based on algebraic de Rham cohomology. Unlike
the l-adic (etale) cohomology, it is well-suited for explicit computations,
particularly over fields of small characteristic. We describe how
to use Monsky-Washnitzer to construct efficient algorithms for computing
zeta functions of varieties over finite fields, using as an example the
case of hyperelliptic curves in odd characteristic.

**April 4
Michael Rubinstein (AIM and University of Texas)**

*Moments of L-functions and Random Matrix Theory*

We present conjectures and heuristics for the full asymptotics of the
moments of L-functions on/at the critical line/point.

**April 11
Noriko Yui (Queens University)**

*Mirror moonshine phenomenon*

B.H. Lian and S.-T. Yau first observed that mirror maps of certain families of Calabi-Yau hypersurfaces are expressed in terms of McKay-Thompson series arising from the representation theory of Monster. This is the so-called {\it mirror moonshine phenomenon}. In this talk, I will give more examples of families of Calabi-Yau threefolds in support of the mirror moonshine phenomenon. Examples include Calabi-Yau threefolds with K3 fibrations. This is a preliminary report on a joint work with Ling Long.

**NOTE: Ken Ono (University of Wisconsin) will be speaking in
the colloquium**

**April 12
Joseph Hundley (Columbia University)**

**NOTE: This is an additional seminar, on Friday at 1:25 in 115 McAllister**

*Siegel zeros of Eisenstein series on GL(n)*

Let E(z,s) be the usual, non-holomorphic Eisenstein series defined on
the upper half plane. By considering the Fourier expansion of E(z,s) it
may be readily verified that for all y sufficiently large, E(z,s) has a
zero in the interval (1-(1/log y),1). We will generalize this result to
a fairly broad class of Eisenstein series defined on GL(n,R).

**April 18
Michael Filaseta (University of South Carolina)**

*Applications of Pad\'e Approximations of $(1-z)^{k}$ to Number
Theory*

Pad\'e approximations of $(1-z)^{k}$ have been used to tackle a variety
of different problems in Number Theory. These uses include results associated
with the prime factorization of $n(n+1)$, inverse Galois theory, the Ramanujan-Nagell
equation and its generalizations, other diophantine equations, irrationality
measures, $k$-free numbers in short intervals, powerfree values of polynomials
and binary forms, and the $abc$-conjecture. The goal will be to discuss
results obtained from Pad\'e approximations of $(1-z)^{k}$ and to give
an indication, at least in some cases, as to how Pad\'e approximations
enter into these investigations.

**April 25
Yuri Zarhin (PSU)**

*Hyperelliptic jacobians without complex multiplication and doubly
transitive Galois groups*