# Algebra and Number Theory Seminar, Spring 2002

## Thursdays, 11:15-12:05, room 201 EE West

January 24         Alexander Retakh (Yale)

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