Number theory  seminar  will meet  2 more times this week :
 
Thursday,  March.20,1:55- 2:45 pm  in LIT 339.
Friday ,  March.21 ,1:55- 2:45 pm  in   LIT 339.

SPEAKER:, Professor Peter Sarnak ( Princeton University and The Institute
for Advanced Study).

 TITLE: "Sieves, the generalized Ramanujan Conjectures and expander
graphs (part 2 and part 3)"

ABSTRACT: We review various classical problems concerning the
existence of primes or numbers with few prime factors as well
as some of the developments towards resolving these long standing
questions. We then put these problems in a natural and general
geometric/group theoretic context of actions by morphisms on
affine n-space and indicate what can be established there. The
methods used to develop a combinatorial sieve in this context
involve automorphic forms and especially the generalized Ramanujan
Conjectures, expander graphs and unexpectedly, arithmetic
combinatorics. Applications to classical problems such as the
divisibilty of areas of Pythagorean triangles and of curvatures
of circles in integral Apollonian packings will be given. In the
first lecture we will give a general overview (for a general audience),
in the second lecture* we discuss the interesting special cases for
which approximations to the general Ramanujan Conjectures can be
used effectively in the analysis and in the third lecture* we
explain the role of arithmetic combinatorics in dealing with
the general problem.

Number theory  seminar  will meet this Tuesday,  March.18,
 
1:55- 2:45 pm  in LIT 368.


SPEAKER:, Hershel Farkas , Hebrew University, Jerusalem

TITLE:   A generalization of Thomae's formula to Z_n curves


ABSTRACT:
A z_n curve is the Riemann surface of the algebraic curve
W^n = \prod_{i=1}^{rn}(z-\lambda_i)
where $ \lamda_i \neq \lambda_j $ when $i \neq j$.
When n=2 the surface is hyperelliptic and Thomae derived a formula
relating the theta constants with integer characteristic and polynomials
in the variables $ \lambda_i$. Bershadsky and Radul extended Thomae's
formula to z_n curves and this was later reworked by Nakayashiki.
I have discovered a more transparent proof which follows from 
elementary principles and gives a more explicit formula.

The formulae should be understood as relating the transcendental
parameters
of a compact Riemann surface to the algebraic parameters, the
$\lambda_i$.
We show that there is a collection of points of order 2n in the Jacobi
variety
of the surface and polynomials in the variables $ \lambda_i $
such that the ratio of the associated theta constants to the polynomials
is constant.

These polynomials are given quite explicitly in terms of a matrix but 
are never the less quite complicated. More interesting to us is the 
fact that we can actually write equations for powers of the variables 
$ \lambda_i$ in terms
of quotients of products of theta constants.

We shall concentrate in the talk on one example and then time permitting
will
try to explain the general situation. Our motivation for doing this 
work comes from the fact there has been a resurgence of interest among 
mathematical physicists in this topic.

 Number theory  seminar
 will meet this Tuesday,  Feb.19,
 1:55- 2:45 pm  in LIT 368.

 SPEAKER: Alexander T. Berkovich  (UF)

TITLE: "Dyson, his rank and much more" ( Part 2)

ABSTRACT: I will continue talking about my current research project.
In particular , I plan to discuss new iterative process : Bailey with 
massage
The talk will be accessible to non-experts.

 Number theory  seminar
 will meet this Tuesday,  Feb.12,
 1:55- 2:45 pm  in LIT 368.

SPEAKER: Professor Roger Baker (BYU)

TITLE: "A mean square asymptotic formula sums
        of two k-th powers"

ABSTRACT: The number of representations of an integer
as a sum of two relatively prime k-th powers (where
k is at least 3) has summatory function T(x)
approximately equal to a constant multiple of x^{2/k}.
Let the remainder on subtracting this main term be
E(x). We give an asymptotic formula for the mean
square of E(x), subject to the existence of a suitable
zero free strip for the Riemann zeta function.
Unfortunately, no unconditional result of this
kind is known.

The talk will be accessible to non-experts.

Number theory  seminar
will meet this Tuesday,  Feb.5,
1:55- 2:45 pm  in LIT 368,

Title:

Dyson, his rank and much more .

Speaker:       Alexander  T. Berkovich. (Math. Dept. UF  )

Abstract: I will talk about my current research project.