September 13
Scott Parsell (PSU)
Irrational Linear
Forms in Prime Variables
September 20 Mark Watkins (PSU)
The work of Alain Connes on the Riemman Hypothesis
We review Weil's "explicit
formula" for the zeros of the zeta-function, and describe recent work of
Connes which gives a spectral interpretation for the critical zeros, and
shows the Riemann Hypothesis to be equivalent to a trace formula on the
noncommutative space of Adele classes.
September 27 Robert Vaughan (PSU)
A matrix connected
with the Riemann Hypothesis
October 4
No Seminar This Week
October 11 Helena Verrill (Hannover University)
Algorithms for higher weight modular symbols for Gamma_0(N)
Modular symbols were
invented by Manin and Shokurov in order to carry out explicit computations
with modular forms. Much computational work has been done particularly
in the weight 2 case by Cremona, Merel, and more recently by William Stein.
I will describe algorithms for working with modular symbols of even weight
greater than two. These methods generalise some of the methods of
Cremona and Merel. In particular I will describe "transportable"
modular symbols, and an algorithm for computing the intersection pairing
for modular symbols of even weight k>2. I will also mention an application
to compute the intermediate Jacobian of certain rigid Calabi-Yau threefolds.
Some of this is joint work with William Stein.
October 19
Morley Davidson (Kent State)
NOTE: Special
Day, Time and Place: Friday, 2:30, Boucke 306.
Cyclotomic Properties of Partition Polynomials
This talk describes the results of separate joint work with George Andrews,
Stephen Gagola, and Jeffrey Keen. We define the $n^{\text{th}}$ partition
polynomial $\wp_{n}(x)$ as the characteristic polynomial of Euler's linear
recursion of order $\omega(n):=(3n^{2}-n)/2$ for the partition function
$p(m)$ with $m < \omega(-n).$ This polynomial sequence may be generated
by setting $\wp_{0}(x):=0$ and using the recursion
$$\wp_{n}(x)= x^{3n-2}\wp_{n-1}(x) + (-1)^{n-1}(x^{2n-1}-1),$$
although we have a formula based on an identity of Shanks which better
clarifies the structure of $\wp_{n}(x)$. On the basis of both symbolic
algebra and numerical experiments, we arrived at a number of conjectures
regarding cyclotomic properties of $\wp_{n}(x).$ While certain of the key
analytic conjectures remain open, we have found proofs of several if not
most of our algebraic conjectures. We will discuss three different such
theorems, an example of which is the following: $x^{n}-1$ divides $\wp_{n}(x)$
for all $n;$ when $n$ is a prime exceeding 5, the quotient polynomial has
maximal coefficient 2 and minimal coefficient -2, whereas if $n>1$ factors
as $2^{a}3^{b},$ then these coefficients are $\pm 1.
October 25 Jeff Lagarias (AT&T Labs Research) : COLLOQUIUM
Some integer permutation problems from the bottom of the Erdös barrel
We discuss certain permutations {a(n): n \ge 1} of the nonnegative
integers which have restrictions on the greatest common divisors
(a(n), a(n+1)) of consecutive terms. Such questions were first raised
and studied by Paul Erdös, Robert Freud and Norbert Hegyvari in 1983.
We describe work on the original problem of Erdös, Freud, and Hegyvari
on making all the gcd's large, and recent work of permutations with gcd's
> 1 constructed by a greedy algorithm, done jointly with Eric Rains and
Neil J. A. Sloane.
November 1 Yi Ouyang (University of Toronto)
The Universal Norm Distribution and its Application
The theory of universal ordinary distribution plays an important role in number theory, It is closely related to, the circular units in the $1$-dimensional case and the elliptic and modular units in the $2$-dimensional case. Recently Anderson proposed a double complex method to study the universal ordinary distribution.
In this talk, we will generalize the universal ordinary distibution
to the universal norm distribution and use Anderson's method to study it.
Besides the special case of ordinary distribution, there are many other
important cases of the universal norm distribution, e.g., the weight ordinary
distribution, the universal Euler system. We obtain some interesting
results in the index calculation. We also relate its group cohomology
to the inductive procedure of the Euler system.
November 8 Kevin Ford (University of Illinois at Urbana-Champaign)
The prime number race
We will survey many problems, results and conjectures concerning the
relative magnitudes of the functions $\pi(x;q,a)$ for fixed $q$.
Here $\pi(x;q,a)$ is the number of primes $\le x$ in the progression $a$
modulo $q$. It is known that for fixed $q$, all of the functions
$\pi(x;q,a)$ with $\text{gcd}(a,q)=1$ are asymptotic to $x/(\phi(q)\log
x)$, but curious inequities occur. For instance, $\pi(x;4;3) > \pi(x;4,1)$
for "most" x. The behavior of such inequities is closely linked to
the distribution of non-trivial zeros of Dirichlet L-functions.
November 15 Michael Knapp (University of Rochester)
Artin's Conjecture on Forms in Many Variables
Consider a system of homogeneous polynomials in many variables with
integral coefficients. A conjecture attributed to Artin states that
this system will have a nontrivial simultaneous zero in p-adic integers
for every prime p provided only that the number of variables is sufficiently
large in terms of the degrees of the polynomials, and proposes a specific
bound on how many variables suffice. We will begin this talk by discussing
the extent to which the conjecture is true and mentioning some related
problems. Then we will specialize to the case in which all of the
polynomials are additive (ie. have no cross terms) and discuss some recent
results in this situation.
November 29 Eric Freeman (Institute for Advanced Study, Princeton)
Systems of Diophantine equations and inequalities
A Diophantine inequality is an inequality of the type $|F(\bf x)|< \epsilon$, where $F(\bf x)$ is a polynomial with real coefficients, and where we seek an integral vector solution $\bf x$. We consider certain combined systems, comprised of both Diophantine equations and inequalities. By considering these systems, we are able to show that certain systems of diagonal Diophantine inequalities of even degree have solutions.
The talk will include a presentation of the above work, but we will
also spend a significant portion of the time discussing some related results
and ideas.
December 6 Robert Vaughan (PSU)
Conference Report and Open Problem Session
I will start by giving a brief report on the talks given at the Richard
Hall retirement meeting at the University of York in October, and then
start an open problem session by describing some conjectures. Please
bring along your conjectures and be prepared to say something about them!