
 

 
Abstract: 
The asymptotics of the number of partitions of a large number into primes and related matters will be discussed.  

 

Rational curves on singular surfaces and a geometric approach to the Jacobian Conjecture  
Abstract: 
I will outline a geometric approach to the twodimensional
Jacobian Conjecture using the methods of the MoriReid's Minimal Model
Program. In particular, I will explain the connections to the work of Keel
and McKernan on the log del Pezzo surfaces. I will explain the background
theory and will try to avoid the technical details. No prior familiarity
with the Minimal Model Program  

 

Number Theoretic Properties of Wronskians of
AndrewsGordon Series  
Abstract: 
We consider the arithmetic properties of quotients of Wronskians in certain normalizations of the AndrewsGordon qseries \prod_{1\leq n\not \equiv 0,\pm
i\pmod{2k+1}}\frac{1}{1q^n}.  

 

Polynomial parametrization for the solutions of
Diophantine equations and arithmetic groups  
Abstract: 
A polynomial parametrization for the group of integer
two by two matrices with determinant one is given, solving an old
open problem of Skolem and Beurkers. It follows that, for many
Diophantine equations, the integer solutions and the primitive solutions
admit polynomial parametrizations.  

 

 

 
Abstract: 
Another RankinSelberg talk. What's different about
this one is that we consider a group which is quasisplit but
notnecessarily  

 

 
Abstract: 
Let G be a reductive padic group. Continuing work of J. Bernstein, we (i.e. P.Baum, R.Plymen, and AM Aubert) conjecture that the admissible dual of G has in a natural way the structure of a countable disjoint union of complex affine algebraic varieties. The conjecture states what these varieties are. Each is an "extended quotient" for an action of a finite group on a complex torus.  

 

Small Gaps Between Products of Two Primes  
Abstract: 
The techniques that Goldston, Pintz, and Yildirim recently
used to prove the existence of short gaps between primes can be applied to
other sequences. For example, one can apply these techniques to the
sequence of numbers that are products of exactly two primes.
Using this, we can prove that there are infinitely many integers n
such that at least two of the numbers n, n+2, n+6 are products of exactly
two primes. The same can be done for more general linear forms;
e.g., there are infintely many n such at least two of 42n+1, 44n+1,
45n+1 are products of exactly two primes. This in turn leads to simple
proofs of HeathBrown's theorem that d(n)=d(n+1) infinitely often
and of SchlagePuchta's theorem that omega(n)=omega(n+1) infinitely often.
With other choices of linear forms, we can sharpen this to d(n)=d(n+1)=24
and omega(n)=omega(n+1)=3 infinitely often. This is joint work with
D. Goldston, J. Pintz, and C. Yildirim.  

 

 

Irreducible polynomials in two variables
and polynomial ranges over finite fields  
Abstract: 
We discuss the following problem. Let f(x,y) be
a polynomial with complex coefficients. Assume
that f(x,y) is not of the form g(h(x,y)) with a
univariant polynomial g of degree > 1. Then the
polynomial f(x,y)  c can be reducible for only finitely
many This has applications to the image of f(x,y) modulo
primes p (in the case when f(x,y) has integer
coefficients).  

David Farmer (AIM)  

Lfunctions and modular
forms  
Abstract: 
I will discuss the general structure of Lfunctions and
describe what is known about the connection between Lfunctions and
modular forms.  

Robert Vaughan (PSU)  

Diophantine Approximation on Planar
Curves  
Abstract: 
 

 

Hilbert's Tenth Problem for function fields over
$p$adic fields  
Abstract: 
Hilbert's Tenth Problem in its original form
was to find an algorithm to decide, given a polynomial equation
$f(x_1,\dots,x_n)=0$ with coefficients in the ring $\mathbf{Z}$ of integers, whether it has a solution with $x_1,\dots,x_n \in \mathbf{Z}$. Matiyasevich proved that no such algorithm exists, i.e. Hilbert's Tenth Problem is undecidable. Since then, analogues of this problem have been studied by asking the same question for polynomial equations with coefficients and solutions in other commutative rings. Let $k$ be a subfield of a $p$adic field. We will prove that Hilbert's Tenth Problem for function fields of varieties over $k$ of dimension $\geq 1$ is undecidable.  

 

Multiclock control shift register sequences  
Abstract: 
 

 

Nonvanishing of p(n) modulo 3  
Abstract: 
Let p(n) be the ordinary
partition function. It is wellknown that p(n) satisfies interesting
congruences properties. The most famous examples are the
Ramanujan congruences (for example, p(5n+4) = 0 mod 5). Later,
Ahlgren and Ono showed that p(n) satisfies similar linear congruences
modulo M for every M coprime to 6. In contrast, little is known
about p(n) mod 2 or mod 3. In fact, it is not yet known whether 3
divides p(n) infinitely often (though this is certainly believed to be
true). In this talk, we use modular Galois representations to show that for all integers r and s with s positive, #{n < X : n = r mod 3^s with p(n) <> 0 mod 3} >> sqrt{X}/log X.  

 

A Variance for kfree numbers in arithmetic
progressions  
Abstract: 
We obtain an asymptotic formula for the variance over residue classes of the error term for the standard approximation for the number of kfree numbers in an initial segment. This improves significantly on earlier estimates of Brudern, Croft, Granville, Orr, Perelli, Vaughan, Warlimont and Wooley. 