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Abstract: |
The asymptotics of the number of partitions of a large number into primes and related matters will be discussed. | |
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Rational curves on singular surfaces and a geometric approach to the Jacobian Conjecture | |
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Abstract: |
I will outline a geometric approach to the two-dimensional
Jacobian Conjecture using the methods of the Mori-Reid'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 | |
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Number Theoretic Properties of Wronskians of
Andrews-Gordon Series | |
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Abstract: |
We consider the arithmetic properties of quotients of Wronskians in certain normalizations of the Andrews-Gordon q-series \prod_{1\leq n\not \equiv 0,\pm
i\pmod{2k+1}}\frac{1}{1-q^n}. | |
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Polynomial parametrization for the solutions of
Diophantine equations and arithmetic groups | |
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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. | |
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Abstract: |
Another Rankin-Selberg talk. What's different about
this one is that we consider a group which is quasi-split but
not-necessarily | |
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Abstract: |
Let G be a reductive p-adic group. Continuing work of J. Bernstein, we (i.e. P.Baum, R.Plymen, and A-M 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. | |
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Small Gaps Between Products of Two Primes | |
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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 Heath-Brown's theorem that d(n)=d(n+1) infinitely often
and of Schlage-Puchta'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. | |
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Irreducible polynomials in two variables
and polynomial ranges over finite fields | |
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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). | |
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David Farmer (AIM) | |
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L-functions and modular
forms | |
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Abstract: |
I will discuss the general structure of L-functions and
describe what is known about the connection between L-functions and
modular forms. | |
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Robert Vaughan (PSU) | |
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Diophantine Approximation on Planar
Curves | |
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Hilbert's Tenth Problem for function fields over
$p$-adic fields | |
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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. | |
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Multi-clock control shift register sequences | |
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Non-vanishing of p(n) modulo 3 | |
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Abstract: |
Let p(n) be the ordinary
partition function. It is well-known 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. | |
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A Variance for k-free numbers in arithmetic
progressions | |
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Abstract: |
We obtain an asymptotic formula for the variance over residue classes of the error term for the standard approximation for the number of k-free numbers in an initial segment. This improves significantly on earlier estimates of Brudern, Croft, Granville, Orr, Perelli, Vaughan, Warlimont and Wooley. |