# Thursdays, 11:15-12:05, 106 McAllister

January 12

Robert Vaughan (PSU)

Title:

The number of partitions into primes

Abstract:

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

January 19

Alexander Borisov (PSU)

Title:

Rational curves on singular surfaces and a geometric approach to the Jacobian Conjecture

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
will be assumed.

January 26

Eric Mortenson (PSU)

Title:

Number Theoretic Properties of Wronskians of Andrews-Gordon Series

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}.

We  determine the vanishing of such Wronskians, a result whose proof reveals many partition identities. For example, if P_{b}(a;n) denotes the number of partitions of n into parts which are not congruent to 0, \pm a\pmod b, then for every positive integer n we have

P_{27}(12; n) = P_{27}(6;n-1)  +  P_{27}(3;n-2).

We also show that these quotients classify supersingular elliptic curves in characteristic p. More precisely, if 2k+1=p, where
p \ge  q 5 is prime, and the quotient is non-zero, then it is essentially the locus of characteristic p supersingular j-invariants.

February 2

Leonid Vaserstein (PSU)

Title:

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.

February 9

No seminar this week

February 16

Joe Hundley (PSU)

Title:

Spin L-functions of quasi-split D4

Abstract:

Another Rankin-Selberg talk.  What's different about this one is that we consider a group which is quasi-split but not-necessarily
split, and focus on the unramified local computations.  As we'll see, locally there are essentially three ways for our group to be quasi-split, leading to three variations which, individually and in their relationships to one another make for a very pleasing overall picture.

February 23

Paul Baum (PSU)

Title:

On the Heck algebra for reductive p-adic groups: A geometric conjecture

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.

March 2

Sid Graham (Central Michigan University)

Title:

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  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.

March 9

No seminar, Spring break

March 16

Leonid Vaserstein and Alex Borisov (PSU)

Title:

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
numbers c.

This has applications to the image of  f(x,y) modulo primes p (in the case when  f(x,y) has integer coefficients).

March 23

David Farmer (AIM)

Title:

L-functions and modular forms

Abstract:

I will discuss the general structure of L-functions and describe what is known about the connection between L-functions and modular forms.

March 30

Robert Vaughan (PSU)

Title:

Diophantine Approximation on Planar Curves

Abstract:

In joint work with Sanju Velani the convergence theory for the set of simultaenously psi-approximable points lying on a planar curve is established.  Our results complement the divergence theory developed by Beresnevich, Dickson, Velani and Vaughan and completes the general metric theory for planar curves.  The proof combines harmonic analysis with an elementary estimate of Huxley.

April 6

Kirsten Eisentraeger (University of Michigan)

Title:

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.

April 13

Guo-zhen Xiao (Xidian University)

Title:

Multi-clock control shift register sequences

Abstract:

 We propose a new clock-control-model which can be put into practice easily. The sequences produced with this model have a long period and larger linear complexity. Key words: stream cipher, linear complexity, clock-control sequences

April 20

Matthew Boylan (University of  South Carolina)

Title:

Non-vanishing of p(n) modulo 3

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.

April 27

Robert Vaughan (PSU)

Title:

A Variance for k-free 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 k-free numbers in an initial segment.  This improves significantly on earlier estimates of Brudern, Croft, Granville, Orr, Perelli, Vaughan, Warlimont and Wooley.