Speaker: Jorge Jimenez-Urroz (University of Barcelona)

Title: Small gaps between primes and almost primes; Recent work

of Goldston, Pintz and Yildirim.

Abstract. Recently Goldston, Pintz and Yildirim have proven

one of the most important open problems on the distribution of

primes. Namely that the difference between consecutive primes

can be arbitrarily smaller than the average. The talk will serve to

describe a little bit the proof of this important theorem.

Speaker: Jeremy Rouse (UW-Madison) Jan. 31 (regular seminar)

Title: Almost-Square Primes

Abstract:

An old but difficult problem in number theory is to determine if there

are polynomials f(x) that represent infinitely many prime numbers.

Dirichlet settled the degree 1 case in 1837, but no substantial progress

has been made on the degree 2 case. In particular, it is not known if

there are infinitely many n so that n^2 + 1 is prime. We consider primes

of the form p = a^2 + b^2 where b is small (less than C*log(p)). Assuming

GRH, we can show there are at least (3/4)*sqrt(x) such between x and 2x

(provided x is big enough), and as a corollary there are two such primes

that are very close ( less than C^2*log^2 x). This talk will be quite

expository and will be accessible to students with minimal background.

Speaker: Karl Mahlburg(UW-Madison) Feb. 7

Title: Some properties of partitions by Atkin and Swinnerton-Dyer (1953)

Abstract: In this paper, the authors prove Dyson's conjecture that the partition

rank proves the Ramanujan congruences modulo 5 and 7 by decomposing the

partitions into classes of equal size. They achieve this by proving

identities for the generating functions, largely through the application

of a striking vanishing criterion for analytic functions with a certain

transformation property.

Speaker: Masataka Chida (Tohoku Univ. and UW-Madison) Feb. 14

Title: "Indivisibility of orders of Selmer groups for modular forms".

Abstract:

For a prime p, James, Kohnen and Ono considerd the p-indivisibility of orders

of Selmer (or Tate-Shafarevich) groups of elliptic curves over Q.

For any elliptic curve E, they proved that the number of

{D: |D|<X, Sel_p(E(D))={0}} is greater than c\sqrt X/log(X) for all $X$ sufficiently large,

where Sel_p(E(D)) is the p-primary part of Selmer group of the D-quadratic twist of E.

We will discuss the generalization of this result to higher weight modular forms.

Speaker: Harris Nover (UW-Madison) Feb. 21

Title: Introduction to Class Field Theory, Infinite Galois Theory and Pro-p Groups

Abstract: Class field theory concerns itself with, among other things, the classification of abelian extensions of global fields. As

well as being one of the major results of early 20th century number theory, it is a key tool in nearly all branches of modern

number theory. Meanwhile, the study of profinite extensions of number fields, such as the algebraic closure of Q, began in

earnest in the latter half of the last century, with the study of pro-p extensions and the corresponding pro-p Galois groups

playing a major role. In this talk, we will provide a brief overview of these topics at a very introductory level.

Speaker: Sharon

Title: Fourier Expansion of vector valued Poincare serie

Abstract: Modular forms appear quite often in our Number Theory Seminar. (Recall recent talks by Jeremy Rouse, Ben Kane, and Karl

Mahlburg, to name a few.)

Here we look at some related types of functions that also appear on occasion, namely Maass forms and Poincare series. In

particular, we will look at relationships that occur between these functions when we extend their definitions to vector-valued

versions.

For reference, see the papers:

"On vector-valued modular forms and their Fourier coefficients" (2003)

and "Vector-valued modular forms and Poincare series" (2004)

by Marvin Knopp and Geoffrey Mason (2003)

Speaker: Mike Woodbury (UW-Madison) March 7

Title:

"Non-Vanishing of L-functions, Waldspurger's Theorem and Ranks of

Elliptic Curves"

Abstract:

I will first discuss some results about the group of rational points

on elliptic curves, particularly ones relating to the rank. Then,

motivated by the Birch Swinnerton-Dyer conjecture, I will discuss the

role of L-functions and the use of modular forms and Waldspurger's

theorem in Ono and Skinner's work on the vanishing of L-functions to

get results on the ranks of elliptic curves in families of quadratic

twists.

Speaker: Frank Thorne (UW-Madison) March 29

Title: Introduction to Sieve Methods

Abstract: We will give an overview of some elementary sieve methods and some

of their applications in number theory. In particular we will discuss the small sieve method of Selberg, and give a few general

results as well as a couple of applications. We will conclude with a brief overview of how the Selberg sieve figures in

Goldston, Graham, Pintz, and Yildirim's recent results in the direction of proving the twin prime conjecture, and hopefully

provide some background for Prof. Graham's talk on 3/31.

Speaker: Zhi-Wei Sun (Nanking Univ/UC-Irvine) April 4

Title: SOME CURIOUS RESULTS ON BERNOULLI AND EULER POLYNOMIALS

Abstract: In this talk we tell the story how the developments of some curious

identities concerning Bernoulli (and Euler) polynomials finally led to the

following unified symmetric relation (of Z. W. Sun and H. Pan): If n is a

positive integer, r + s + t = n and x + y + z = 1, then we have

r s t

x y n

+ s t r

y z n

+ t r s

z x n

= 0

where

s t

x y n

:=

n Xk=0

(1)ks

k t

n kBnk(x)Bk(y).

It is interesting to compare this with the easy identity

0 =

r s t

r s t

z x y

= r

s t

x y + s

t r

y z + t

r s

z x .

We will also talk about some explicit congruences for Bernoulli and

Euler polynomials, and a q-analogue of Stern~{!/~}s result concerning Euler

numbers.

Speaker: Adrian Diaconu (U. Minnesota), April 18

Title: INTEGRAL MOMENTS AND BOUNDS FOR AUTOMORPHIC L-FUNCTIONS

Abstract. Motivated by the Lindel\"of Hypothesis, integral moments of

automorphic L-functions were subject of intensive study over the past 90

years. In 1918, Hardy and Littlewood obtained an asymptotic formula for the

second integral moment of the Riemann zeta-function, and about 8 years later,

Ingham obtained a similar result for the fourth integral moment of the same

function. Subsequently, Good, Heath-Brown, Ivic, Iwaniec, Jutila, Motohashi

and others made important contributions to the subject, notably, asymptotic

formulae with good error terms were obtained for the second integral moment

of all GL_2 automorphic L-functions defined over \Bbb{Q}.

The main objective in this talk is to present a new method

of obtaining asymptotic formulae for the second integral moments

of GL_2 automorphic L-functions. The approach being adelic, it has the

advantage of working over any number field F. While for F = \Bbb{Q}, one

recovers the classical results, the asymptotic

formulae I am going to present are new for any other number

field F\ne \Bbb{Q}. Finally, I will indicate how to obtain good error terms

in the asymptotics, this allowing to break convexity bounds in t aspect. This

is joint work with Paul Garrett.

Speaker: Matt Darnall (UW-Madison) April 25

Title: Roth's Theorem

Abstract: Following Hindry and Silverman, I will outline the proof of Roth's

theorem about the approximation of algebraic numbers by rational numbers.

Speaker: Edray Goins (Purdue U.)

Title: Extending the Serre-Faltings Method for $\mathbb Q$-Curves

Abstract: In this talk, we consider a method for calculating modular forms associated to elliptic curves with a rational point

of order $\ell = 2, \, 3$. We discuss a variant of the Serre-Faltings method which considers the symmetric square

representations. As an application, we show that certain $\mathbb Q$-curves with reducible mod 3 representations are modular.