Seminars Sorted by Series

This is an exposition of work on Artin's Conjecture on the zeros of p-adic forms.A variety of lines of attack are described ,going back to 1945.However there is particular emphasis on recent developments concerning quartic forms on the one hand ,and...

The Schmidt subspace theorem is a deep generalization of Roth's theorem in Diophantine approximation to the setting of hyperplanes in projective space. Another well-known generalization of Roth's theorem is the theorem of Wirsing, which extends Roth...

We improve Kolyvagin's upper bound on the order of the p-primary part of the Shafarevich-Tate group of an elliptic curve of rank one over a quadratic imaginary field. In many cases, our bound is precisely the one predicted by the Birch and...

A sequence of primes $p_1, \dotsc , p_k$ is a prime chain if $p_{j+1} \equiv 1 \pmod{p_j}$ for each $j$. For example: 3, 7, 29, 59. We describe new estimates for counts of prime chains satisfying various properties, e.g. the number of chains with $p...

An overview of methods and results concerning small gaps between primes, between almost primes, and the consequences on values taken by certain arithmetical functions will be presented. The material is from the works with Goldston, Graham and Pintz...

In the first half of the talk I will give details of my joint work with Henryk Iwaniec. We use half-dimensional sieve to obtain a lower bound for the density of rational points on the cubic Chatelet surface. The cubic Chatelet surface can also be...

For the last 5 years or so Terry Tao and I have been working on a programme to prove certain conjectures of Hardy and Littlewood concerning the number of primes vectors p = (p_1, . . . ,p_n) in some box which satisfy the equation Ap = b . The number...

A bounded Apollonian circle packing (ACP) is an ancient Greek construction which is made by repeatedly inscribing circles into the triangular interstices in a Descartes configuration of four mutually tangent circles. Remarkably, if the original four...

A result of Kim-Sarnak (2003) gives the best known bounds towards the Ramanujan conjecture for Maass forms. The technique employed has not, until now, been made to apply to general GL2 cusp forms over number fields whose unit group is infinite. In...

In this joint work with Stephan Baier, we prove a subconvexity bound for Godement-Jacquet L-functions associated with Maass forms for SL(3,Z). The bound arrives from extending a method of M. Jutila (with new ingredients and innovations) on...

We discuss the question of quantitative bounds on the sup-norm of automorphic cusp forms. We present an improvement on a recent result by Blomer-Holowinsky on Hecke-Maass forms on $X_0(N)$ with large level $N$. Analogous results are then established...

Let K/Q be an extension of number fields. The Hasse norm theorem states that when K is cyclic any non-zero element of Q can be represented as a norm from K globally if and only if it can be represented everywhere locally. In this talk I will discuss...

I first present an algorithm to compute the truncated theta function in poly-log time. The algorithm is elementary and suited for computer implementation. The algorithm is a consequence of the periodicity of the complex exponential, and the self...

Let E be an elliptic curve over Q and let Q(E[n]) be its n-th division field. In 1972, Serre showed that if E is without complex multiplication, then the Galois group of Q(E[n])/Q is as large as possible, that is, GL_2(Z/n Z), for all integers n...

In this talk I will construct a class of probabilistic random Euler products to model the behavior of L-functions in the strip 1/2 Re(s) 1. We then deduce results on the distribution of extreme values of several families of L-functions, including...

The Ramanujan conjecture states that for a holomorphic cusp form $f(z) =\sum_{n \in N} \lambda_f(n)e(nz)$ of weight $k$, the coefficients $\lambda_f(n)$ satisfy the bound $|\lambda_f(n)| \ll_\epsilon n^{(k−1)/2+\epsilon}$. In the case where $k$ is...

An important theme in arithmetic combinatorics, which is closely related to the ergodic-theoretic project of understanding characteristic factors, is higher-order Fourier analysis. It has been well known for a long time that various norms defined in...

We shall discuss joint work with I Z Ruzsa in which it is shown that if A is a subset of {1,..,N} such that its difference set contains no number of the form $p-1$ for $p$ a prime, then $|A|=O(N\exp(-c\sqrt{4}{\log N}))$ for some absolute $c>0$.

Let A be subset of {1,...,n}. We say that A is square sum-free if the sum of any two different elements of A is not a square. Erdos and Sarkozy asked whether a square sum-free set can have more than n(1/3+epsilon) elements (motivated by the sequence...

This will be a survey of old and new problems and results in additive number theory.

In 1977 Szemeredi proved that any subset of the integers of positive density contains arbitrarily long arithmetic progression. A couple of years later Furstenberg gave an ergodic theoretic proof for Szemeredi's theorem. At around the same time...

Recent work of Alon-Shapira and Rodl-Schacht has demonstrated that every hereditary graph and hypergraph property is testable with one-sided error. This result appears definitive, but there are some subtleties to it that I will present here. For...

Let Q(n,p) denote the adjacency matrix of the Erdos-Renyi graph G(n,p), that is to say a symmetric matrix whose entries above the main diagonal are independently set to 1 with probability p and 0 with probability 1-p. We will examine the behavior of...

For a probability distribution X over a finite set, let D(X) denote the L_2-distance of X from the uniform distribution. Let X, Y be probability distributions over the finite group G and let Z be their G-convolution. Inspired by recent work of...

The aim is to present some of the more technical aspects of my joint project with Tim Gowers regarding the true complexity of a system of linear quations. Using so-called "quadratic Fourier analysis", we determined a necessary and sufficient...

Let $G$ be a finite Abelian group, say $Z/NZ$. A set $\Lambda = \{ lambda_1, \dots, \lambda_{m} \}$ is called {\it dissociated} if any equality $\sum_{i=1}^m \varepsilon_i \lambda_i = 0$, where $\varepsilon_i \in \{ 0,\pm 1 \}$ implies that all $...

We will study the following problem: Given $n$ subsets $A_1, A_2,\ldots, A_n\subset \mathbb{F}_q$ of a finite field $\mathbb{F}_q$ with $q$ elements. Let $|A_1|\cdot |A_2|\cdot\ldots dot|A_n|>q^{1+\varepsilon}$ for some $\varepsilon>0,$ one needs to...