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54 KAM Mathematical Colloquium

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Prof. Ben Green

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University of Bristol

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ARITHMETIC PROGRESSIONS IN THE PRIMES

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February 9, 2005

Lecture room S5, II. floor
10:30 AM

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Abstract

I will outline the recent proof that there are arbitrarily long arithmetic progressions of primes, which is joint work with Terry Tao.

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THE GOWERS $U^3$ NORM AND ITS APPLICATIONS

10. 2. 2005 v 10:30, S5

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Abstract

The Gowers $U^3$ norm is one of a series of norms on the space $\mathbf{C}(G)$ of complex valued functions on a
finite abelian group $G$. These norms are called the Gowers $U^d$ norms. The Gowers $U^{k-1}$ norm is
relevant to the study of arithmetic progressions of length $k$. The $U^2$ norm is well understood, largely because
the $U^2$ norm of a function $f$ is just the $L^4$ norm of the Fourier transform of $f$. I will discuss the $U^3$ norm,
attempting to answer the following questions:
\begin{itemize}
\item What is the $U^3$ norm?
\item Why is it useful for studying arithmetic progressions of length 4?
\item When is the $U^3$ norm of $f$ large?
\item How can this help us count the number of 4-term arithmetic progressions of primes less than $N$?