Oxford University

I will describe a sketch of the proof of Grothendieck conjecture on fundamental groups.
 

This is the first talk of the workshop organised by F. Brown, M. Kim and D. Rössler on Beilinson's approach to p-adic Hodge theory. 

In this talk, we shall give the definition and recall various properties of the cotangent complex, which was originally defined by L. Illusie in his monograph "Complexe cotangent et déformations" (Springer LNM 239, 1971).

I will explain how to find an explicit embedding of the Kummer variety of a higher genus curve into projective space and discuss applications of such an embedding to the study of rational points on Jacobians of curves, as well as the original curves.

Part of the series 'What do historians of mathematics do?' 

" Over the last four centuries a huge amount of mathematics has been published.  Most of it has, however, had little or no influence.  By way of contrast, some mathematics, although unpublished in its time, has had great influence.  My hope is to illustrate this with discussion of manuscript sources and resources that have survived from Thomas Harriot (c.1560--1621), Isaac Newton (1642--1727) and Évariste Galois (1811--1832)."

Using the "trichotomy principle" by Boris Zilber I will give model theoretic proofs of appropriate versions of Torelli theorem and Borel-Tits theorem. The first one has interesting applications to anabelian geometry, I won't assume any prior knowledge in model theory.

Consider the following question. Given a $k$-colouring of the positive integers, must there exist a solution to $x+y=z^2$ with $x,y,z$ all the same colour (and not all equal to 2)? Using $10$ colours a counterexample can be given to show that the answer is "no". If one instead asks the same question over $\mathbb{Z}/p\mathbb{Z}$ for some prime $p$, the answer turns out to be "yes", provided $p$ is large enough in terms of the number of colours used.  I will talk about how to prove this using techniques developed by Ben Green and Tom Sanders. The main ingredients are a regularity lemma, a counting lemma and a Ramsey lemma.

Many theorems and conjectures in prime number theory are equivalent to finding solutions to certain linear equations in primes -- witness Goldbach's conjecture, the twin prime conjecture, Vinogradov's theorem, finding k-term arithmetic progressions, etcetera. Classically these problems were attacked using Fourier analysis -- the 'circle' method -- which yielded some success, provided that the number of variables was sufficiently large. More recently, a long research programme of Ben Green and Terence Tao introduced two deep and wide-ranging techniques -- so-called 'higher order Fourier analysis' and the 'transference principle' -- which reduces the number of required variables dramatically. In particular, these methods give an asymptotic formula for the number of k-term arithmetic progressions of primes up to X. In this talk we will give a brief survey of these techniques, and describe new work of the speaker, partially ongoing, which applies the Green-Tao machinery to count prime solutions to certain linear inequalities in primes -- a 'higher order Davenport-Heilbronn method'.