I also study algebraic varieties in finite and mixed characteristics. Finite characteristics are universes in which the rules of arithmetic are modified by choosing a prime number p, and setting it to zero. Although classical geometry of varieties does not make sense in finite and mixed characteristics, the etale topology provides a suitable alternative, allowing us to gain much valuable insight into the behaviour of the Galois group. This is an area which I find fascinating, as much topological intuition still works in contexts far removed from real and complex geometry.

Indeed, many results in complex geometry have been motivated by phenomena observed in finite characteristic. Moduli spaces parametrise classes of geometric objects, and can themselves often be given geometric structures, similar to those of algebraic varieties. This structure tends to misbehave at points parametrising objects with a lot of symmetry.

To obviate this difficulty, algebraic geometers work with moduli stacks, which parametrise the symmetries as well as the objects. Sometimes the symmetries can themselves have symmetries and so on, giving rise to infinity stacks. Usually, the dimension of a moduli stack can be calculated by naively counting the degrees of freedom in defining the geometric object it parametrises. However, the space usually contains singularities points where the space is not smooth , and regions of different dimensions.

### Publications of Douglas C. Ravenel

Partially inspired by ideas from theoretical physics, it has been conjectured that every moduli stack can be extended to a derived moduli stack, which would have the expected dimension, but with some of the dimensions only virtual. Extending to these virtual dimensions also removes the singularities, a phenomenon known as hidden smoothness.

Different classification problems can give rise to the same moduli stack, but different derived moduli stacks. Much of my work will be to try to construct derived moduli stacks for a large class of problems. This has important applications in algebraic geometry, as there are many problems for which the moduli stacks are unmanageable, but which should become accessible using derived moduli stacks. I will also seek to investigate the geometry and behaviour of derived stacks themselves.

A common thread through the various aspects of my project will be to find ways of applying powerful ideas and techniques from a branch of topology, namely homotopy theory, in contexts where they would not, at first sight, appear to be relevant. Second descent and rational points on Kummer varieties Proceedings of the London Mathematical Society , to appear pdf , arXiv.

The abstract cotangent complex and Quillen cohomology of enriched categories with Joost Nuiten and Matan Prasma , Journal of Topology , 11 3 , , p.

### Programme Theme

Geometry and arithmetic of certain log K3 surface Annales de l'Institut Fourier , to appear pdf , arXiv. An integral model structure and truncation theory for coherent group actions with Matan Prasma , The Israel Journal of Mathematics , to appear pdf , arXiv. On the fibration method for zero-cycles and rational points with Olivier Wittenberg , Annals of Mathematics , 1 , , p.

The Grothendieck construction for model categories with Matan Prasma , Advances in Mathematics , , , p.

## Stable Homotopy Theory – University of Copenhagen

Algebraic and Geometric Topology , 15 4 , , p. The section conjecture for graphs and conical curves We show that the finite descent obstruction controls the existence of rational points on normal crossing singular curves whose components are all of genus 0, by relating the problem to a fix point property of pro-finite groups acting on pro-finite trees pdf , arXiv. Second 2-descent and rational points on Kummer surfaces , Rational points and algebraic geometry, Luminy September pdf. Higher additivity, higher monoids and the universal property of finite spans. Rational points on fibered varieties , Interactions between arithmetic and homotopy, Royal Imperial College, London, September pdf.

## Ben Williams

The descent-fibration method for integral points , Arithmetic geometry, Chow groups and rational points, The Euler Institute, St. Petersburg, June pdf. The descent-fibration method for integral points , Heilbronn seminar, university of Bristol, Bristol, May pdf. From linear equations in primes to the fibration method , The intercity number theory seminar, university of Groningen, October The section conjecture for graphs and applications for singular curves , London-Paris number theory seminar, Royal Imperial College, London, June Rational points on elliptic surfaces , notes for minicourse given at the IHP, spring pdf.

Arizona Winter School Selected projects by J. Some solutions pdf.

Simplicial homotopy theory , Master course, Radboud university pdf. General relativity , HUJI pdf. Elliptic regularity , HUJI pdf. Quasi-categories , Caesarea pdf. Complex cobordism and formal group laws , Caesarea pdf. Classification of framed manifolds , MIT Kan seminar pdf. Loop structures on the 3-sphere , MIT babytop seminar pdf.