Homotopy Theory and Arithmetic Geometry – Motivic and Diophantine Aspects, 1st ed. 2021 LMS-CMI Research School, London, July 2018 Lecture Notes in Mathematics Series, Vol. 2292

The contributions to this volume are based on original lectures by leading researchers at the LMS-CMI Research School on ?Homotopy Theory and Arithmetic Geometry - Motivic and Diophantine Aspects? and the Nelder Fellow Lecturer Series, which both took place at Imperial College London in the summer of 2018. The contribution by Brazelton, based on the lectures by Wickelgren, provides an introduction to arithmetic enumerative geometry, the notes of Cisinski present motivic sheaves and new cohomological methods for intersection theory, and Schlank?s contribution gives an overview of the use of étale homotopy theory for obstructions to the existence of rational points on algebraic varieties. Finally, the article by Asok and Østvær, based in part on the Nelder Fellow lecture series by Østvær, gives a survey of the interplay between motivic homotopy theory and affine algebraic geometry, with a focus on contractible algebraic varieties.

Now a major trend in arithmetic geometry, this volume offers a detailed guide to the fascinating circle of recent applications of homotopy theory to number theory. It will be invaluable to research students entering the field, as well as postdoctoral and more established researchers.

Ambrus Pál received his Ph.D. at Columbia University, New York. After visiting positions at the Institute for Advanced Study in Princeton, McGill University in Montréal and the IHES in Paris, he started to work at Imperial College London, United Kingdom, where he currently is an associate professor. His original area of research is the arithmetic of function fields. Over time his interests shifted towards other areas of arithmetic geometry, most notably p-adic cohomology. He is also interested in the arithmetic aspects of homotopy theory, for example he developed simplicial homotopy theory for algebraic varieties over real closed fields. With his former PhD student Christopher Lazda he also published an extensive research monograph in the Springer series Algebra and Applications entitled "Rigid cohomology over Laurent series fields" in which a new theory of p-adic cohomology for varieties over Laurent series fields in positive characteristic based on Berthelot's theory of rigid cohomology is developed.

Presents the state of the art in applications of homotopy theory to arithmetic geometry

A unique collection of original lecture notes aimed at research students

Contains lectures on étale and motivic homotopy theory, arithmetic enumerative geometry, and motives