Overview
- 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
Part of the book series: Lecture Notes in Mathematics (LNM, volume 2292)
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Table of contents (5 chapters)
Keywords
- Etale homotopy
- Rational points
- Infinity topoi
- Shape theory
- Brauer-Manin obstruction
- Motivic homotopy
- Enumerative geometry
- Motivic degree
- Milnor number
- Intersection theory
- Grothendieck-Verdier duality
- Etale motives
- Grothendieck-Lefschetz trace formula
- Contractible algebraic varieties
- Unstable homotopy
- Stable homotopy
About this book
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.
Editors and Affiliations
About the editors
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.
Bibliographic Information
Book Title: Homotopy Theory and Arithmetic Geometry – Motivic and Diophantine Aspects
Book Subtitle: LMS-CMI Research School, London, July 2018
Editors: Frank Neumann, Ambrus Pál
Series Title: Lecture Notes in Mathematics
DOI: https://doi.org/10.1007/978-3-030-78977-0
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
Softcover ISBN: 978-3-030-78976-3Published: 30 September 2021
eBook ISBN: 978-3-030-78977-0Published: 29 September 2021
Series ISSN: 0075-8434
Series E-ISSN: 1617-9692
Edition Number: 1
Number of Pages: IX, 218
Topics: Algebraic Geometry, Number Theory, Algebraic Topology