Homotopy Theory and Arithmetic Geometry – Motivic and Diophantine Aspects

1. Front Matter

   Pages i-ix

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2. Homotopy Theory and Arithmetic Geometry—Motivic and Diophantine Aspects: An Introduction

   • Frank Neumann, Ambrus Pál

   Pages 1-9

3. An Introduction to \(\mathbb {A}^1\)-Enumerative Geometry

   • Thomas Brazelton

   Pages 11-47

4. Cohomological Methods in Intersection Theory

   • Denis-Charles Cisinski

   Pages 49-105

5. Étale Homotopy and Obstructions to Rational Points

   • Tomer M. Schlank

   Pages 107-143

6. \({\mathbb A}^1\)-homotopy Theory and Contractible Varieties: A Survey

   • Aravind Asok, Paul Arne Østvær

   Pages 145-212

7. Back Matter

   Pages 213-218

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This book provides an introduction to state-of-the-art applications of homotopy theory to arithmetic geometry.

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.

• 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

• School of Computing and Mathematical Sciences, University of Leicester, Leicester, UK

  Frank Neumann

• Department of Mathematics, Imperial College London, London, UK

  Ambrus Pál

Frank Neumann received his Ph.D. at the University of Göttingen. After holding a postdoctoral position at the University of Göttingen and an EU Marie Curie fellowship at the Centre de Recerca Matemática (CRM) in Barcelona, he started to work at the University of Leicester, United Kingdom, where he currently is an associate professor. His original area of research is algebraic topology and in particularly homotopy theory. Over time his interests shifted towards interactions between algebraic topology and algebraic geometry and his current work is especially on the homotopy theory and arithmetic of moduli stacks. His research has direct links with mathematical physics. He was also a visiting professor at TIFR in Mumbai, IMPA in Rio de Janeiro, Isaac Newton Institute for Mathematical Sciences in Cambridge, CRM in Barcelona, Steklov Institute Moscow, CIMAT in Guanajuato, and the University of Chicago.

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.

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