Homotopy Theory and Arithmetic Geometry – Motivic and Diophantine Aspects
LMS-CMI Research School, London, July 2018
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Beschreibung
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.
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.
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
Autor*in
Frank Neumann
Themen in »Homotopy Theory and Arithmetic Geometry – Motivic and Diophantine Aspects«
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