Homotopy Theory And Arithmetic Geometry Motivic And Diophantine Aspects

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Homotopy Theory and Arithmetic Geometry – Motivic and Diophantine Aspects

Frank Neumann,Ambrus Pál
Homotopy Theory and Arithmetic Geometry Motivic and Diophantine Aspects Book PDF

Author : Frank Neumann,Ambrus Pál
Publisher : Springer Nature
Page : 223 pages
File Size : 42,8 Mb
Release : 2021-09-29
Category : Mathematics
ISBN : 9783030789770

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Homotopy Theory and Arithmetic Geometry – Motivic and Diophantine Aspects by Frank Neumann,Ambrus Pál Pdf

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.

Homotopy Theory and Arithmetic Geometry - Motivic and Diophantine Aspects

Frank Neumann,Ambrus Pál
Homotopy Theory and Arithmetic Geometry Motivic and Diophantine Aspects Book PDF

Author : Frank Neumann,Ambrus Pál
Publisher : Unknown
Page : 0 pages
File Size : 52,6 Mb
Release : 2021
Category : Electronic
ISBN : 3030789780

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Homotopy Theory and Arithmetic Geometry - Motivic and Diophantine Aspects by Frank Neumann,Ambrus Pál Pdf

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.

Motivic Homotopy Theory and Refined Enumerative Geometry

Federico Binda,Marc Levine,Manh Toan Nguyen,Oliver Röndigs
Motivic Homotopy Theory and Refined Enumerative Geometry Book PDF

Author : Federico Binda,Marc Levine,Manh Toan Nguyen,Oliver Röndigs
Publisher : American Mathematical Soc.
Page : 267 pages
File Size : 44,7 Mb
Release : 2020-03-09
Category : Education
ISBN : 9781470448981

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Motivic Homotopy Theory and Refined Enumerative Geometry by Federico Binda,Marc Levine,Manh Toan Nguyen,Oliver Röndigs Pdf

This volume contains the proceedings of the Workshop on Motivic Homotopy Theory and Refined Enumerative Geometry, held from May 14–18, 2018, at the Universität Duisburg-Essen, Essen, Germany. It constitutes an accessible yet swift introduction to a new and active area within algebraic geometry, which connects well with classical intersection theory. Combining both lecture notes aimed at the graduate student level and research articles pointing towards the manifold promising applications of this refined approach, it broadly covers refined enumerative algebraic geometry.

Triangulated Categories of Mixed Motives

Denis-Charles Cisinski,Frédéric Déglise
Triangulated Categories of Mixed Motives Book PDF

Author : Denis-Charles Cisinski,Frédéric Déglise
Publisher : Springer Nature
Page : 406 pages
File Size : 41,6 Mb
Release : 2019-11-09
Category : Mathematics
ISBN : 9783030332426

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Triangulated Categories of Mixed Motives by Denis-Charles Cisinski,Frédéric Déglise Pdf

The primary aim of this monograph is to achieve part of Beilinson’s program on mixed motives using Voevodsky’s theories of A1-homotopy and motivic complexes. Historically, this book is the first to give a complete construction of a triangulated category of mixed motives with rational coefficients satisfying the full Grothendieck six functors formalism as well as fulfilling Beilinson’s program, in particular the interpretation of rational higher Chow groups as extension groups. Apart from Voevodsky’s entire work and Grothendieck’s SGA4, our main sources are Gabber’s work on étale cohomology and Ayoub’s solution to Voevodsky’s cross functors theory. We also thoroughly develop the theory of motivic complexes with integral coefficients over general bases, along the lines of Suslin and Voevodsky. Besides this achievement, this volume provides a complete toolkit for the study of systems of coefficients satisfying Grothendieck’ six functors formalism, including Grothendieck-Verdier duality. It gives a systematic account of cohomological descent theory with an emphasis on h-descent. It formalizes morphisms of coefficient systems with a view towards realization functors and comparison results. The latter allows to understand the polymorphic nature of rational mixed motives. They can be characterized by one of the following properties: existence of transfers, universality of rational algebraic K-theory, h-descent, étale descent, orientation theory. This monograph is a longstanding research work of the two authors. The first three parts are written in a self-contained manner and could be accessible to graduate students with a background in algebraic geometry and homotopy theory. It is designed to be a reference work and could also be useful outside motivic homotopy theory. The last part, containing the most innovative results, assumes some knowledge of motivic homotopy theory, although precise statements and references are given.

Elements of Homotopy Theory

George W. Whitehead
Elements of Homotopy Theory Book PDF

Author : George W. Whitehead
Publisher : Springer Science & Business Media
Page : 764 pages
File Size : 52,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461263180

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Elements of Homotopy Theory by George W. Whitehead Pdf

As the title suggests, this book is concerned with the elementary portion of the subject of homotopy theory. It is assumed that the reader is familiar with the fundamental group and with singular homology theory, including the Universal Coefficient and Kiinneth Theorems. Some acquaintance with manifolds and Poincare duality is desirable, but not essential. Anyone who has taught a course in algebraic topology is familiar with the fact that a formidable amount of technical machinery must be introduced and mastered before the simplest applications can be made. This phenomenon is also observable in the more advanced parts of the subject. I have attempted to short-circuit it by making maximal use of elementary methods. This approach entails a leisurely exposition in which brevity and perhaps elegance are sacrificed in favor of concreteness and ease of application. It is my hope that this approach will make homotopy theory accessible to workers in a wide range of other subjects-subjects in which its impact is beginning to be felt. It is a consequence of this approach that the order of development is to a certain extent historical. Indeed, if the order in which the results presented here does not strictly correspond to that in which they were discovered, it nevertheless does correspond to an order in which they might have been discovered had those of us who were working in the area been a little more perspicacious.

Homotopy Theory via Algebraic Geometry and Group Representations

Mark E. Mahowald
Homotopy Theory via Algebraic Geometry and Group Representations Book PDF

Author : Mark E. Mahowald
Publisher : American Mathematical Soc.
Page : 394 pages
File Size : 44,8 Mb
Release : 1998
Category : Geometry, Algebraic
ISBN : 9780821808054

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Homotopy Theory via Algebraic Geometry and Group Representations by Mark E. Mahowald Pdf

The academic year 1996-97 was designated as a special year in Algebraic Topology at Northwestern University (Evanston, IL). In addition to guest lecturers and special courses, an international conference was held entitled "Current trends in algebraic topology with applications to algebraic geometry and physics". The series of plenary lectures included in this volume indicate the great breadth of the conference and the lively interaction that took place among various areas of mathematics. Original research papers were submitted, and all submissions were refereed to the usual journal standards.

Introduction to Homotopy Theory

Martin Arkowitz
Introduction to Homotopy Theory Book PDF

Author : Martin Arkowitz
Publisher : Springer Science & Business Media
Page : 352 pages
File Size : 52,6 Mb
Release : 2011-07-25
Category : Mathematics
ISBN : 9781441973290

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Introduction to Homotopy Theory by Martin Arkowitz Pdf

This is a book in pure mathematics dealing with homotopy theory, one of the main branches of algebraic topology. The principal topics are as follows: Basic Homotopy; H-spaces and co-H-spaces; fibrations and cofibrations; exact sequences of homotopy sets, actions, and coactions; homotopy pushouts and pullbacks; classical theorems, including those of Serre, Hurewicz, Blakers-Massey, and Whitehead; homotopy Sets; homotopy and homology decompositions of spaces and maps; and obstruction theory. The underlying theme of the entire book is the Eckmann-Hilton duality theory. The book can be used as a text for the second semester of an advanced ungraduate or graduate algebraic topology course.

Motivic Homotopy Theory

Bjorn Ian Dundas,Marc Levine,P.A. Østvær,Oliver Röndigs,Vladimir Voevodsky
Motivic Homotopy Theory Book PDF

Author : Bjorn Ian Dundas,Marc Levine,P.A. Østvær,Oliver Röndigs,Vladimir Voevodsky
Publisher : Springer
Page : 226 pages
File Size : 47,6 Mb
Release : 2009-09-02
Category : Mathematics
ISBN : 3540830987

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Motivic Homotopy Theory by Bjorn Ian Dundas,Marc Levine,P.A. Østvær,Oliver Röndigs,Vladimir Voevodsky Pdf

This book is based on lectures given at a summer school on motivic homotopy theory at the Sophus Lie Centre in Nordfjordeid, Norway, in August 2002. Aimed at graduate students in algebraic topology and algebraic geometry, it contains background material from both of these fields, as well as the foundations of motivic homotopy theory. It will serve as a good introduction as well as a convenient reference for a broad group of mathematicians to this important and fascinating new subject. Vladimir Voevodsky is one of the founders of the theory and received the Fields medal for his work, and the other authors have all done important work in the subject.

Homotopy Theory with Bornological Coarse Spaces

Ulrich Bunke,Alexander Engel
Homotopy Theory with Bornological Coarse Spaces Book PDF

Author : Ulrich Bunke,Alexander Engel
Publisher : Springer Nature
Page : 248 pages
File Size : 41,8 Mb
Release : 2020-09-03
Category : Mathematics
ISBN : 9783030513351

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Homotopy Theory with Bornological Coarse Spaces by Ulrich Bunke,Alexander Engel Pdf

Providing a new approach to assembly maps, this book develops the foundations of coarse homotopy using the language of infinity categories. It introduces the category of bornological coarse spaces and the notion of a coarse homology theory, and further constructs the universal coarse homology theory. Hybrid structures are introduced as a tool to connect large-scale with small-scale geometry, and are then employed to describe the coarse motives of bornological coarse spaces of finite asymptotic dimension. The remainder of the book is devoted to the construction of examples of coarse homology theories, including an account of the coarsification of locally finite homology theories and of coarse K-theory. Thereby it develops background material about locally finite homology theories and C*-categories. The book is intended for advanced graduate students and researchers who want to learn about the homotopy-theoretical aspects of large scale geometry via the theory of infinity categories.

Motivic Homotopy Theory

Bjorn Ian Dundas,Marc Levine,P.A. Østvær,Oliver Röndigs,Vladimir Voevodsky
Motivic Homotopy Theory Book PDF

Author : Bjorn Ian Dundas,Marc Levine,P.A. Østvær,Oliver Röndigs,Vladimir Voevodsky
Publisher : Springer Science & Business Media
Page : 228 pages
File Size : 51,7 Mb
Release : 2007-07-11
Category : Mathematics
ISBN : 9783540458975

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Motivic Homotopy Theory by Bjorn Ian Dundas,Marc Levine,P.A. Østvær,Oliver Röndigs,Vladimir Voevodsky Pdf

This book is based on lectures given at a summer school on motivic homotopy theory at the Sophus Lie Centre in Nordfjordeid, Norway, in August 2002. Aimed at graduate students in algebraic topology and algebraic geometry, it contains background material from both of these fields, as well as the foundations of motivic homotopy theory. It will serve as a good introduction as well as a convenient reference for a broad group of mathematicians to this important and fascinating new subject. Vladimir Voevodsky is one of the founders of the theory and received the Fields medal for his work, and the other authors have all done important work in the subject.

Lecture Notes on Motivic Cohomology

Carlo Mazza,Vladimir Voevodsky,Charles A. Weibel
Lecture Notes on Motivic Cohomology Book PDF

Author : Carlo Mazza,Vladimir Voevodsky,Charles A. Weibel
Publisher : American Mathematical Soc.
Page : 240 pages
File Size : 47,6 Mb
Release : 2006
Category : Mathematics
ISBN : 0821838474

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Lecture Notes on Motivic Cohomology by Carlo Mazza,Vladimir Voevodsky,Charles A. Weibel Pdf

The notion of a motive is an elusive one, like its namesake "the motif" of Cezanne's impressionist method of painting. Its existence was first suggested by Grothendieck in 1964 as the underlying structure behind the myriad cohomology theories in Algebraic Geometry. We now know that there is a triangulated theory of motives, discovered by Vladimir Voevodsky, which suffices for the development of a satisfactory Motivic Cohomology theory. However, the existence of motives themselves remains conjectural. This book provides an account of the triangulated theory of motives. Its purpose is to introduce Motivic Cohomology, to develop its main properties, and finally to relate it to other known invariants of algebraic varieties and rings such as Milnor K-theory, etale cohomology, and Chow groups. The book is divided into lectures, grouped in six parts. The first part presents the definition of Motivic Cohomology, based upon the notion of presheaves with transfers. Some elementary comparison theorems are given in this part. The theory of (etale, Nisnevich, and Zariski) sheaves with transfers is developed in parts two, three, and six, respectively. The theoretical core of the book is the fourth part, presenting the triangulated category of motives. Finally, the comparison with higher Chow groups is developed in part five. The lecture notes format is designed for the book to be read by an advanced graduate student or an expert in a related field. The lectures roughly correspond to one-hour lectures given by Voevodsky during the course he gave at the Institute for Advanced Study in Princeton on this subject in 1999-2000. In addition, many of the original proofs have been simplified and improved so that this book will also be a useful tool for research mathematicians. Information for our distributors: Titles in this series are copublished with the Clay Mathematics Institute (Cambridge, MA).

Modular Forms and Fermat’s Last Theorem

Gary Cornell,Joseph H. Silverman,Glenn Stevens
Modular Forms and Fermat s Last Theorem Book PDF

Author : Gary Cornell,Joseph H. Silverman,Glenn Stevens
Publisher : Springer Science & Business Media
Page : 592 pages
File Size : 47,8 Mb
Release : 2013-12-01
Category : Mathematics
ISBN : 9781461219743

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Modular Forms and Fermat’s Last Theorem by Gary Cornell,Joseph H. Silverman,Glenn Stevens Pdf

This volume contains the expanded lectures given at a conference on number theory and arithmetic geometry held at Boston University. It introduces and explains the many ideas and techniques used by Wiles, and to explain how his result can be combined with Ribets theorem and ideas of Frey and Serre to prove Fermats Last Theorem. The book begins with an overview of the complete proof, followed by several introductory chapters surveying the basic theory of elliptic curves, modular functions and curves, Galois cohomology, and finite group schemes. Representation theory, which lies at the core of the proof, is dealt with in a chapter on automorphic representations and the Langlands-Tunnell theorem, and this is followed by in-depth discussions of Serres conjectures, Galois deformations, universal deformation rings, Hecke algebras, and complete intersections. The book concludes by looking both forward and backward, reflecting on the history of the problem, while placing Wiles'theorem into a more general Diophantine context suggesting future applications. Students and professional mathematicians alike will find this an indispensable resource.

Lectures on Homotopy Theory

R.A. Piccinini
Lectures on Homotopy Theory Book PDF

Author : R.A. Piccinini
Publisher : Elsevier
Page : 292 pages
File Size : 47,6 Mb
Release : 1992-01-21
Category : Mathematics
ISBN : 0080872824

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Lectures on Homotopy Theory by R.A. Piccinini Pdf

The central idea of the lecture course which gave birth to this book was to define the homotopy groups of a space and then give all the machinery needed to prove in detail that the nth homotopy group of the sphere Sn, for n greater than or equal to 1 is isomorphic to the group of the integers, that the lower homotopy groups of Sn are trivial and that the third homotopy group of S2 is also isomorphic to the group of the integers. All this was achieved by discussing H-spaces and CoH-spaces, fibrations and cofibrations (rather thoroughly), simplicial structures and the homotopy groups of maps. Later, the book was expanded to introduce CW-complexes and their homotopy groups, to construct a special class of CW-complexes (the Eilenberg-Mac Lane spaces) and to include a chapter devoted to the study of the action of the fundamental group on the higher homotopy groups and the study of fibrations in the context of a category in which the fibres are forced to live; the final material of that chapter is a comparison of various kinds of universal fibrations. Completing the book are two appendices on compactly generated spaces and the theory of colimits. The book does not require any prior knowledge of Algebraic Topology and only rudimentary concepts of Category Theory are necessary; however, the student is supposed to be well at ease with the main general theorems of Topology and have a reasonable mathematical maturity.

Homotopy Theory: Relations with Algebraic Geometry, Group Cohomology, and Algebraic $K$-Theory

Paul Gregory Goerss,Stewart Priddy,International Conference on Algebraic Topology
Homotopy Theory Relations with Algebraic Geometry Group Cohomology and Algebraic K Theory Book PDF

Author : Paul Gregory Goerss,Stewart Priddy,International Conference on Algebraic Topology
Publisher : American Mathematical Soc.
Page : 520 pages
File Size : 46,7 Mb
Release : 2004
Category : Homotopy theory
ISBN : 9780821832851

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Homotopy Theory: Relations with Algebraic Geometry, Group Cohomology, and Algebraic $K$-Theory by Paul Gregory Goerss,Stewart Priddy,International Conference on Algebraic Topology Pdf

As part of its series of Emphasis Years in Mathematics, Northwestern University hosted an International Conference on Algebraic Topology. The purpose of the conference was to develop new connections between homotopy theory and other areas of mathematics. This proceedings volume grew out of that event. Topics discussed include algebraic geometry, cohomology of groups, algebraic $K$-theory, and $\mathbb{A 1$ homotopy theory. Among the contributors to the volume were Alejandro Adem,Ralph L. Cohen, Jean-Louis Loday, and many others. The book is suitable for graduate students and research mathematicians interested in homotopy theory and its relationship to other areas of mathematics.

Rational Points and Arithmetic of Fundamental Groups

Jakob Stix
Rational Points and Arithmetic of Fundamental Groups Book PDF

Author : Jakob Stix
Publisher : Springer
Page : 257 pages
File Size : 40,5 Mb
Release : 2012-10-19
Category : Mathematics
ISBN : 9783642306747

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Rational Points and Arithmetic of Fundamental Groups by Jakob Stix Pdf

The section conjecture in anabelian geometry, announced by Grothendieck in 1983, is concerned with a description of the set of rational points of a hyperbolic algebraic curve over a number field in terms of the arithmetic of its fundamental group. While the conjecture is still open today in 2012, its study has revealed interesting arithmetic for curves and opened connections, for example, to the question whether the Brauer-Manin obstruction is the only one against rational points on curves. This monograph begins by laying the foundations for the space of sections of the fundamental group extension of an algebraic variety. Then, arithmetic assumptions on the base field are imposed and the local-to-global approach is studied in detail. The monograph concludes by discussing analogues of the section conjecture created by varying the base field or the type of variety, or by using a characteristic quotient or its birational analogue in lieu of the fundamental group extension.