It has been known for some time that geometries over finite fields, their automorphism groups and certain counting formulae involving these geometries have interesting guises when one lets the size of the field go to 1. On the other hand, the nonexistent field with one element, F1
, presents itself as a ghost candidate for an absolute basis in Algebraic Geometry to perform the Deninger–Manin program, which aims at solving the classical Riemann Hypothesis.
This book, which is the first of its kind in the F1
-world, covers several areas in F1
-theory, and is divided into four main parts – Combinatorial Theory, Homological Algebra, Algebraic Geometry and Absolute Arithmetic.
Topics treated include the combinatorial theory and geometry behind F1
, categorical foundations, the blend of different scheme theories over F1
which are presently available, motives and zeta functions, the Habiro topology, Witt vectors and total positivity, moduli operads, and at the end, even some arithmetic.
Each chapter is carefully written by experts, and besides elaborating on known results, brand new results, open problems and conjectures are also met along the way.
The diversity of the contents, together with the mystery surrounding the field with one element, should attract any mathematician, regardless of speciality.
Keywords: The field with one element, F1
-geometry, combinatorial F1-geometry, non-additive category, Deitmar scheme, graph, monoid, motive, zeta function, automorphism group, blueprint, Euler characteristic, K-theory, Grassmannian, Witt ring, noncommutative geometry, Witt vector, total positivity, moduli space of curves, operad, torificiation, Absolute Arithmetic, counting function, Weil conjectures, Riemann Hypothesis

lgli/K:\!genesis\0day\kolxoz\83\M_Mathematics\MA_Algebra\MAg_Algebraic geometry\Thas K. (ed.) Absolute arithmetic and F1-geometry (EMS, 2016)(ISBN 9783037191576)(O)(399s)_MAg_.pdf

lgrsnf/K:\!genesis\0day\kolxoz\83\M_Mathematics\MA_Algebra\MAg_Algebraic geometry\Thas K. (ed.) Absolute arithmetic and F1-geometry (EMS, 2016)(ISBN 9783037191576)(O)(399s)_MAg_.pdf

lgli/M_Mathematics/MA_Algebra/MAg_Algebraic geometry/Thas K. (ed.) Absolute arithmetic and F1-geometry (EMS, 2016)(ISBN 9783037191576)(O)(399s)_MAg_.pdf

nexusstc/Absolute Arithmetic and F1-geometry/fc0f4e5a2e26a2bc5cbf147948755a5e.pdf

zlib/Mathematics/Koen Thas, Koen Thas (eds.)/Absolute Arithmetic and F1-geometry_3374003.pdf

Absolute Arithmetic And F1-geometry (ems European Mathematical Society)

Absolute Arithmetic and F1 Geometry

Koen Thas; Société mathématique européenne

Koen Thas; European Mathematical Society

Koen Thas, editor

European Mathematical Society, Thomas Hintermann

Amer Mathematical Society

publisher not identified

EMS Press

European Mathematical Society - EMS - Publishing House GmbH, Zürich, Switzerland, 2016

Place of publication not identified, 2016

Zürich, Switzerland, Switzerland, 2016

Zuerich, Switzerland, 2016

Zürich, cop. 2016

Providence, 2016

kolxoz -- 83

lg2132299

producers:
pdfTeX-1.40.14

{"isbns":["3037191570","3037196572","9783037191576","9783037196571"],"last_page":397,"publisher":"European Mathematical Society","series":"EMS European Mathematical Society"}

Includes bibliographical references and index.

It has been known for some time that geometries over finite fields, their automorphism groups and certain counting formulae involving these geometries have interesting guises when one lets the size of the field go to 1. On the other hand, the nonexistent field with one element, $\mathbb F_1$, presents itself as a ghost candidate for an absolute basis in Algebraic Geometry to perform the Deninger–Manin program, which aims at solving the classical Riemann Hypothesis. This book, which is the first of its kind in the $\mathbb F_1$-world, covers several areas in $\mathbb F_1$-theory, and is divided into four main parts – Combinatorial Theory, Homological Algebra, Algebraic Geometry and Absolute Arithmetic. Topics treated include the combinatorial theory and geometry behind $\mathbb F_1$, categorical foundations, the blend of different scheme theories over $\mathbb F_1$ which are presently available, motives and zeta functions, the Habiro topology, Witt vectors and total positivity, moduli operads, and at the end, even some arithmetic. Each chapter is carefully written by experts, and besides elaborating on known results, brand new results, open problems and conjectures are also met along the way. The diversity of the contents, together with the mystery surrounding the field with one element, should attract any mathematician, regardless of speciality.

It has been known for some time that geometries over finite fields, their automorphism groups and certain counting formulae involving these geometries have interesting guises when one lets the size of the field go to 1. On the other hand, the nonexistent field with one element, F1, presents itself as a ghost candidate for an absolute basis in Algebraic Geometry to perform the Deninger–Manin program, which aims at solving the classical Riemann Hypothesis. This book, which is the first of its kind in the F1-world, covers several areas in F1-theory, and is divided into four main parts – Combinatorial Theory, Homological Algebra, Algebraic Geometry and Absolute Arithmetic. Topics treated include the combinatorial theory and geometry behind F1, categorical foundations, the blend of different scheme theories over F1 which are presently available, motives and zeta functions, the Habiro topology, Witt vectors and total positivity, moduli operads, and at the end, even some arithmetic. Each chapter is carefully written by experts, and besides elaborating on known results, brand new results, open problems and conjectures are also met along the way. The diversity of the contents, together with the mystery surrounding the field with one element, should attract any mathematician, regardless of speciality.
Erscheinungsdatum: 01.07.2016

It has been known for some time that geometries over finite fields, their automorphism groups and certain counting formulae involving these geometries have interesting guises when one lets the size of the field go to 1. On the other hand, the nonexistent field with one element, F1, presents itself as a ghost candidate for an absolute basis in Algebraic Geometry to perform the Deninger-Manin program, which aims at solving the classical Riemann Hypothesis. This book, which is the first of its kind in the F1-world, covers several areas in F1-theory and is divided into four main parts: Combinatorial Theory, Homological Algebra, Algebraic Geometry and Absolute Arithmetic. Topics treated include the combinatorial theory and geometry behind F1, categorical foundations, the blend of different scheme theories over F1 which are presently available, motives and zeta functions, the Habiro topology, Witt vectors and total positivity, moduli operads, and at the end, even some arithmetic. Each chapter is carefully written by an expert. In addition to elaborating on known results, the authors introduce brand-new results, open problems and conjectures. The diversity of the contents, together with the mystery surrounding the field with one element, should attract any mathematician, regardless of speciality

A casual preface 6
Table of Contents 8
Combinatorial Theory 10
The Weyl functor—Introduction to Absolute Arithmetic Koen Thas 12
Homological Algebra 46
Belian categories Anton Deitmar 48
Algebraic Geometry 90
The combinatorial-motivic nature of F1-schemes Koen Thas 92
A blueprinted view on F1-geometry Oliver Lorscheid 170
Absolute geometry and the Habiro topology Lieven Le Bruyn 230
Witt vectors, semirings, and total positivity James Borger 282
Moduli operad over F1 Yuri I. Manin and Matilde Marcolli 340
Absolute Arithmetic 372
A taste of Weil theory in characteristic one Koen Thas 374
List of Contributors 396

Content: The Weyl functor : introduction to absolute arithmetic / Koen Thas --
Belian categories / Anton Deitmar --
The combinatorial-motivic nature of F1-schemes / Koen Thas --
A blueprinted view on F1-geometry / Oliver Lorscheid --
Absolute geometry and the Habiro topology / Lieven Le Bruyn --
Witt vectors, semirings, and total positivity / James Borger --
Moduli operad over F1 / Yuri I. Manin and Matilde Marcolli --
A taste of Weil theory in characteristic one / Koen Thas.

2017-10-15