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Absolute Arithmetic and F1-Geometry

Thas, Koen,

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Absolute Arithmetic and F1-Geometry

Record Type:	Electronic resources : Monograph/item
Title/Author:	Absolute Arithmetic and F1-Geometry/ Koen Thas
other author:	Thas, Koen,
Published:	Zuerich, Switzerland :European Mathematical Society Publishing House, : 2016,
Description:	1 online resource (397 pages)
Subject:	Combinatorics & graph theory -
Online resource:	https://doi.org/10.4171/157
Online resource:	https://www.ems-ph.org/img/books/thas_mini_2016.jpg
ISBN:	9783037196571

Absolute Arithmetic and F1-Geometry
Absolute Arithmetic and F1-Geometry[electronic resource] /Koen Thas - Zuerich, Switzerland :European Mathematical Society Publishing House,2016 - 1 online resource (397 pages)

The Weyl functor. Introduction to Absolute Arithmetic /Koen Thas --

Restricted to subscribers:https://www.ems-ph.org/ebooks.php

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.

ISBN: 9783037196571

Standard No.: 10.4171/157doiSubjects--Topical Terms:

3480941
Combinatorics & graph theory

Absolute Arithmetic and F1-Geometry
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505 0 0 $t The Weyl functor. Introduction to Absolute Arithmetic / $r Koen Thas -- $t Belian categories / $r Anton Deitmar -- $t The combinatorial-motivic nature of $\mathbb F_1$-schemes / $r Koen Thas -- $t A blueprinted view on $\mathbb F_1$-geometry / $r Oliver Lorscheid -- $t Absolute geometry and the Habiro topology / $r Lieven Le Bruyn -- $t Witt vectors, semirings, and total positivity / $r James Borger -- $t Moduli operad over $\mathbb F_1$ / $r Yuri I. Manin, Matilde Marcolli -- $t A taste of Weil theory in characteristic one / $r Koen Thas.
506 1 $a Restricted to subscribers: $u https://www.ems-ph.org/ebooks.php
520 $a 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.
650 0 7 $a Combinatorics & graph theory $2 bicssc $3 3480941
650 0 7 $a Combinatorics $2 msc $3 3480913
650 0 7 $a Number theory $v Problems, exercises, etc. $3 1232300
650 0 7 $a Commutative rings and algebras $2 msc $3 3480799
650 0 7 $a Algebraic geometry $2 bicssc $3 3480846
700 1 $a Thas, Koen, $e editor. $3 3481123
856 4 0 $u https://doi.org/10.4171/157
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