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100   $a20160630j20160725  fy             0
101 0 $aeng
135   $aurnn|mmmmamaa
181   $ctxt$2rdacontent
182   $cc$2rdamedia
183   $acr$2rdacarrier
200 10$aAbsolute Arithmetic and $\mathbb F_1$-Geometry$b[electronic resource] /$fKoen Thas
210 3 $aZuerich, Switzerland $cEuropean Mathematical Society Publishing House$d2016
215   $a1 online resource (397 pages)
311   $a3-03719-157-0 
327   $tThe Weyl functor. Introduction to Absolute Arithmetic /$rKoen Thas --$tBelian categories /$rAnton Deitmar --$tThe combinatorial-motivic nature of $\mathbb F_1$-schemes /$rKoen Thas --$tA blueprinted view on $\mathbb F_1$-geometry /$rOliver Lorscheid --$tAbsolute geometry and the Habiro topology /$rLieven Le Bruyn --$tWitt vectors, semirings, and total positivity /$rJames Borger --$tModuli operad over $\mathbb F_1$ /$rYuri I. Manin, Matilde Marcolli --$tA taste of Weil theory in characteristic one /$rKoen Thas.
330   $aIt 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.
606   $aCombinatorics & graph theory$2bicssc
606   $aCombinatorics$2msc
606   $aNumber theory$2msc
606   $aCommutative rings and algebras$2msc
606   $aAlgebraic geometry$2msc
615 07$aCombinatorics & graph theory
615 07$aCombinatorics
615 07$aNumber theory
615 07$aCommutative rings and algebras
615 07$aAlgebraic geometry
686   $a05-xx$a11-xx$a13-xx$a14-xx$2msc
701   $aThas$b Koen$f1977-$0726617
801  0$bch0018173
906   $aBOOK
912   $a9910153279503321
996   $aAbsolute Arithmetic and$92564511
997   $aUNINA