LEADER 03318nam 22004695a 450 001 9910153279503321 005 20160630234501.0 010 $a3-03719-657-2 024 70$a10.4171/157 035 $a(CKB)3340000000002766 035 $a(CH-001817-3)204-160630 035 $a(PPN)194913732 035 $a(EXLCZ)993340000000002766 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 LEADER 04856nam 22007455 450 001 996466663503316 005 20200701010914.0 010 $a3-642-23669-3 024 7 $a10.1007/978-3-642-23669-3 035 $a(CKB)3390000000021680 035 $a(SSID)ssj0000609363 035 $a(PQKBManifestationID)11433923 035 $a(PQKBTitleCode)TC0000609363 035 $a(PQKBWorkID)10625703 035 $a(PQKB)11413119 035 $a(DE-He213)978-3-642-23669-3 035 $a(MiAaPQ)EBC3070395 035 $a(PPN)159084725 035 $a(EXLCZ)993390000000021680 100 $a20120104d2012 u| 0 101 0 $aeng 135 $aurnn|008mamaa 181 $ctxt 182 $cc 183 $acr 200 10$aComplex Monge?Ampère Equations and Geodesics in the Space of Kähler Metrics$b[electronic resource] /$fedited by Vincent Guedj 205 $a1st ed. 2012. 210 1$aBerlin, Heidelberg :$cSpringer Berlin Heidelberg :$cImprint: Springer,$d2012. 215 $a1 online resource (VIII, 310 p. 4 illus.) 225 1 $aLecture Notes in Mathematics,$x0075-8434 ;$v2038 300 $aBibliographic Level Mode of Issuance: Monograph 311 $a3-642-23668-5 320 $aIncludes bibliographical references. 327 $a1.Introduction -- I. The Local Homogenious Dirichlet Problem.-2. Dirichlet Problem in Domains of Cn -- 3. Geometric Maximality -- II. Stochastic Analysis for the Monge-Ampère Equation -- 4. Probabilistic Approach to Regularity -- III. Monge-Ampère Equations on Compact Manifolds -- 5.The Calabi-Yau Theorem -- IV Geodesics in the Space of Kähler Metrics -- 6. The Riemannian Space of Kähler Metrics -- 7. MA Equations on Manifolds with Boundary -- 8. Bergman Geodesics. 330 $aThe purpose of these lecture notes is to provide an introduction to the theory of complex Monge?Ampère operators (definition, regularity issues, geometric properties of solutions, approximation) on compact Kähler manifolds (with or without boundary). These operators are of central use in several fundamental problems of complex differential geometry (Kähler?Einstein equation, uniqueness of constant scalar curvature metrics), complex analysis and dynamics. The topics covered include, the Dirichlet problem (after Bedford?Taylor), Monge?Ampère foliations and laminated currents, polynomial hulls and Perron envelopes with no analytic structure, a self-contained presentation of Krylov regularity results, a modernized proof of the Calabi?Yau theorem (after Yau and Kolodziej), an introduction to infinite dimensional riemannian geometry, geometric structures on spaces of Kähler metrics (after Mabuchi, Semmes and Donaldson), generalizations of the regularity theory of Caffarelli?Kohn?Nirenberg?Spruck (after Guan, Chen and Blocki) and Bergman approximation of geodesics (after Phong?Sturm and Berndtsson). Each chapter can be read independently and is based on a series of lectures by R. Berman, Z. Blocki, S. Boucksom, F. Delarue, R. Dujardin, B. Kolev and A. Zeriahi, delivered to non-experts. 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