06168nam a22004095a 4500001001100000003001200011005001700023006001900040007001500059008004100074020001800115024002100133040001400154072001700168084003600185245012000221260008200341300003400423336002600457337002600483338003600509347002400545490005400569505267500623506006603298520200503364650002705369650003805396650005505434650004005489650002805529700003205557700003705589700003205626856003205658856006805690182-140804CH-001817-320140804234500.0a fot ||| 0|cr nn mmmmamaa140804e20140901sz fot ||| 0|eng d a978303719649670a10.4171/1492doi ach0018173 7aPBFL2bicssc a12-xxa06-xxa13-xxa14-xx2msc10aValuation Theory in Interactionh[electronic resource] /cAntonio Campillo, Franz-Viktor Kuhlmann, Bernard Teissier3 aZuerich, Switzerland :bEuropean Mathematical Society Publishing House,c2014 a1 online resource (670 pages) atextbtxt2rdacontent acomputerbc2rdamedia aonline resourcebcr2rdacarrier atext filebPDF2rda0 aEMS Series of Congress Reports (ECR) ;x2523-515X00tA study of irreducible polynomials over henselian valued fields via distinguished pairs /rKamal Aghigh, Anuj Bishnoi, Sudesh K. Khanduja, Sanjeev Kumar --tOn fields of totally $\mathfrak{S}$-adic numbers. With an appendix by Florian Pop /rLior Bary-Soroker, Arno Fehm --tInfinite towers of Artin-Schreier defect extensions of rational function fields /rAnna Blaszczok --tA refinement of Izumi's Theorem /rSébastien Boucksom, Charles Favre, Mattias Jonsson --tMultivariable Hodge theoretical invariants of germs of plane curves. II /rPierrette Cassou-Noguès, Anatoly Libgober --tExistence des diviseurs dicritiques, d’après S.S. Abhyankar /rVincent Cossart, Mickaël Matusinski, Guillermo Moreno-Socías --tInvariants of the graded algebras associated to divisorial valuations dominating a rational surface singularity /rVincent Cossart, Olivier Piltant, Ana J. Reguera --tAn introduction to $C$-minimal structures and their cell decomposition theorem /rPablo Cubides Kovacsics --tValuation semigroups of Noetherian local domains /rSteven Dale Cutkosky --tAdditive polynomials over perfect fields /rSalih Durhan --tOn $\mathbb{R}$-places and related topics /rDanielle Gondard-Cozette --tExtending valuations to formal completions /rFrancisco Javier Herrera Govantes, Miguel Ángel Olalla Acosta, Mark Spivakovsky, Bernard Teissier --tExtending real valuations to skew polynomial rings /rÁngel Granja, M. C. Martínez, C. Rodríguez --tStratifications in valued fields /rImmanuel Halupczok --tImaginaries and definable types in algebraically closed valued fields /rEhud Hrushovski --tDefects of algebraic function fields, completion defects and defect quotients /rFranz-Viktor Kuhlmann, Asim Naseem --tOn generalized series fields and exponential-logarithmic series fields with derivations /rMickaël Matusinski --tJet schemes of rational double point singularities /rHussein Mourtada --tValuations centered at a two-dimensional regular local domain: infima and topologies /rJosnei Novacoski --tReduction of local uniformization to the rank one case /rJosnei Novacoski, Mark Spivakovsky --tLittle survey on large fields - old & new /rFlorian Pop --tQuasi-valuations -- topology and the weak approximation theorem /rShai Sarussi --tOverweight deformations of affine toric varieties and local uniformization /rBernard Teissier --tDetecting valuations using small Galois groups /rAdam Topaz --tTruncation in Hahn fields /rLou van den Dries --tThe ergodicity of 1-Lipschitz transformations on 2-adic spheres /rEkaterina Yurova --tRamification of higher local fields approaches and questions /rLiang Xiao, Igor Zhukov.1 aRestricted to subscribers:uhttps://www.ems-ph.org/ebooks.php aHaving its classical roots, since more than a century, in algebraic number theory, algebraic geometry and the theory of ordered fields and groups, valuation theory has seen an amazing expansion into many other areas in recent decades. Moreover, having been dormant for a while in algebraic geometry, it has now been reintroduced as a tool to attack the open problem of resolution of singularities in positive characteristic and to analyse the structure of singularities. Driven by this topic, and by its many new applications in other areas, also the research in valuation theory itself has been intensified, with a particular emphasis on the deep open problems in positive characteristic.
As important examples for the expansion of valuation theory, it has become extremely useful in the theory of complex dynamical systems, and in the study of non-oscillating trajectories of real analytic vector fields in three dimensions. Analogues of the Riemann-Zariski valuation spaces have been found to be the proper framework for questions of intersection theory in algebraic geometry and in the analysis of singularities of complex plurisubharmonic functions. In a different direction, the relation between Berkovich geometry, tropical geometry and valuation spaces, on the one hand, and the geometry of arc spaces and valuation spaces, on the other, have begun to deepen and clarify.
Ever since its beginnings, valuation theory and Galois theory have grown closely together and influenced each other. Arguably, studying and understanding the extensions of valuations in algebraic field extensions is one of the most important questions in valuation theory, whereas using valuation theory is one of he most important tools in studying Galois extensions of fields, as well as constructing field extensions with given properties.
The well established topic of the model theory of valued fields is also being transformed, in particular through the study of valued fields with functions and operators, a...07aFields & rings2bicssc07aField theory and polynomials2msc07aOrder, lattices, ordered algebraic structures2msc07aCommutative rings and algebras2msc07aAlgebraic geometry2msc1 aCampillo, Antonio,eeditor.1 aKuhlmann, Franz-Viktor,eeditor.1 aTeissier, Bernard,eeditor.40uhttps://doi.org/10.4171/149423cover imageuhttps://www.ems-ph.org/img/books/campillo_mini.gif