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The Basic Theory of Real Closed Spaces

Author : Niels Schwartz

Publisher : American Mathematical Soc.

Page : 134 pages

File Size : 36,85 MB

Release : 1989

Category : Commutative algebra

ISBN : 0821824600

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Book Description

Much in the same way as classical algebraic varieties are generalized by the theory of schemes, locally semi-algebraic spaces are generalized by a class of locally ringed spaces, called real closed spaces. The underlying spaces of affine real closed spaces are real spectra of rings, the structure sheaves are called real closed sheaves. With these spaces a theory can be developed which is very similar to the theory of schemes. There is a natural functor from the category of semi-algebraic spaces to the category of real closed spaces. Via this functor properties of semi-algebraic spaces and their corresponding real closed spaces can be compared.



Weakly Semialgebraic Spaces

Author : Manfred Knebusch

Publisher : Springer

Page : 396 pages

File Size : 23,4 MB

Release : 2006-11-14

Category : Mathematics

ISBN : 3540460896

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Book Description

The book is the second part of an intended three-volume treatise on semialgebraic topology over an arbitrary real closed field R. In the first volume (LNM 1173) the category LSA(R) or regular paracompact locally semialgebraic spaces over R was studied. The category WSA(R) of weakly semialgebraic spaces over R - the focus of this new volume - contains LSA(R) as a full subcategory. The book provides ample evidence that WSA(R) is "the" right cadre to understand homotopy and homology of semialgebraic sets, while LSA(R) seems to be more natural and beautiful from a geometric angle. The semialgebraic sets appear in LSA(R) and WSA(R) as the full subcategory SA(R) of affine semialgebraic spaces. The theory is new although it borrows from algebraic topology. A highlight is the proof that every generalized topological (co)homology theory has a counterpart in WSA(R) with in some sense "the same", or even better, properties as the topological theory. Thus we may speak of ordinary (=singular) homology groups, orthogonal, unitary or symplectic K-groups, and various sorts of cobordism groups of a semialgebraic set over R. If R is not archimedean then it seems difficult to develop a satisfactory theory of these groups within the category of semialgebraic sets over R: with weakly semialgebraic spaces this becomes easy. It remains for us to interpret the elements of these groups in geometric terms: this is done here for ordinary (co)homology.



Manis Valuations and Prüfer Extensions I

Author : Manfred Knebusch

Publisher : Springer

Page : 276 pages

File Size : 38,95 MB

Release : 2004-10-19

Category : Mathematics

ISBN : 3540456252

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Book Description

The present book is devoted to a study of relative Prüfer rings and Manis valuations, with an eye to application in real and p-adic geometry. If one wants to expand on the usual algebraic geometry over a non-algebraically closed base field, e.g. a real closed field or p-adically closed field, one typically meets lots of valuation domains. Usually they are not discrete and hence not noetherian. Thus, for a further develomemt of real algebraic and real analytic geometry in particular, and certainly also rigid analytic and p-adic geometry, new chapters of commutative algebra are needed, often of a non-noetherian nature. The present volume presents one such chapter.



Sum of Even Powers of Real Linear Forms

Author : Bruce Arie Reznick

Publisher : American Mathematical Soc.

Page : 169 pages

File Size : 17,73 MB

Release : 1992

Category : Mathematics

ISBN : 0821825232

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Book Description

This work initiates a systematic analysis of the representation of real forms of even degree as sums of powers of linear forms and the resulting implications in real algebraic geometry, number theory, combinatorics, functional analysis, and numerical analysis. The proofs utilize elementary techniques from linear algebra, convexity, number theory, and real algebraic geometry and many explicit examples and relevant historical remarks are presented.



Real and Etale Cohomology

Author : Claus Scheiderer

Publisher : Springer

Page : 300 pages

File Size : 32,84 MB

Release : 2006-11-15

Category : Mathematics

ISBN : 3540487972

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Book Description

This book makes a systematic study of the relations between the étale cohomology of a scheme and the orderings of its residue fields. A major result is that in high degrees, étale cohomology is cohomology of the real spectrum. It also contains new contributions in group cohomology and in topos theory. It is of interest to graduate students and researchers who work in algebraic geometry (not only real) and have some familiarity with the basics of étale cohomology and Grothendieck sites. Independently, it is of interest to people working in the cohomology theory of groups or in topos theory.



Logic Colloquium 2007

Author : Françoise Delon

Publisher : Cambridge University Press

Page : 282 pages

File Size : 27,44 MB

Release : 2010-06-07

Category : Mathematics

ISBN : 1139488937

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Book Description

The Annual European Meeting of the Association for Symbolic Logic, also known as the Logic Colloquium, is among the most prestigious annual meetings in the field. The current volume, Logic Colloquium 2007, with contributions from plenary speakers and selected special session speakers, contains both expository and research papers by some of the best logicians in the world. This volume covers many areas of contemporary logic: model theory, proof theory, set theory, and computer science, as well as philosophical logic, including tutorials on cardinal arithmetic, on Pillay's conjecture, and on automatic structures. This volume will be invaluable for experts as well as those interested in an overview of central contemporary themes in mathematical logic.



Model Theory of Fields

Author : David Marker

Publisher : CRC Press

Page : 172 pages

File Size : 32,86 MB

Release : 2005-12-15

Category : Mathematics

ISBN : 1439864411

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Book Description

The model theory of fields is a fascinating subject stretching from Tarski's work on the decidability of the theories of the real and complex fields to Hrushovksi's recent proof of the Mordell-Lang conjecture for function fields. This volume provides an insightful introduction to this active area, concentrating on connections to stability theory.



Ordered Algebraic Structures

Author : Jorge Martínez

Publisher : Springer Science & Business Media

Page : 288 pages

File Size : 37,7 MB

Release : 2012-12-06

Category : Mathematics

ISBN : 9400924720

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Book Description

Proceedings of the Caribbean Mathematics Foundation Conference, held in Curaçao, August 1988



Ordered Algebraic Structures

Author : W.C. Holland

Publisher : Springer Science & Business Media

Page : 334 pages

File Size : 21,36 MB

Release : 2012-12-06

Category : Mathematics

ISBN : 9401156409

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Book Description

The conference on Ordered Algebraic Structures held in Curat;ao, from the 26th of June through the 30th of June, 1995, at the Avila Beach Hotel, marked the eighth year of ac tivities by the Caribbean Mathematics Foundation (abbr. CMF), which was the principal sponsor of this conference. CMF was inaugurated in 1988 with a conference on Ordered Algebraic Structures. During the years between these two conferences the field has changed sufficiently, both from my point of view and, I believe, that of my co-organizer, W. Charles Holland, to make one wonder about the label "Ordered Algebraic Structures" itself. We recognized this from the start, and right away this conference carried a subtitle, or, if one prefers, an agenda: we concentrated on the one hand, on traditional themes in the theory of ordered groups, including model-theoretic aspects, and, on the other hand, on matters in which topology (more precisely C(X)-style topology) and category theory would play a prominent role. Plainly, ordered algebra has many faces, and it is becoming increas ingly difficult to organize an intimate conference, such as the ones encouraged in the series sponsored by CMF, in this area on a broad set of themes. These proceedings reflect, accurately we think, the spirit of the conferees, but it is not a faithful record of the papers presented at the conference.



Real Algebraic Geometry

Author : Michel Coste

Publisher : Springer

Page : 425 pages

File Size : 19,91 MB

Release : 2006-11-15

Category : Mathematics

ISBN : 3540473378

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Book Description

Ten years after the first Rennes international meeting on real algebraic geometry, the second one looked at the developments in the subject during the intervening decade - see the 6 survey papers listed below. Further contributions from the participants on recent research covered real algebra and geometry, topology of real algebraic varieties and 16thHilbert problem, classical algebraic geometry, techniques in real algebraic geometry, algorithms in real algebraic geometry, semialgebraic geometry, real analytic geometry. CONTENTS: Survey papers: M. Knebusch: Semialgebraic topology in the last ten years.- R. Parimala: Algebraic and topological invariants of real algebraic varieties.- Polotovskii, G.M.: On the classification of decomposing plane algebraic curves.- Scheiderer, C.: Real algebra and its applications to geometry in the last ten years: some major developments and results.- Shustin, E.L.: Topology of real plane algebraic curves.- Silhol, R.: Moduli problems in real algebraic geometry. Further contributions by: S. Akbulut and H. King; C. Andradas and J. Ruiz; A. Borobia; L. Br|cker; G.W. Brumfield; A. Castilla; Z. Charzynski and P. Skibinski; M. Coste and M. Reguiat; A. Degtyarev; Z. Denkowska; J.-P. Francoise and F. Ronga; J.M. Gamboa and C. Ueno; D. Gondard- Cozette; I.V. Itenberg; P. Jaworski; A. Korchagin; T. Krasinksi and S. Spodzieja; K. Kurdyka; H. Lombardi; M. Marshall and L. Walter; V.F. Mazurovskii; G. Mikhalkin; T. Mostowski and E. Rannou; E.I. Shustin; N. Vorobjov.