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Zbl 1153.01333
Vershik, A.M.; Gel'fand, I.M.; Gindikin, S.G.; Kirillov, A.A.; Litvinov, G.L.; Molchanov, V.F.; Neretin, Yu A.; Retakh, V.S.
Mark Iosifovich Graev (to his 85th brithday). (English. Russian original)
[J] Russ. Math. Surv. 63, No. 1, 173-188 (2008); translation from Usp. Mat. Nauk. 63, No. 1, 169-182 (2008). ISSN 0036-0279

MSC 2000:: *01A70 Biographies, obituaries, personalia, bibliographies
00A15 Bibliographies

Keywords: with bibliography

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2
Zbl 1138.14001
Gelfand, I.M.; Kapranov, M.M.; Zelevinsky, A.V.
Discriminants, resultants, and multidimensional determinants. Reprint of the 1994 edition. (English)
[B] Modern Birkhäuser Classics. Boston, MA: Birkhäuser. x, 523~p. EUR~34.90/net; SFR~59.90; \$~39.95; \sterling~27.00 (2008). ISBN 978-0-8176-4770-4/hbk

The book under review is the unaltered reprint of the original edition first published in [Mathematics: Theory\,\& Applications. Boston, MA: Birkhäuser (1994; Zbl 0827.14036)]. Therefore, as for its precise contents, we may refer to the exhaustive and enthusiastic review by W. Bruns, in which the book's outstanding significance as an unique standard-text of modern combinatorial algebra and elimination theory was both predicted and profoundly substantiated. Reviewers of other leading journals have unanimously expressed the same overwhelming appreciation for the authors' remarkable revival and expansion of the classical theory-discriminants, resultants, and determinants in the context of contemporary algebraic geometry, homological algebra; and combinatorial geometry. In fact, the book has developed into an indispensable modern classic, during the past 15 years, and it has initiated a true avalanche of related research activities due to its pioneering and inspiring impact. The steadily large number of quotations, which seems to be ever-flowing, be speaks the undiminished pacemaking role of this unrivalled source book in various areas of current mathematical research, ranging from algebraic geometry and the general theory of hypergeometric functions up to their recent applications in mathematical physics.\par Now as before this utmost lucid and comprehensive monograph presents a unique blend of classical 19th century mathematics on the one hand, and very recent developments in algebraic geometry (Chow varieties, toric varieties, projective varieties, and hypersurfaces), combinatorial geometry and algebra (polytopes and their triangulations, aymmetric functions, rings of invariants, and special rings in local algebra), homological algebra (discriminantal complexes, resolutions, and differential complexes), and microlocal analysis ($D$-modules, perverse sheaves, and $D$-equivalence) on the other.\par It is and remains this broader, unifying approach combining the classical heritage and the powerful abstract viewpoint of the subject that makes the book under review so timeless, attractive, enlightening, inspiring and virtually invaluable -- a characteristic that will persist for further decades in the future. Besides, the comparatively inexpensive reprint of this classic in paperback form must be seen as another rewarding service to the mathematical community as a whole.
[Werner Kleinert (Berlin)]

MSC 2000:: *14-02 Research monographs (algebraic geometry)
14M12 Determinantal varieties
14M25 Toric varieties, etc.
13D25 Complexes
13C40 Linkage, complete intersections and determinantal ideals
52B20 Lattice polytopes (convex geometry)
14F10 Special sheaves

Keywords: textbook (algebraic geometry); elimination theory; determinantal varieties; polytopes; $D$-modules; holonomic sytems; perverse sheaves; hypersurfaces

Citations: Zbl 0827.14036

Cited in: Zbl 1245.14061 Zbl 1217.47103 Zbl 1159.52016

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3
Zbl 1158.00300
Gelfand, Israel M.
Mathematics as an adequate language. (English)
[A] Etingof, Pavel (ed.) et al., The unity of mathematics. In honor of the ninetieth birthday of I. M. Gelfand. Papers from the conference held in Cambridge, MA, USA, August 31--September 4, 2003. Boston, MA: Birkhäuser. Progress in Mathematics 244, xv-xxii (2006). ISBN 0-8176-4076-2/hbk

Contents: 0. Introduction; 1. Noncommutative Multiplication; 2. Addition and Multiplication; 3. Geometry; 4. Fourier Transforms, Analytic Functionals, and Hypergeometric Functions; 5. Applied Mathematics, Blow-up, and PDE's.

MSC 2000:: *00A05 General mathematics

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	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

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