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The problem consists in this: To establish rigorously and with an exact determination of the limits of their validity those geometrical numbers which
Schubert especially has determined on the basis of the so-called principle of special position, or conservation of number, by means of the enumerative
54:
The additive structure of the ring H*(G/P) is given by the basis theorem of
Schubert calculus due to Ehresmann, Chevalley, and Bernstein-Gel'fand-Gel'fand, stating that the classical Schubert classes on G/P form a free basis of the cohomology ring H*(G/P). The remaining problem of expanding products
42:
Schubert calculus is the intersection theory of the 19th century, together with applications to enumerative geometry. Justifying this calculus was the content of
Hilbert's 15th problem, and was also the major topic of the 20 century algebraic geometry. In the course of securing the foundations of
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Although the algebra of today guarantees, in principle, the possibility of carrying out the processes of elimination, yet for the proof of the theorems of enumerative geometry decidedly more is requisite, namely, the actual carrying out of the process of elimination in the case of equations of
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related the problem to the determination of the cohomology ring H*(G/P) of a flag manifold G/P, where G is a Lie group and P a parabolic subgroup of G.
62:
While enumerative geometry made no connection with physics during the first century of its development, it has since emerged as a central element of
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530:
S. Kleiman, Intersection theory and enumerative geometry: A decade in review, Proc. Symp. Pure Math., 46:2, Amer. Math. Soc. (1987), 321-370.
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b) Special presentations of the Chow rings of flag manifolds have been worked out by Borel, Marlin, Billey-Haiman and Duan-Zhao, et al.;
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31:
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Mathematical developments arising from
Hilbert problems (Proc. Sympos. Pure Math., Northern Illinois Univ., De Kalb, Ill., 1974)
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special form in such a way that the degree of the final equations and the multiplicity of their solutions may be foreseen.
691:
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553:, Proc. Sympos. Pure Math., vol. XXVIII, Providence, R. I.: American Mathematical Society, pp. 445–482,
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630:
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S. Kleiman, Book review on “Intersection Theory by W. Fulton”, Bull. AMS, Vol.12, no.1(1985), 137-143.
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c) Major enumerative examples of
Schubert have been verified by Aluffi, Harris, Kleiman, XambĂł, et al.
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134:
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111:
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According to Van der
Waerden and André Weil Hilbert problem fifteen has been solved. In particular,
107:
478:
H. Duan; X. Zhao (2020). "On
Schubert's Problem of Characteristic". In J. Hu; et al. (eds.).
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Kleiman, Steven L. (1976), "Problem 15: rigorous foundation of
Schubert's enumerative calculus",
513:
485:
399:
327:
260:
99:
479:
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Waerden, B. L. van der (1930). "Topologische Begründung des Kalküls der abzählenden
Geometrie".
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by
Schubert, and regarded by him as "the main theoretic problem of enumerative geometry".
304:
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H. Schubert, Lösung des Charakteristiken-Problems für lineare Räume beliebiger Dimension
484:. Springer Proceedings in Mathematics & Statistics. Vol. 332. pp. 43–71.
188:(1901), 44-63 and 213-237. Published in English translation by Dr. Maby Winton Newson,
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a) Schubert's characteristic problem has been solved by Haibao Duan and Xuezhi Zhao;
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Schubert Calculus and Its Applications in Combinatorics and Representation Theory
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285:, Student Mathematical Library, vol. 32, American Mathematical Society,
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Pragacz, Piotr (1997), "The status of Hilbert's Fifteenth Problem in 1993",
349:. Proceedings of Symposia in Pure Mathematics. Vol. 56. pp. 1–26.
345:
Chevalley, C. (1994). "Sur les décompositions cellulaires des espaces G/B".
462:, Student Mathematical Library, vol. 32, American Mathematical Society
434:, Mitteilungen der Mathematische Gesellschaft in Hamburg 1 (1886), 134-155.
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of Schubert classes as linear combinations of basis elements was called
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F. Sottile, Schubert calculus, Springer Encyclopedia of Mathematics
118:, and in particular its algorithmic aspects are still of interest.
114:). It was a precursor of several more modern theories, for example
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The entirety of the original problem statement is as follows:
347:
Algebraic Groups and Their Generalizations: Classical Methods
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Manin, Ju. I. (1969), "On Hilbert's fifteenth problem",
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of a linear subspace in projective space with a given
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I.N. Bernstein; I.M. Gel'fand; S.I. Gel'fand (1973).
380:"Schubert cells and cohomology of the spaces G/P"
585:Hilbert's Problems (Polish) (Międzyzdroje, 1993)
445:https://projecteuclid.org/euclid.bams/1183552346
305:"Sur la topologie de certains espaces homogenes"
587:, Warsaw: Polsk. Akad. Nauk, pp. 175–184,
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420:H. Schubert, Kalkül der abzählenden Geometrie
190:Bulletin of the American Mathematical Society
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570:, Izdat. “Nauka”, Moscow, pp. 175–181,
121:The objects introduced by Schubert are the
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176:Hilbert, David, "Mathematische Probleme"
102:introduced in the nineteenth century by
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26:set out in a list compiled in 1900 by
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459:Enumerative Geometry and String Theory
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106:to solve various counting problems of
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16:On Schubert's enumerative calculus
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283:Foundations of algebraic geometry
98:Schubert calculus is a branch of
182:Archiv der Mathematik und Physik
396:10.1070/RM1973v028n03ABEH001557
207:10.1090/S0002-9904-1902-00923-3
32:Schubert's enumerative calculus
180:, (1900), pp. 253-297, and in
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790:Unsolved problems in geometry
422:, 1879, Leipzig: B.G. Teubner
568:Hilbert's problems (Russian)
500:10.1007/978-981-15-7451-1_4
79:calculus developed by him.
20:Hilbert's fifteenth problem
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57:the characteristic problem
34:on a rigorous foundation.
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133:defined by conditions of
30:. The problem is to put
456:Katz, Sheldon (2006),
303:Ehresmann, C. (1934).
116:characteristic classes
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384:Russian Math. Surveys
178:Göttinger Nachrichten
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43:intersection theory,
112:enumerative geometry
539:10.1090/pspum/046.2
355:10.1090/pspum/056.1
108:projective geometry
785:Algebraic geometry
780:Hilbert's problems
631:Hilbert's problems
249:10.1007/BF01782350
141:. For details see
100:algebraic geometry
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509:978-981-15-7450-4
281:Weil, A. (1962),
197:Web-viewable text
94:Schubert calculus
88:Schubert calculus
70:Problem statement
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24:Hilbert problems
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127:locally closed
123:Schubert cells
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131:Grassmannian
125:, which are
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38:Introduction
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390:(3): 1–26.
774:Categories
491:1912.10745
161:References
129:sets in a
49:André Weil
518:209444479
404:250748691
265:177808901
237:Math. Ann
135:incidence
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