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The Schubert calculus Higham, David Paul

Abstract

An enumerative problem asks the following type of question; how many figures (lines, planes, conies, cubics, etc.) meet transversely (or are tangent to) a certain number of other figures in general position? The last century saw the development of a calculus for solving this problem and a large number of examples were worked out by Schubert, after whom the calculus is named. The calculus, however, was not rigorously justified, most especially its main principle whose modern interpretation is that when conditions of an enumerative problem are varied continuously then the number of solutions in the general case is the same as the number of solutions in the special case counted with multiplicities. Schubert called it the principle of conservation of number. To date the principle has been validated in the case where the figures are linear spaces in complex projective space, but only isolated cases have been solved where the figures are curved. Hilbert considered the Schubert calculus of sufficient importance to request its justification in his fifteenth problem. We trace the first foundation of the calculus due primarily to Lefschetz, van der Waerden and Ehresmann. The introduction is historical, being a summary of Kleiman's expository article on Hilbert's fifteenth problem. We describe the Grassmannian and its Schubert subvarieties more formally and describe explicitly the homology of the Grassmannian which gives a foundation for the calculus in terms of algebraic cycles. Finally we compute two examples and briefly mention some more recent developments.

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