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  "body": "\n## Problem\n\nGive a rigorous foundation for the enumerative calculus of Hermann Schubert.\nDetermine exactly the range of validity of his methods, and check the numbers of [[enumerative geometry|ZPZA1FKZZC]] that he and his contemporaries found.\n\n## Remarks\n\nHilbert posed this problem in his list of 23 problems presented in 1900.\nSchubert had computed many enumerative numbers with symbolic rules, but these rules lacked proofs, and some classical answers were later found to need correction.\nThe modern foundations are [[intersection theory|4HV4WTAX9P]] and the cohomology of Grassmannians and flag varieties, in which [[Schubert calculus|FWGHMADJR4]] becomes a set of theorems.\nMuch of the problem is regarded as settled in this way, but checking all of Schubert's numbers is a separate task, and the problem is often listed as partially solved.\n\n## Explanations\n\n### L1 human\n\nIn the nineteenth century, geometers counted figures, such as lines meeting other lines or conics touching other conics, with clever but unproven rules.\nThe rules usually gave right answers, but sometimes wrong ones, and nobody knew why.\nHilbert asked for a solid foundation that explains when the rules work.\nThe answer came through the theory of intersections and the topology of Grassmannians, developed during the twentieth century.\n",
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      "first_stated": "1900",
      "posed_by": "David Hilbert"
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    "name": "Hilbert's fifteenth problem",
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  "msc": [
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  "name": "Hilbert's fifteenth problem",
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