Abstract

Hilbert's 15th problem is an inspiring and far-reaching one.lt promoted the enumerativegeometry of the 19th century growing into the algebraic geometry founded by Van der Waerden andAndre Weil, and integrated Schubert calculus into many branches of contemporary mathematicsHowever, despite great many achievements in the 20th century the part of the problem on findingan effective rule performing the calculus has been stagnant for a long time, notably, the Schubert'sproblem of characteristics, or the Weil's problem on the intersection theory of flag manifolds. Thistalk surveys the background, content, and solution of the 15th problem. Our main results are aunified formula for the characteristics, and a systematic description of the intersection theory of flagmanifolds. We illustrate the effectiveness of the formula and the algorithm by explicit examples.