The University of Michigan Mathematics Colloquium
"The question of how many solutions of real equations can be real is still very much open, particularly for enumerative problems" (W. Fulton, Introduction to Intersection Theory in Algebraic Geometry, p. 55). Let F be one of the fields R (real numbers) or C (complex numbers). Consider the following problem of enumerative geometry: Given mp subspaces of dimension p in general position in a vector space of dimension m+p over F, how many subspaces of dimension m intersect all these given subspaces non-trivially? When F=C, the answer was obtained by H. Schubert in 1886. It is equal to the number of Standard Young Tableaux of shape p times m. The main ingredient in Schubert's argument is the "Principle of Conservation of a Number": the number of complex solutions (counting multiplicity) of a system of algebraic equations does not change when the parameters of the system vary continuously. In this talk, a survey of recent results for the case F=R, obtained by A. Gabrielov and the speaker, will be given. The Principle of Conservation of a Number does not hold for real solutions. To find a substitute, we introduce a kind of topological degree, which estimates from below the number of real solutions, and show that under some conditions this degree does not change when the parameters of a system vary continuously. This permits to compute the degree by considering some degenerate systems. In particular, for the problem of Schubert stated above, we obtain: If m+p is odd then there are real solutions. Moreover, we obtain an explicit lower estimate for the number of such solutions: it is the number of Shifted Standard Young Tableaux with p rows, (m+p-1)/2 cells in the top row, and (m-p+1)/2 cells in the bottom row. The method has applications to linear control theory which will also be briefly discussed. Prerequisites: calculus and linear algebra.