Towards a probabilistic Schubert calculus

Hermann Schubert developed in the 19th century a calculus for answering enumerative questions in algebraic geometry, e.g., ''How many lines intersect four curves of degrees d_1,...,d_4 in three-dimensional space in general position?''. In his 15th problem, Hilberts asked for a rigorous foundation of Schubert's enumerative calculus, which led to important progress in algebraic geometry and topology (intersection theory of the Grassmannians). However, Schubert calculus only yields the typical number of complex solutions. Is there a meaningful way to speak about the typical number of REAL solutions?

We shall outline a way to do so, by assuming that the given objects (the four curves in the above example) are randomly rotated and to inquire about the expected number of real solutions. The approach blends ideas from real algebraic geometry with integral geometry and the theory of random polytopes.

(Joint work with Antonio Lerario.)