Antonio Lerario

Enumerative geometry deals with the problem of counting ("enumerating") geometric objects satisfying some constraint on their arrangement. For example: "how many lines in three-space intersect at the same time four given lines?". (Another example: "how many lines are there on a cubic surface in three-space?") For the first example, the answer is two if we are allowed to look for complex lines, but it depends on the four given lines if we search for real lines. (For the second example the answer over the complex numbers is 27.) In the complex framework this type of questions can be answered using a beautiful, sophisticated technique called Schubert calculus: it is the study of the way cycles intersect in complex Grassmannians. Unfortunately over the reals this technique loses its power: this is the old problem of finding real solutions to real equations, for which the number of complex solutions only gives upper bounds. In this talk I will adopt a probabilistic approach, trying to address questions like: "how many real lines in three-space intersect four given random lines?". The technique that I will introduce, which will produce an expected answer over the reals, when adapted to the complex setting gives the generic answer and offers a new point of view on the classical problem. This is based on joint works with P. Burgisser and with S. Basu, E. Lundberg and C. Peterson.