Boris Shapiro University of Stockholm

CONVEX CURVES AND REAL SCHUBERT CALCULUS

Abstract. A real projective curve γ : S^{1} → RP^{n} is called convex if any projective hyperplane intersects it at most n times counted with multiplicities. Convexity of curves is closely related to the classical notion of Tschebychev systems of functions. About 7 years ago we (together with Michael Shapiro) realized that convex curves are important in real Schubert calculus whose major problem is to find collections of real projective subspaces of some fixed dimension having the maximal possible number of real projective subspaces of the complementary dimension intersecting all of them. The purpose of this talk is to present new results in this area together with the existing numerical evidence and to formulate a variety of related conjectures. No preliminary knowledge exceeding some acquaintance with Grassmannians is required.