Convex geometry and random matrices in high dimensions


AbstractsRamon van Handel (Princeton University) Title: Around the AlexandrovFenchel inequality Abstract: The isoperimetric theorem states that the ball minimizes surface area among all bodies of the same volume. For convex bodies, however, volume and surface area are merely two examples of a large family of natural geometric parameters called mixed volumes that arise as coefficients of the volume polynomial. Mixed volumes were discovered by Minkowski in a seminal 1903 paper that laid much of the foundation for modern convex geometry. In particular, Minkowski, Alexandrov and Fenchel discovered a remarkable set of quadratic inequalities between mixed volumes that constitute a farreaching generalization of the classical isoperimetric theorem. The theory of these inequalities and their applications in characterized by unexpected connections with various questions in geometry, analysis, algebra, and combinatorics, and features some longstanding open problems. My aim in these lectures is to introduce some of the problems, connections, and recent progress on this topic.
Mark Rudelson (University of Michigan) Title: On the delocalization of the eigenvectors of random matrices Abstract: Consider a random matrix with i.i.d. normal entries. Since its distribution is invariant under rotations, any normalized eigenvector is uniformly distributed over the unit sphere. For a general distribution of the entries, this is no longer true. Yet, if the size of the matrix is large, the eigenvectors are distributed approximately uniformly. This property, called delocalization, can quantified in various senses. In these lectures, we will discuss recent results on delocalization for general random matrices. 