## Universality laws in geometric random matrix theory

#### Joel TroppCalifornia Institute of TechnologyComputing and Mathematical Sciences

A basic problem in geometry is to understand the probability that a uniformly random subspace of a given codimension intersects a fixed convex set. The hitting probability exhibits a phase transition as the codimension of the subspace increases. That is, the probability changes rapidly from one to zero when the codimension reaches the "statistical dimension," a geometric invariant of the convex set.

The focus of this talk is a new universality law in random matrix theory connected to this geometric problem. For a fixed convex set, the location of the phase transition is *universal* over a large class of random subspaces that are constructed as the kernels of random matrices.

On joint works with Dennis Amelunxen, Martin Lotz, Michael McCoy, and Samet Oymak.

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