On the thin-shell conjecture for the Schatten classes
- Series
- Analysis Seminar
- Time
- Wednesday, November 2, 2016 - 14:05 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Beatrice-Helen Vritsiou – University of Michigen – vritsiou@umich.edu
The thin-shell or variance conjecture asks whether the
variance of the Euclidean norm,
with respect to the uniform measure on an isotropic convex body, can be
bounded from above by an absolute constant times the mean of the
Euclidean norm (if the
answer to this is affirmative, then we have as a consequence that most
of the mass of the isotropic convex body is concentrated in an annulus
with very small width, a "thin shell''). So far all the general bounds
we know depend on the dimension of the bodies, however for a few special
families of convex bodies, like the $\ell_p$ balls, the conjecture has
been resolved optimally. In this talk, I will talk about another family of
convex bodies, the unit balls of the
Schatten classes (by this we mean spaces of square matrices with
real, complex or
quaternion entries equipped with the $\ell_p$-norm of their singular
values, as well as their subspaces of self-adjoint matrices).In a
joint work with Jordan Radke, we verified the conjecture for the
operator norm (case of $p = \infty$) on all three general spaces of
square matrices, as well as for complex self-adjoint matrices, and
we also came up with a necessary condition for the conjecture to be true
for any of the other p-Schatten norms on these spaces. I will discuss
how one can obtain these results: an essential step in the proofs is
reducing the
question to corresponding variance estimates with respect to the joint
probability density of the singular values of the matrices.Time
permitting, I will also talk about a different method to obtain such
variance estimates that allows to verify the variance conjecture for the
operator norm on the remaining spaces as well.