A Summer Seminar: Dr. Apoorva Khare

Title: Entrywise positivity preservers: covariance estimation, metric geometry, and a novel graph invariant

Abstract:
Which functions preserve positive semidefiniteness (psd) when applied entrywise to psd matrices? This question was asked exactly 100 years ago by Polya and Szego, and has a long history beginning with Schur, Schoenberg, and Rudin. At the same time, it has also recently received renewed attention due to applications in high-dimensional statistics.

After explaining some of these developments, I will present a selection of results on entrywise positivity preservers. We begin with the early work of Schoenberg, which led from metric (spherical) geometry to matrix positivity and its preservers. Next come some of the contributions of Loewner, Karlin, and their students: FitzGerald, Horn, Micchelli, and Pinkus, on entrywise maps in all dimensions and in a fixed dimension. I will end with some recent results that describe novel connections to symmetric function theory and combinatorics. (These last are based on joint works with Alexander Belton, Dominique Guillot, Mihai Putinar, Bala Rajaratnam, and Terence Tao.)