Analysis seminar - Khare

Title: Entrywise positivity preservers: analysis, metric geometry, novel graph invariant

Speaker: Apoorva Khare (Indian Institute of Science, Bangalore, India) 

Abstract:

Which functions preserve positive semidefiniteness (psd) when applied entrywise to psd matrices? This question has a long history beginning with Schur, Schoenberg, and Rudin, and has also recently received renewed attention due to applications in high-dimensional statistics. However, effective characterizations of entrywise functions preserving positivity in a fixed dimension remain elusive to date.

I will present four aspects of the study of this question, beginning with (1) the "all dimensions" case that connects to analysis and measures. This is followed by (2) the origins of this question, via positive definite functions on Euclidean spheres - which connect, remarkably, to multiple celebrated problems. 

The other two aspects are in fixed dimension, including (3) Loewner's necessary condition, whose modern analogue led to the unexpected appearance of Schur polynomials from positivity preservers. Finally, (4) I will describe a recent novel graph invariant that naturally arises from positivity preservers, and whose value is currently known only for chordal and bipartite graphs. (Partly based on joint works with Alexander Belton, Dominique Guillot, Mihai Putinar, Bala Rajaratnam, and Terence Tao.)