Georgia Institute of Technology College of Sciences

Let $A = (a_{ij})$ be an $n \times n$ matrix with entries in a field $\mathbb{F}$ and let $f$ be a function defined on $\mathbb{F}$. The function naturally induces an entrywise transformation of $A$ via $f[A] := (f(a_{ij}))$. The study of such entrywise transforms that preserve various forms of matrix positivity has a rich and long history since the seminal work of Schoenberg. In this talk, I will discuss recent developments in the setting that the underlying field $\mathbb{F}$ is the real field, the complex field, and finite fields. I will also highlight some interesting connections between these problems with arithmetic combinatorics, finite geometry, and graph theory. Joint work with Dominique Guillot, Himanshu Gupta, and Prateek Kumar Vishwakarma.