Research Projects

Total positivity is a widely studied and evergreen topic in mathematics. For almost a century, totally positive (TP) and totally non-negative (TN) matrices and kernels have appeared in diverse areas in pure and applied mathematics, including analysis, approximation theory, cluster algebras, combinatorics, differential equations, Gabor analysis, integrable systems, matrix theory, probability and statistics, and representation theory. This notion of positivity was first studied by Fekete-Polya in 1912 in terms of the variation diminishing property of one sided Polya frequency sequences. An interesting problem in the literature has been that of understanding the preservers of various structures, including forms of positivity. In 2023, Belton-Guillot-Khare-Putinar classified the composition operators preserving TN and TP for various classes of structured kernels, including continuous Hankel kernels on an interval, Polya frequency functions, and Polya frequency sequences. One of the main goals of this project is to study the linear preservers of several families of structured TN and TP kernels. In particular, the PI is interested to classify linear preservers of Polya frequency sequences, Polya frequency functions, as well as real rooted polynomials. From the algebraic side, the PI is interested to classify automorphisms of the semigroup of TP matrices of fixed dimension, as well as of the semigroup of TP kernels. Another theme of investigation in this proposal involves entrywise preservers of positive semidefiniteness-related properties. Functions preserving Loewner positivity (positive semidefiniteness) when acting entrywise on positive semidefinite matrices have a long history in the literature. A well known result of Schoenberg and of Rudin says that functions preserving Loewner positivity in all dimensions are analytic with nonnegative Taylor coefficients. A natural follow-up question is to classify Loewner positive functions in a fixed dimension n. This challenging problem is still open for every n greater than 2. Recently Guillot-Khare-Rajaratnam, Belton-Guillot-Khare-Putinar investigated this question by applying additional constraints either on functions or on matrices (or both). The second goal of this project is to investigate entrywise powers and functions that preserve positivity, monotonicity and super-additivity in a fixed dimension with additional sparsity constraints imposed by graph structure. Moreover, the PI is also interested to connect this to well studied combinatorial parameters.