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# {Seminar} @ CDS: 04th May : ” Computing Determinantal Representations of Hyperbolic Polynomials, and Degeneracy Loci of Matrices”

## 04 May @ 5:00 PM -- 6:00 PM

**Department of Computational and Data Sciences**

**Department Seminar**

**SPEAKER** : Dr. Papri Dey, University of Missouri, Columbia.

**TITLE** : “Computing Determinantal Representations of Hyperbolic Polynomials, and Degeneracy Loci of Matrices.”

**Date & Time** : May 04, 2021, 05:00 PM.

**Venue** : Online

**ABSTRACT**

At first, I shall introduce the notion of real stable and hyperbolic polynomials which are two important generalizations of real-rooted (all of its roots are real) univariate polynomials, to multivariate polynomials. I shall show-case many different problems (seemingly unrelated) from various fields which exploit these notions to prove long-term conjectures and unsolved open problems, in particular, in theoretical computer science (existence of Ramanujan (spectral Expander) graphs, lower bound to the number of perfect matchings in a graph via permanent matrix-van der Waerden conjecture), machine learning (determinantal point process (DPP), strongly Rayleigh distribution), optimization (semidefinite and hyperbolic programming).

Determinantal polynomials are the polynomials which can be expressed as the determinant of some linear matrix polynomials with symmetric/Hermitian coefficient matrices such that the linear span of these coefficient matrices contains a positive definite matrix. These determinantal polynomials are real stable as well as hyperbolic polynomials. Then I shall briefly discuss the algorithms to solve determinantal representation problems proposed throughout my research work which have been successfully implemented in Macaulay2 as a software package. Not all hyperbolic polynomials are determinantal, so I shall also talk about intersection method to testing hyperbolicity property of a multivariate polynomial via sum-of-squares relaxation (solving semi-definite programming) by translating the hyperbolicity condition into different types of nonnegative conditions.

Finally, I shall report on my two current projects. One is on degeneracy loci of matrices and its correspondence with determinantal representation problem and its importance in phase retrieval, and matrix completion which often arises in real life problems such as Netflix problem. The other one is focused on solving three fundamental computational problems (Jordan forms, spectral factorization, and approximation to determinantal representations of hyperbolic polynomials) in linear algebra, control theory and real algebraic geometry. At the end, a more generalized notion, Lorentzian or log-concave polynomials, matroids and tropicalization will be mentioned to conclude my talk.

**BIOGRAPHY**

Papri Dey is a University Post-Doctoral fellow in the Department of Mathematics, University of Missouri, Columbia. She has received several research fellowships to continue her collaborative research work in India, Germany and USA. She has been awarded with the prestigious Simons-Berkeley fellowship (Microsoft Research Fellow) for conducting her research in the area of Geometry of Polynomials at Simons Institute for the Theory of Computing, UC Berkeley and semester postdoctoral fellowship in the area of Nonlinear Algebra at the Institute for Computational and Experimental Research in Mathematics (ICERM) at Brown University. Besides, she worked as a Visiting Scientist at Indian Statistical Institute Kolkata and Max Planck Institute for Mathematics in the Sciences Leipzig, Germany, and a CSIR fellow for her PhD in the Department of Electrical Engineering at IIT Bombay. She completed her B.Sc and M.Sc in Mathematics at the University of Calcutta. Her research interests include convex algebraic geometry, nonlinear algebra and computational algebra. She has developed a software package in Macaulay2, a software system devoted to supporting research in algebraic geometry and commutative algebra, as a part of a collaborative research with Georgia Tech. She gave invited talks in several universities including UC Berkeley, Brown University, University of Pittsburgh, University of Missouri, Otto-Von Guericke university, TU Dortmund, Max Planck Institute for Mathematics in the Sciences Leipzig, KUMUNU 2019 (plenary speaker), University of Nebraska, AMS Spring Southeastern Sectional Meeting. She is also an invited speaker in the forthcoming SIAM Conference on Applied Algebraic Geometry 2021.

**Host Faculty**: Prof. Murugesan Venkatapathi