Presenter: Dr. Papri Dey

Description: Lorentzian polynomials, introduced independently by Brändén and Huh, and by Anari, Oveis Gharan, and Vinzant (under the name of completely log-concave polynomials), generalize hyperbolic polynomials while maintaining log-concavity. These polynomials are specific to the nonnegative orthant. In this talk, I will discuss Lorentzian polynomials defined on proper convex cones, known as -Lorentzian polynomials, and demonstrate the equivalence between -Lorentzian polynomials and -completely log- concave (CLC) polynomials. For a self-dual cone , we establish a connection between -Lorentzian polynomials and -positive linear maps, which were studied in the context of the generalized Perron- Frobenius theorem. Based on this connection, we provide a simplified characterization of -Lorentzian polynomials. Furthermore, I will illustrate a method for constructing a proper convex cone for a given - CLC polynomial. Utilizing the corresponding cone, I will highlight its applications in optimization and dynamical systems through examples. Finally, I will conclude with a discussion of open problems for future research. This talk is based on our joint paper on -Lorentzian polynomials with Greg Blekherman and ongoing follow-up project.

Bio: Dr. Papri Dey served as a Visiting Assistant Professor in the School of Mathematics at Georgia Tech from June 2022 to July 2024. Prior to that, she held Postdoctoral fellowships at the University of Missouri, Columbia, UC Berkeley, and Brown University. She has also worked as a Visiting Scientist at the Indian Statistical Institute in Kolkata, and the Max Planck Institute in Leipzig, Germany. Dr. Dey completed her PhD in Electrical Engineering at IIT Bombay in 2017. Her research interests include applied algebraic geometry, nonlinear algebra, and algebraic combinatorics, with applications in optimization, and data science. She collaborated on the development of a software package, DeterminantalRepresentations.m2, in Macaulay2.

Hosted by: UCSC Applied Mathematics Department.