Real algebraic and combinatorial structures in matrix spaces

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This project aims to apply mathematical tools to study objects from optimization, engineering, and statistics. The research involved will lead to a better understanding of the underlying mathematics and further the development and analysis of algorithms. Development of such theory benefits both the mathematical and broader scientific communities and strengthens the connections between them. Another important part of this project involves mentoring undergraduate and graduate students. Participation in this project will expose students to a wide range of fields in mathematics, impress upon them the importance of interdisciplinary research, and train them in the theory and computational skills needed to carry it out.The broad goals of the proposed research are to develop computational techniques in real algebraic geometry and combinatorics in order to study objects arising in optimization and other applications. Real algebraic geometry is the study of solutions to polynomial equations and inequalities, which can describe a vast array of systems from engineering and science. Tropical geometry is a powerful tool for extracting discrete data from real algebraic sets as well as for constructing examples of real algebraic sets with desired properties. The proposed research applies these powerful mathematical tools to problems in linear algebra and convex optimization. Stable polynomials, determinants, and linear spaces of matrices are fundamental objects in these fields. A better understanding of their underlying geometry would advance these fields and the connections between them.