The goal in this proposal is to develop the mathematical foundations and associated computational methods for the study of convex sets in real algebraic geometry. This work requires a combination of ideas and mathematical tools from optimization, analysis, algebra and combinatorics. The proposed program will lead not only to theoretical insights, but also to new algorithms and software that will enable novel applications in mathematics, engineering, and beyond. The work is organized in five main thrusts: semidefinite programming and sums of squares, convex semi-algebraic sets, sparsity and graphical structure, numerical polynomial optimization and applications, and deformations and variation of parameters. The PIs will focus on the development of a comprehensive theory and practical new algorithms for convex sets defined by polynomial inequalities. Specific problems and techniques include the formulation of semidefinite descriptions of convex hulls of real algebraic varieties, determinantal representations of hyperbolic polynomials, sparse polynomials and their symmetries, tropical geometry and homotopy techniques, and geometric programming.
Many areas in mathematics, as well as applications in engineering, finance and the sciences, require a thorough understanding of convex sets. This is a class of geometric shapes, with several different but complementary interpretations. The goal in this project is to achieve a better understanding of how these geometric properties emerge from their algebraic descriptions in terms of polynomial equations, and the corresponding computational implications. One of the main motivations is the possibility of applying these results in the context of optimization. The proposed research will contribute to existing knowledge, both in algebraic-geometric techniques as well as in mathematical optimization. It will create synergies between different branches of applied mathematics, and their engineering and scientific applications (e.g., in computational biology and statistical modeling). Successful completion of this project should contribute to the availability of efficient and reliable computational tools for solving polynomial systems, which have clear technological and economic interest. Other key features of this proposal include its integration with curriculum development, undergraduate research projects, training of graduate students and postdocs, and the development of new software tools for computational optimization.
