This project investigates links between optimization problems arising in applications and the classical mathematical area of real algebraic geometry. Nonnegative polynomials and sums of squares are the key objects linking optimization and algebraic geometry. The PI will carry out research that sheds light on the computational quality of sums-of-squares methods while building new connections with algebraic geometry. Such connections between mathematics, engineering, and natural sciences enrich mathematics by bringing new types of questions, new perspectives, and new directions of research. The PI will also organize two summer workshops for graduate students centered around student presentations on applications of algebraic geometry. The participating students will be exposed during their graduate studies to perspectives from several different scientific fields, while learning from their own peers.
The PI will investigate the connection between nonnegative polynomials and sums of squares in all of its aspects: algebraic, algorithmic, analytical. The PI will also study the closely related topic of real symmetric tensor decompositions. Understanding nonnegativity and its relation with sums of squares is one of the basic challenges of real algebraic geometry. Sums of squares methods have applications in diverse areas such as control and optimization, robotics, and complexity theory. Nonnegativity and sums of squares also have intrinsic connections to classical topics in algebraic geometry. The fundamental research that PI will carry out will lead to furthering these connections, while also improving the understanding of the computational quality of sums of squares algorithms. The PI will also research fundamental geometric aspects of real symmetric tensor decomposition, which are not nearly as well understood as for complex tensors.
