CAREER: Nonnegative Polynomials, Sums of Squares and Real Symmetric Tensor Decompositions

本计画探讨最佳化问题在应用与真实的代数几何的经典数学领域之间的连结。非负多项式与平方和是联系最优化与代数几何的关键问题。PI将开展研究，揭示平方和方法的计算质量，同时与代数几何建立新的联系。数学、工程和自然科学之间的这种联系通过带来新类型的问题、新的视角和新的研究方向来丰富数学。PI还将为研究生组织两个夏季研讨会，围绕代数几何应用的学生演讲。参与的学生将在研究生学习期间接触到来自几个不同科学领域的观点，同时向自己的同龄人学习。PI将研究非负多项式和平方和之间的联系，包括代数，算法，分析等各个方面。PI还将研究与之密切相关的真实的对称张量分解。理解非负性及其与平方和的关系是真实的代数几何的基本挑战之一。平方和方法在控制和优化、机器人和复杂性理论等不同领域都有应用。非负性和平方和也与代数几何中的经典主题有着内在的联系。PI将开展的基础研究将进一步促进这些联系，同时也提高了对平方和算法计算质量的理解。PI还将研究真实的对称张量分解的基本几何方面，这些方面的理解不如复杂的张量。

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.