Symbolic Powers and Lefschetz Properties: Geometric and Homological Aspects

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This research concerns problems in commutative algebra motivated by algebraic geometry. At the heart of a wide array of scientific endeavors is the ubiquitous need to solve polynomial equations. A complementary goal is to find polynomial equations, for example, an equation whose graph passes through a given set of data points. This procedure, termed polynomial interpolation, is a fundamental challenge at the interface of data science, numerical analysis, and algebraic geometry. The investigator will bring methods from commutative and computational algebra to bear on aspects of a higher order version of polynomial interpolation. For example, the situation when the data points exhibit intrinsic symmetry will be elucidated. Additionally, a deeper understanding of the interactions between this topic and emerging techniques in homological algebra will be pursued. The broader impact of this fundamental research lies in the engagement and training of graduate students, software development, and the recruitment, retention, and professional development of junior mathematicians. The research project focuses on two topics which generate current excitement: polynomial interpolation and the algebraic Lefschetz properties. The former theme will be analyzed through the lens of symbolic power ideals, which can be thought of as sets of polynomials that vanish to a certain order on a given algebraic variety. The latter theme constitutes an algebraic abstraction of the Hard Lefschetz Theorem with spectacular applications to several areas of mathematics. The interrelations between these two topics will be thoroughly explored and exploited. One particular direction of investigation is on applications of the algebraic Lefschetz properties to homological algebra, specifically to graded free resolutions. Other directions include applications to the containment problem relating the ordinary and symbolic topologies defined by an ideal. Aspects of this work exhibit relationships to the theory of reflection groups, hyperplane arrangements, convex geometry, and differential graded algebras.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.