Interaction of Commutative Algebra, Valuations, and Geometry

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This project is on the interaction of algebra and geometry. A major focus is on the analysis of singularities and their resolution. Resolution of singularities is the process of smoothing out, by algebraic operations, corners and cusps in a space defined by polynomial equations. This is of importance throughout mathematics and physics and has potential application to engineering. One direct application of research from this project is to the implicitation problem in computer aided geometric design. Commutative Algebra, Valuation Theory, Algebraic Geometry and Convex Geometry are unified in this project. An important focus of the project will be the training of graduate students and the mentoring of young mathematicians from diverse backgrounds. The theory of multiplicities and mixed multiplicities will be developed, by extending the theory from filtrations of m-primary ideals to arbitrary filtrations. Within this theory, the question of upper semicontinuity of multiplicity will be studied. The methods of the convex geometry of Okounkov bodies, commutative algebra and valuation theory will be fundamental in this project. The problem of determining a largest filtration which contains a given filtration and has the same multiplicity will be investigated. The theory of mixed multiplicities of line bundles on a projective variety will be studied, with the goal of characterizing when the Minkowski equality holds. Local uniformization of Abhyankar valuations dominating arbitrary excellent local rings, and the role of defect in local uniformization will be investigated.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.