A Unified Perspective on Singularities in Commutative Algebra and Algebraic Geometry

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This research project brings together different perspectives on singularities in commutative algebra and algebraic geometry. Both algebraic geometry and commutative algebra concern shapes defined by polynomial equations, such as the parabola that is the graph of the equation y = x^2. Not every shape is so smooth; some, such as the graph of the equation y^2 = x^3, have sharp points or other singularities. Study of geometric singularities is important in mathematics and has application to mathematical models for a wide range of important physical, social, and economic systems. Researchers have long known that important types of singularities in classical geometric settings also appear naturally when studying geometric shapes in modular (or clock) arithmetic. In fact, algebraic geometry in the clock arithmetic setting is a key component of modern communications infrastructure. This project aims to develop a theory of singularities in mixed characteristic, a middle ground between the classical and clock arithmetic worlds (that possesses aspects of both). The project also aims to develop geometric applications of this theory and to develop open-source software to study singularities in commutative algebra and algebraic geometry. Techniques developed in number theory and arithmetic geometry have given rise to numerous recent breakthrough results in mixed characteristic commutative algebra. This project aims to use these methods to develop a singularity theory suitable for studying higher dimensional birational algebraic geometry in mixed characteristic, which would also facilitate translating many of the successes of positive characteristic commutative algebra to this setting. The project additionally aims to develop an analog of F-pure or log canonical singularities (and their centers) in mixed characteristic, to study fundamental groups of singularities, and to study boundary divisors in this setting.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.