Algebraic Geometry Close to Characteristic p

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This project supports research in algebraic geometry (which studies solutions to systems of polynomial equations in many variables). This subject is thousands of years old, and provides a systematic language for translating statements in geometry (such as "a doughnut is more curved than a ball") to those in algebra (such as "an elliptic function field is not rational"). A dominant theme of this project is to use this dictionary to better understand the variation in the geometric structure of a solution set of a certain system of equations as one perturbs the coefficients of the defining equations, especially in an arithmetic sense (i.e., in passing from usual arithmetic to modular or ``clockwork'' arithmetic). A better understanding of the geometric structure of solution sets of equations in modular arithmetic is fundamental to many applications of mathematics. The PI shall study the interaction between techniques coming from number theory (such as cohomology theories in p-adic Hodge theory) and algebraic topology (such as topological Hochschild homology) in the context of algebraic varieties over a p-adic field. A deep relationship between the two is expected: they should be related in the same way that motivic cohomology and algebraic K-theory are related (i.e., via an Atiyah-Hirzebruch type spectral sequence). The PI shall also use tools coming from number theory and p-adic geometry (such as perfectoid geometry) to approach problems in commutative algebra.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.