Interactions between p-adic arithmetic geometry and commutative algebra

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The PI proposes to investigate problems lying at the shared boundaries of arithmetic geometry (especially p-adic aspects), algebraic geometry, and commutative algebra. For example, Mel Hochster's direct summand conjecture, which posits the existence of some fundamental properties of regular rings, has been an important open problem in commutative algebra for over four decades; the solution to the equal characteristic case (due to Hochster from the 70s) is responsible for large swathes of modern commutative algebra, while the p-adic case remains tantalizingly open. The PI recently discovered that some ideas from p-adic Hodge theory (due to Faltings) can be used solve certain unknown cases of this conjecture. The PI intends to pursue this direction further by using powerful recent techniques --- chiefly Scholze's beautiful theory of perfectoid spaces --- from the fast evolving subject of p-adic Hodge theory to approach the direct summand conjecture and other purely algebraic problems. Conversely, previous work of the PI on the direct summand conjecture, coupled with some derived algebraic geometry, has recently proven instrumental in arriving at a significant simplification of certain geometric aspects of p-adic Hodge theory; the PI plans to develop the derived aspects more thoroughly to conceptualize the picture better.The study of solutions of polynomials with integer coefficients dates back to antiquity. An extremely useful technique here is to study "approximate" solutions first, i.e., solutions modulo primes, and then modulo powers of primes. The idea is that as the power of prime increases, the approximation becomes better. Grothendieck's revolutionization of mathematics in the last half century not only allows one to not only attach a precise meaning to the previous statement, but also provides a beautiful geometric context --- the world of p-adic geometry ---- to study such approximate solutions. This context has been at the heart of numerous recent advances in mathematics (such as Wiles' proof of Fermat's last theorem and other recent milestones in the Langlands program). The PI plans to contribute further to underlying geometric theory as well as develop applications to purely algebraic problems.