Collaborative Research: Bridging Singularities in Algebra and Geometry Across Characteristics

Commutative algebra studies mathematical structures such as the integers and polynomials, and has broad applications in computer science, engineering, and other areas of mathematics. Execution of the planned research will deepen the theoretical foundations of the field and address problems in related disciplines such as algebraic geometry, which studies the geometric objects associated to polynomials. A central focus of this research is on singularities, or points where the geometric objects behave irregularly (for example, a curve crossing over itself), using techniques from modular arithmetic (also known as clock arithmetic or prime characteristic algebra) and mixed characteristic settings (where a prime is treated as a variable). These approaches, including the use of perfectoid algebras, help bridge distinct mathematical worlds and enhance our understanding of both. In addition, the principal investigators are dedicated to promoting mathematics education, developing future generations of researchers, and assisting in building a strong STEM workforce in the US. Towards these goals, the principal investigators will supervise, train, and mentor graduate students and postdoctoral fellows. The principal investigators will also facilitate seminars and workshops for undergraduate and graduate students. Recent advances, including past efforts of the principal investigators, have inspired a rapidly-emerging theory of singularities in mixed characteristic, bridging the existing notions from classical complex geometry defined using resolutions of singularities with those in positive characteristic commutative algebra that utilize Frobenius splittings and tight closure theory. Establishing finiteness properties is a crucial component in the study of singularities across characteristics. In prime characteristic, the principal investigators will study finite generation of the anticanonical algebra for certain singularities defined by Frobenius, and the existence and properties of boundary divisors in both prime and mixed characteristics, which in turn can be used to prove strong finiteness conditions. The investigators will also research log canonical singularities and ideal closure operations in the complex setting, better definitions of F-pure pairs in prime characteristic, and the theory of singularities outside the F-finite setting in prime characteristic. 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.