Date and Time: 03/07/2025, 14:00--15:00

Location: 25 Park Place Room 1441

Colloquium: Rational powers of ideals: properties and applications

Speaker: Catalin Ciuperca, North Dakota State University

Speaker's website: https://sites.google.com/ndsu.edu/catalin-ciuperca/home

Title: Rational powers of ideals: properties and applications

Abstract: The study of asymptotic properties of ideals is a common theme in commutative algebra where algebraic and geometric properties of an ideal I are revealed by considering the asymptotic behavior of the power ideals I^n. In many instances, considering the asymptotic behavior of the integral closures of the power ideals instead of the regular powers has the advantage of eliminating some of the irrelevant information and simplifying the theoretical approach.

The rational powers of an ideal are a refinement of the integral closures of the power ideals. They capture more information associated with the ideal I while maintaining the theoretical advantage of working with multiplicative filtrations of integrally closed ideals.  In an algebraic context, they were introduced by David Rees in the 70’s; in geometry, about the same time, they also appear (in some disguised form) in the work of Bernard Teissier about singularities.

In this talk we will give an exposition of the theory of rational powers with a focus on their behavior under derivations. We will also present several applications that justify the usefulness of this construction.

Host: Florian Enescu (fenescu@gsu.edu)