Multiplicities and Valuations in Commutative Algebra

This award supports a research project on the interaction of algebra and geometry, and the application of commutative algebra to other areas of mathematics. A motivating problem is the existence of resolution of singularities. To resolve singularities is to smooth out, by algebraic operations, the singularities in a space defined by polynomial or analytic equations. Resolution of singularities is still open in certain cases for spaces of dimension greater than 3. It is of importance in other branches of mathematics, physics and engineering. An important focus of the project will be the training of graduate students and the mentoring of young mathematicians from diverse backgrounds. A major topic to be investigated in this project is properties and applications of filtrations in local rings. Another topic to be investigated is inseparable local uniformization, the local resolution of singularities along a valuation after taking a purely inseparable extension. The PI will also investigate the characterization of good properties of extensions of valuations. The project will explore properties of filtrations of rings, including their analytic spread, Hilbert functions and generalized multiplicities. A particular emphasis will be on divisorial filtrations which although non-Noetherian, share many good properties of the Noetherian I-adic filtrations of an ideal I. Inseparable local uniformization will be investigated. This is a local resolution of singularities, after taking a purely inseparable extension. The PI will also study extensions of valuation rings with the goal of giving valuation theoretic characterizations of when the extensions are almost finite etale, or have related good properties. 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.