Cutkosky proposes to investigate several problems involving the interaction of algebraic geometry and commutative algebra. Asymptotic properties of high powers of ideals will be investigated. Good stable properties tend to hold for high powers. Toroidalization of mappings of algebraic varieties will be studied, continuing previous work of the PI. Ramification of varieties in positive characteristic will be investigated, with an eye towards local uniformization. The author will investigate the related problem of classification of semigroups of valuations on a noetherian ring, and the construction of generating sequences of valuations.
Algebra is of increasing importance in modern science and technology. Cutkosky will investigate several problems of theoretical importance in algebra. Undergraduate and graduate students will be trained and young researchers will be mentored through this project.
The project investigated several fundamental problems in commutative algebra and algebraic geometry. Siignificant contributions were made in the project toward monomialization, including giving a counterexample to local monomialization in positive characteristic. Simple necessary and sufficient conditions are given for the existence of all limits of lengths of graded families of ideals and applications are given of this. Simple necessary and sufficient conditions are given for the existence of all limits of lengths of graded linear series on a projective scheme over an arbitrary field. The research from this work has application to applied mathematics. For instance, the work on monomialization has applications to solving systems of equations, showing convergence of integrals and quantifier elimination. The training of graduate students and post docs has been a significant part of the project. Soumya Sanyal and Pham An Vinh obtained their Ph.D,s under my direction at the University of Missouri on topics related to this research (both in May, 2014). Razieh Ahmadian is well on her way towards completing her Ph.D. on a topic from this research, and I am currently training Arpan Dutta to begin his thesis. Sara Gharabeigi is a post doc at the University of Missouri who is working with me on a project from this research. I have also given several series of lectures for graduate students at international workshops on material from this project. In Septenber 2012, September 2013, and soon to be held in September 2014 we have held the KUMUNU conference at the University of Missouri. The KUMUNU conference is a rotating conference between the University of Kansas, the University of Missouri and the University of Nebraska. These conferences on commutative algebra have had attendence of 60 - 80 participants, mostly graduate students and young researchers from neighboring universities.. They have been joint organized by Hema Srinivasan, Dana Weston and Seven Dale Cutkosky