Commutative Algebra and Algebraic Geometry are among the oldest and yet most active disciplines in mathematics. The fields have strong ties to such diverse areas as complex analysis, topology, and number theory, and are used in a wide variety of applied settings. Applications range from error-correcting codes in computer science and genomics to control theory and modeling in engineering. These fields seek to understand geometric objects (algebraic varieties) given locally as the solutions to polynomial equations. For instance, a plane curve is the zero set of a polynomial in two variables (such as the cusp y^2 = x^3). The richness and simplicity of polynomial equations make algebraic varieties fascinating objects of study. The program of research completed by the PI during his fellowship has led to a deeper understanding of the varieties and singularities in positive characteristic, i.e. over number systems having the property that a prime number vanishes. In particular, these systems include the finite fields at the heart of essentially all electronic computation. Singularities and numerical invariants defined via the Frobenius endomorphism are an important part of the study of Commutative Algebra and Algebraic Geometry in positive characteristic. To that end, the program of research completed by the PI has led to a number of important developments in these areas. These include: adescription of test ideals via alterations, as well as transformation rules for them via birational maps; a uniform description of test ideals (characteristic p > 0) and multiplier ideals (characteristic 0), as well as an important analog of the Nadel vanishing theorem for test ideals up to finite covers; and finally showing the existence of a local numerical invariant called the F-signature which characterizes the maximal asymptotic growth of F- splitting numbers, answering a question that had remained open for over a decade. During his tenure, the PI has also worked extensively to have broader impacts outside his research program. He has remained active in the research community at large, attending numerous conferences and coordinating various seminars. Furthermore, the PI (co)organized three significant research conferences: Relating test ideals and multiplier ideals, American Institute of Mathematics, August 8-12, 2011; AMS Special Session, Birational Geometry and Moduli, Joint Mathematics Meetings in New Orleans, January 5-6 2011; and F-Singularities and Invariants, University of Michigan, May 29-June 1, 2012. This last workshop was partially funded by the NSF (DMS # 1160927). The first year of the fellowship of the PI was spent at the University of Utah, where he met regularly with a number of graduate students, postdocs, and other researchers. He additionally volunteered to run a problem session in a continuing education program for math educators, and led an independent study course for two graduate students (Brian Mann and Sonya Liebman). Since moving to Princeton University for the next two years of the fellowship, the PI coordinated a new teacher training program for graduate students,(co)organized the algebraic geometry seminar, supervised an independent study course for an undergraduate (Sarah Trebat-Lieder), and mentored an undergraduate on a senior research project (Samuel Shiedler). Furthermore, in addition to teaching calculus, the PI designed and implemented a new course titled "Number, Shape, and Symmetry" for non-science majors, and also taught upper division course on Commutative Algebra for mathematics majors.
