Under this award, the PI studied the area of "positivity" in algebraic geometry. This research explored the connection between manifolds - shapes in higher dimensional space - and polynomial equations. The goal was to explore the profound connections between two areas of mathematics: topology and algebra. This theme has a long history in algebraic geometry. Starting with the revolutionary work of Grothendieck, Serre, and Kodaira in the 50s, divisors have played a crucial role in the foundations of the subject. A divisor is a "codimension one" subset of an algebraic variety. In particular, Kodaira's work elucidated the tight links between the cohomology, geometry, and topology of a ample divisor. Over the past 30 years, these ties have been extended to many other kinds of divisors as well. However, for larger codimension subsets, very little is known about this issue. The work of the PI and his coauthors gives new insight into the situation. From the right perspective, there turns out to be many surprising parallels between divisors and higher codimension subsets. This is an active area of research that has many promising applications to other areas of mathematics. Also, the PI completed the project proposed by the award - the study of multiplier ideals from the algebraic and the analytic perspective. The goal was to give algebraic "bounds" or limits on the behavior of analytic multiplier ideals. The key idea was to study the behavior of these ideals under the deformation of a line bundle.
