The investigator will use valuations to undertake a broad and unified, yet detailed study of singularities arising in different mathematical fields, including algebraic geometry, complex analysis and dynamical systems. A cornerstone in the analysis is to encode singularities in terms of functions and measures on suitable valuation spaces, that is, sets of valuations on an appropriate ring. In the case of singularities at the origin in the affine plane, the valuation space has the structure of a tree with infinite and dense branching. In more general cases, it is expected to be a building, or projective limit of simplicial complexes. Being fundamental objects in complex analysis, positive closed currents may be approximated by varieties. As an analogue of Hironaka's theorem, the investigator will study whether every positive closed current admits an approximate resolution of singularities. He will also investigate the corresponding question for dynamical systems defined by holomorphic fixed point germs. Other problems to be addressed are semicontinuity of multiplier ideals and degree growths of iterates of rational maps of projective space. The investigator will design an undergraduate course and continue his development of a masters course to suit the needs at the University of Michigan (UM). Additionally he will engage middle and high school students in mathematical activities, in particular through a summer camp at the UM.
Singularities play a prominent role throughout mathematics, even when the primary objects of study are regular (smooth) objects. An example of a singularity is that of a curve in the plane that does not look like a line on a small scale, but rather as a cross or cusp, or even more complicated. It is known that plane singular curves can be viewed as "shadows" of nonsingular curves in space. This provides a way of understanding complicated object thorugh simpler ones. Other examples of singularities appear in dynamical systems, when studying the speed at which iterative algorithms converge. Working with students at the graduate and undergraduate levels, the investigator will undertake a unified study of singularities in different branches of mathematics. In addition he will develop a new undergraduate course and continue the redesigning of a graduate course. Finally he will work with K-12 students, in particular by continuing the development of a course in the framework of the Michigan Mathematics and Science Scholars summer program for high school students.
