The PI will use the algebraic tool of valuations to undertake a detailed study of problems in complex dynamics and analysis. The key object in the study is a space of valuations on analytic functions in several variables. It has an intricate geometric structure already in the case of two variables, being a tree built up by infinitely many curve segments glued together. In more variables, the objects glued together are higher-dimensional simplices. The PI will do analysis and dynamics on this space, and deduce results on several problems. Among many specific proposed projects, one involves establishing an algebraic version of the Oseledec theorem in nonlinear dynamics. Another project concerns the openness conjecture by Demailly and Koll'ar on singularities in analytic geometry. In both projects, the general plan is to translate the problem to more tractable questions on valuation space. The two-dimensional cases were studied earlier by Favre and the PI.
Singularities appear throughout mathematics, even when the primary objects of study are regular objects. An example from geometry is given by a cone: the points outside the apex are regular, since when magnifying the cone looks like a straight plane there, but the apex is a singularity. Another type of singularity occurs in dynamical systems, when applying iterative algorithms with fast convergence rates. Dynamical systems and complex analysis are established areas of mathematics, with connections to economics, biology and engineering. Working with students at the graduate and undergraduate levels, the PI will undertake a unified study of singularities in several different mathematics fields, including dynamical systems and complex analysis.
