Principal Investigator: Sean T. Paul
This proposal aims to relate the stability of a complex projective variety to intrinsic geometric analysis on the variety. Often, the PI is concerned with the small time asymptotics of complicated rational energy integrals over the variety under study. This idea was initiated by the PI in his dissertation. The energies arise in connection with the problem of establishing the 0th order apriori estimate for solutions to the Monge Ampere equation.
Broadly speaking, there are two ways to mathematically approach, or model, a given problem: continuously, or discretely. Traditionally, these are quite seperate approaches. Analysis (differential equations in particular) is the time honoured subject in the continuous domain, combinatorics (the study of enumerating a finite amount of data) is the hallmark of the discrete approach. In this proposal these two methods come together-the PI will explore the question of how the solution to an equation from analysis might be obtained by a purely finite (but large) collection of data. The equation appeared in Einsteins' theory of Gravitation, whereas the finite set of data arose in the mathematics of the 1800's, and is related to the theory of computer vision.
