Date of Award

Summer 8-12-2011

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Graduate Group

Mathematics

First Advisor

Robert Ghrist

Abstract

This thesis de nes and classi es valuations on de nable functionals. The intrinsic
volumes are valuations on "tame" subsets of R^n, and by easy extension, valuations on functionals on R^n with nitely many level sets, each a "tame" subset of R^n. We extend these valuations, which we call Hadwiger integrals, to de nable functionals on R^n, and present some important properties of the valuations. With the appropriate topologies on the set of de nable functionals, we obtain dual classi cation theorems for general valuations on such functionals. We also explore integral transforms, convergence results, and applications of the Hadwiger integrals.