This research is in algebraic geometry, specifically in the area of resolution of singularities. The goal is to develop a new and unified approach to resolution of singularities by studying the infinitesimal loop space of a singular variety. This involves a revisiting and strengthening of the classical valuation theory, since the infinitesimal loop space is related to the abstract Riemann surface of the function field. Much of the work will be on the theory of valuations. Geometric object arising as the solutions of polynomial equations will in general have singularities (cusps, crossings etc). Methods of removing these singularities is an important subject in algebraic geometry. This proposal seeks to develope a unified approach to this diverse subject.
