This project in the field of algebraic geometry concerns aspects of valuation theory. Valuations appear naturally in many branches of mathematics. Spaces of valuations, as a whole, capture important information, for example in connection with the problem of resolution of singularities. There are many natural ways of looking at valuations and packaging them into spaces, and each provides useful tools for studying algebraic varieties. The focus of the proposed research is on two valuation spaces: the space of arcs and the Berkovich analytification of an algebraic variety. These spaces come equipped with interesting structure and can be used to answer important mathematical questions.
Nash investigated the space of arcs in connection to singularities; the same space serves as the underlying space in motivic integration, and has been applied to study invariants of singularities in the minimal model program. Berkovich's non-Archimedean geometry has been used in a variety of contexts, from p-adic geometry to dynamics, geometric group theory, mirror symmetry, and tropical geometry, and most recently in birational geometry. Many interesting questions about the structure of these spaces remain open, and a better understanding of their geometry will lead to new applications. One specific goal of this project is to study the local rings on the arc space of a variety, a problem motivated by Shokurov's semicontinuity conjecture on minimal log discrepancies and therefore, indirectly, by the conjecture on termination of flips, one of the missing steps in the minimal model program. Another objective proposes a new point of view on motivic integration where Berkovich spaces are used in place of arc spaces. The project also addresses some questions about links of isolated singularities, their contact structure, and their CR structure. While these are apparently unrelated questions, there is an underlying connection between the contact structure of a link and valuation theory.
