Grant: Discovery Grants Program - Individual

Algebraic geometry studies systems of polynomial equations. They appear in all sorts of problems in science and technology, for instance, weather forecasts and crash-test simulations. One of the main obstacles in handling these systems is the existence of singularities. Singularities are points where the solutions of the system of equations do not depend in a well-behaved way on its parameters. Many excellent mathematicians have worked to achieve ""resolution of singularities"", that is, the transformation of a given system into a new system that has no singularities. In 1965, Hironaka obtained resolution of singularities in an important particular case (""characteristic zero""). For this work, he received a Fields Medal (the equivalent of the Nobel Prize for mathematicians). Since then many attempts have been made to settle the remaining case of ""positive characteristic"", but the problem is still open. If one cannot solve a problem globally (resolve all singularities) one tries to solve it locally (getting rid of just one singularity at a time). This problem, called ""local uniformization"", is also wide open in positive characteristic. Its solution is one main goal of my research. It involves valuation theory, an area of algebra that has witnessed important applications in number theory, algebraic geometry, and several other branches of mathematics. When valuations are studied from the point of view of mathematical logic, one has the same phenomenon: while the case of characteristic 0 is well-known, in positive characteristic the general case is open (and has turned out to be a very hard problem). The reason here, as well as for local uniformization, is a nasty valuation theoretical effect which only appears in positive characteristic. Its very descriptive name is: the ""defect"". In the past, several excellent mathematicians have surrendered to the very hard problems connected with it. It is the main goal of my work to tame the defect in order to achieve solutions to the above problems.