Collaborative Research: Differential Methods, Implicitization, and Multiplicities with a View Towards Equisingularity Theory

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This research project concerns the structure of geometric objects that arise as solution sets of systems of polynomial equations in several variables. Such objects, called varieties, play a fundamental role throughout mathematics as well as in applications in science and engineering. Motivated by a question that originates with mathematician Henri Poincaré in 1891, the investigators will study the relationship between local properties of a variety and global features of tangents at points of the variety, as these points vary. They will also work on implicitization, a classical question in pure mathematics that is of much interest to scientists in geometric modelling and computer aided design. Given any geometric object, such as a curve or surface, the goal is to find the system of polynomial equations that has the geometric object as a solution set; knowing these 'implicit' equations provides insight into the geometric object. The PIs will involve graduate students and postdoctoral visitors in the project, and they will facilitate scientific exchange by organizing international programs, conferences, and national online seminars.The investigators will work on projects pertaining to algebraic vector fields, the implicitization problem for Rees rings, equisingularity theory, and residual intersections. The PIs will use tools from commutative algebra to investigate how the types of singularities and the global invariants of a variety are reflected in properties of the vector fields that are tangent to the variety. In particular, they wish to establish a correspondence between the types and constellation of the singularities of a projective plane curve on the one hand and the graded Betti numbers of the module of derivations of its homogeneous coordinate ring on the other. Determining the implicit equations defining the graph and image of a rational map between projective spaces is a classical problem in elimination theory. The PIs will concentrate on the case of Cremona maps, where the implicit equations of the graph also provide a parametrization of the inverse map. For dominant rational maps they will investigate the relationship between the projective degrees of the rational map and the number and bidegrees of the equations defining the graph. In equisingularity theory, one seeks fiberwise multiplicity-based criteria for a family of analytic spaces to be Whitney equisingular and hence topologically trivial. The PIs plan to devise such a criterion for analytic spaces with arbitrary singularities by using a new notion of multiplicity inspired by intersection theory. Graduate students will be supported as part of the project.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.