Studies on Local Cohomology, Derivations, Integral Dependence, and Blowup Algebras

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This project is in commutative algebra, but has also been inspired by algebraic geometry. Systems of polynomial equations in several variables occur in many applications in science and technology, for instance in engineering, computer science, cryptography, coding theory, robotics, pattern recognition, and theoretical physics. Commutative algebra and algebraic geometry are concerned with the qualitative study of such systems of polynomial equations. This project focuses on the algebraic approach. One of the goals of the project is to understand the vectors that are tangent to solution sets of systems of polynomial equations. Another objective is to find systems of polynomial equations describing a given geometric object. The latter has applications to computer graphics, algebraic statistics, and rigidity of structures. The PI intends to involve undergraduate students, graduate students, and postdoctoral fellows in her research. She will continue to organize national and international meetings. Throughout her activities, the PI will continue to promote underrepresented groups in mathematics. The first objective of this research is to relate degrees of vector fields on projective space to invariants of curves, or varieties, that they leave invariant. This is a difficult problem that has been studied for over a century, mainly from the point of view of complex analysis, dynamical systems, algebraic and differential geometry. The PI will investigate this question using tools from commutative algebra. The second objective is to find criteria for a variety in projective space to be a set-theoretic complete intersection. When there is only one non vanishing local cohomology module the PI believes that the property of being a set-theoretic complete intersection is encoded in the structure of this module. The third objective is to prove a numerical characterization of integral dependence of modules using a notion of multiplicity that arises in intersection theory. Theorems of this kind have a bearing on equisingularity theory, in fact they lead to fiber-wise numerical conditions for a family of analytic spaces to be Whitney equisingular, hence topologically trivial. The last objective is to study the implicit equations defining the graph and the image of rational maps between projective spaces.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.