The International Research Fellowship Program enables U.S. scientists and engineers to conduct nine to twenty-four months of research abroad. The program's awards provide opportunities for joint research, and the use of unique or complementary facilities, expertise and experimental conditions abroad.
This award will support a twenty-four-month research fellowship by Dr. Max D. Wakefield to work with Dr. Hiroaki Terao at Hokkaido University in Sapporo, Japan. Support for this project comes from the East Asia and Pacific Program of NSF's Office of International Science and Engineering (OISE).
The focus of this program is the interaction of algebra, geometry, and combinatorics in an arrangement of hyperplanes. A key example of this connection of different fields through an arrangement of hyperplanes is a fundamental theorem proved by Hiroaki Terao in 1983. The theorem exhibits a deep relationship between the module of derivations, an algebraic and geometric object, and the intersection lattice, a combinatorial object, of a free arrangement of hyperplanes. This theorem helped motivate Professor Terao to conjecture that the intersection lattice determines the freeness of the arrangement. This conjecture is completely understood in dimension one and two, but is unknown for dimensions three and higher. During this program the principle investigator will study the module of derivations of arrangements varying through a moduli space determined by the intersection lattice. One objective of this program is to write equations for the image of a map from this moduli space to the many degeneration varieties. An arrangement is free if and only if the Jacobian algebra (the polynomial ring modulo the Jacobian ideal) is Cohen-Macaulay. Another objective of this project is to compute invariants such as Cohen-Macaulay type and Castelnuovo-Mumford regularity of the Jacobian algebra. Arrangements of specific importance are Coxeter arrangements and complex reflection arrangements. An additional interest in this project is the apolar algebra of an arrangement of hyperplanes. It contains all the information of the arrangement and has similar characteristics of the module of derivations. Another goal of this project is to characterize arrangements whose apolar algebra is a complete intersection. The Host and principal investigator expect to find information about the module of derivations, apolar algebra, and other related objects through the support of this program.
