International Research Fellowship Program: Algebra, Geometry, and Combinatorics: Arrangements of Hyperplanes

国际研究奖学金计划使美国科学家和工程师能够在国外进行9到24个月的研究。该项目的奖项提供了联合研究的机会，并利用国外独特或互补的设施、专业知识和实验条件。该奖项将支持Max D.Wakefield博士与日本札幌市北海道大学的Hiroaki Terao博士合作的为期24个月的研究奖学金。该项目得到了美国国家科学基金会国际科学与工程办公室(OISE)东亚和太平洋计划的支持。该计划的重点是超平面排列中代数、几何和组合学的相互作用。通过超平面的排列将不同的场联系起来的一个关键例子是Hiroaki Terao在1983年证明的一个基本定理。该定理展示了导子模(代数和几何对象)与超平面自由排列的交格(组合对象)之间的深层次关系。这个定理促使Terao教授推测，交点格子决定了排列的自由度。这一猜想在一维和二维完全可以理解，但对于三维和更高维则未知。在这个项目中，主要的研究人员将研究通过交点格确定的模空间变化的排列的导子的模。这个程序的一个目标是为从这个模空间到许多退化变种的映射的映象写出方程。一个排列是自由的当且仅当雅可比代数(模为雅可比理想的多项式环)是Cohen-Macaulay。这个项目的另一个目标是计算雅可比代数的不变量，如Cohen-Macaulaytype和Castelnuovo-Mumford正则性。特别重要的安排是Coxeter安排和复杂的反射安排。这个项目的另一个兴趣是超平面排列的非极代数。它包含了排列的所有信息，并具有类似于派生模的特征。这个项目的另一个目标是刻画非极代数是完全交的排列。主持人和主要研究人员希望通过这个程序的支持，找到关于导子模、非极代数和其他相关对象的信息。

0600893WakefieldThe 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 relationshipbetween 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 conjectureis 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 isto 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-Macaulaytype 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 throughthe support of this program.