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

[00:06 . 89]韦克菲尔德国际研究奖学金计划使美国科学家和工程师能够到国外进行9至24个月的研究。该计划的奖励为联合研究提供了机会，并利用独特或互补的设施、专业知识和国外的实验条件。该奖项将支持Max D. Wakefield博士与Hiroaki Terao博士在日本札幌北海道大学开展为期24个月的研究。本项目由美国国家科学基金会国际科学与工程办公室（OISE）东亚和太平洋项目提供支持。这个程序的重点是在超平面的安排代数，几何和组合的相互作用。通过超平面的排列来连接不同场的一个关键例子是Hiroaki Terao在1983年证明的一个基本定理。该定理展示了派生模（一个代数和几何对象）与交格（一个超平面自由排列的组合对象）之间的深刻关系。这个定理促使寺尾教授推测，交点晶格决定了排列的自由度。这个猜想在一维和二维中是完全可以理解的，但在三维和更高的维度中是未知的。在这个项目中，主要研究者将研究由交点格决定的模空间中变化的排列的导数的模。这个程序的一个目标是写出从这个模空间到许多退化变体的映射的像的方程。当且仅当雅可比代数（多项式环模雅可比理想）是Cohen-Macaulay时，排列是自由的。该项目的另一个目标是计算雅可比代数的Cohen-Macaulaytype和Castelnuovo-Mumford正则性等不变量。特别重要的安排是考克斯特安排和复杂的反思安排。这个项目的另一个兴趣是超平面排列的极代数。它包含了排列的所有信息，并具有与推导模块相似的特征。这个项目的另一个目标是表征其极代数是完全交集的排列。主持人和主要研究者希望通过该程序的支持找到有关衍生模块，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.