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

国际研究奖学金计划：代数、几何和组合学：超平面的排列

• 批准号: 0600893
• 负责人: Max Wakefield
• 金额: --
• 依托单位: Wakefield Max D
• 依托单位国家: 美国
• 项目类别: Fellowship
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-09-15 至 2009-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0600893&HistoricalAwards=false
• 关键词: International; Research; Fellowship; Program; Algebra

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.

0600893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在国外进行9到24个月的研究。该计划的奖项为联合研究以及使用国外独特或互补的设施、专业知识和实验条件提供了机会。该奖项将支持Max D.博士为期24个月的研究奖学金。韦克菲尔德将与日本札幌的北海道大学的寺尾弘明博士合作。该项目由美国国家科学基金会国际科学与工程办公室（OISE）的东亚和太平洋计划提供支持。该计划的重点是代数、几何和组合学在超平面排列中的相互作用。通过超平面的排列将不同场连接起来的一个关键例子是寺尾宏明在1983年证明的一个基本定理。该定理揭示了导子模（代数和几何对象）与超平面自由排列的交格（组合对象）之间的深刻关系。这个定理帮助激发寺尾教授猜想交叉格子决定了排列的自由度。这个概念在一维和二维中是完全理解的，但在三维和更高的维度中是未知的。在这个程序中，主要研究者将研究通过由交叉晶格确定的模空间变化的安排的导子模块。这个程序的一个目的是写一个映射的图像从这个模空间的许多退化品种的方程。一个排列是自由的当且仅当雅可比代数（多项式环模雅可比理想）是科恩-麦考利。该项目的另一个目标是计算Jacobian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射安排。在这个项目中的另一个兴趣是安排超平面的非极代数。它包含了所有的信息的安排，并具有类似的特点，模块的推导。这个项目的另一个目标是表征安排的非极代数是一个完整的交叉。主持人和主要研究者希望通过这个程序的支持找到关于导子模块、非极性代数和其他相关对象的信息。