CAREER: Invariant Theory, Algorithms and Applications

职业：不变理论、算法和应用

• 批准号: 0349019
• 负责人: Harm Derksen
• 金额: $ 40万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0349019&HistoricalAwards=false
• 关键词: CAREER; Invariant; Theory; Algorithms; Applications

DMS-0349019Harm DerksenThis is a CAREER award investigating Invariant Theory, Algorithms, and Applications. The PI will build up a research group working on algorithms in algebra. He will work on algorithms for various difficult problems in algebra and the connection between complexitytheory and Invariant Theory. The PI will work with Jessica Sidman on Castelnuovo-Mumford regularity and applications to various conjectures. The PI will work with Jerzy Weyman on quiver representations, invariant theory and applications. They also plan to write an introductory book on quiver representations. A graduatestudent will work on the connection between semi-invariants for quivers, combinatorics and exact sequences of abelian p-groups. In addition, the PI will also work on the new notion of "black box algebras". These algebras can be used to describe classical invariant theory. There are many interesting applications, such as PI-theory, and universal formulas for decompositions of tensor products of representations of simple lie algebras. Courses will be developed in the area of algorithms in algebra.In Invariant Theory one studies algebraic quantities which remain unchanged after certain transformations. For example, consider a 3-dimensional space. Consider a point (x,y,z) under rotation around the z-axis. Two fundamental invariants are the distance to the z-axis, and the z-coordinate. These quantities remain unchanged after any rotation around the z-axis. Every algebraic invariant can be expressed in terms of these two fundamental invariants. A famous theorem of Hilbert from 1890 tells us that given a collection of transformations (satisfying certain hypotheses) of a space of arbitrary dimension, there exists a finite set of fundamental invariants such that every invariantcan be expressed in these fundamental invariants. This proposalsupports the study of applications of Invariant Theory to several areasin mathematics. It also supports research on algorithms forvarious problems in Invariant Theory.

DMS-0349019 Harm Derksen这是一个研究不变理论、算法和应用的职业奖项。PI将建立一个研究小组，致力于代数算法的研究。他将致力于解决代数中各种困难问题的算法，以及复杂性理论和不变量理论之间的联系。PI将与杰西卡·西德曼合作研究Castelnuovo-Mumford正则性，并将其应用于各种猜想。PI将与Jerzy Weyman在箭图表示、不变量理论和应用方面进行合作。他们还计划写一本关于箭袋表示法的介绍性书籍。毕业生将研究箭图的半不变量、组合学和交换p-群的精确序列之间的联系。此外，PI还将研究“黑盒代数”的新概念。这些代数可以用来描述经典不变理论。它有许多有趣的应用，如PI-理论，以及用于单李代数表示的张量积的分解的通用公式。课程将在代数算法领域展开。在不变量理论中，我们研究的是在某些变换后保持不变的代数量。例如，考虑一个3维空间。考虑绕z轴旋转的点(x，y，z)。两个基本不变量是到z轴的距离和z坐标。这些量在绕z轴旋转后保持不变。每个代数不变量都可以用这两个基本不变量来表示。1890年希尔伯特的一个著名定理告诉我们，给定任意维空间的一组变换(满足某些假设)，存在一个有限的基本不变量集，使得每个基本不变量都可以用这些基本不变量来表示。这一建议支持了不变量理论在数学中几个领域的应用研究。它还支持对不变量理论中各种问题的算法研究。