CAREER: Test Ideals and the Geometry of Projective Varieties in Positive Characteristic

职业：检验理想和正特征中射影多样性的几何

• 批准号: 1252860
• 负责人: Karl Schwede
• 金额: $ 44.26万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-09-01 至 2014-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1252860&HistoricalAwards=false
• 关键词: CAREER; Test; Ideals; Geometry; Projective

The Principal Investigator will explore questions in characteristic p 0 algebraic geometry and commutative algebra. Explicitly, he proposes to apply methods of commutative algebra to study adjoint line bundles in characteristic p 0, especially to produce global sections. Such line bundles are crucial when classifying the central objects of algebraic geometry. Over the complex numbers, the standard techniques to study adjoint line bundles include Kodaira-type vanishing theorems. These theorems are unavailable in characteristic p 0, thus methods of the Frobenius morphism and test ideals from commutative algebra will be employed instead. The PI also proposes to use methods of algebraic geometry to study questions related to singularities in commutative algebra. The Principal Investigator will also run several mathematics camps for high school students. He will employ students to help develop software related to test ideals and singularities in characteristic p 0. Finally he will continue to organize conferences.Algebraic geometry is a very active field of mathematics. Explicitly, it is the study of geometric objects (called algebraic varieties) made up of the solutions to polynomial equations (for example, the parabola is the solution to y = x^2). Characteristic p 0 algebraic geometry is the study of these equations and their solutions when the underlying rules of arithmetic have changed. For example, instead of simply 4 + 4 = 8, in characteristic p = 5, once we get to 5 we start counting at 1 again so that 4 + 4 = 3 (like counting hours of a clock). This type of arithmetic and related algebra and geometry is heavily employed in modern cryptography and coding theory. The running of camps, conferences and the development of software will also help train the next generation of researchers.

首席研究员将探讨特征p 0代数几何和交换代数的问题。明确地，他提出了用交换代数的方法来研究特征p 0中的伴随线束，特别是生成全局截面。这样的线束在对代数几何的中心对象进行分类时是至关重要的。在复数上，研究伴随线束的标准方法包括kodaira型消失定理。这些定理在特征p 0中是不可用的，因此将采用交换代数中的Frobenius态射和检验理想的方法来代替。PI还建议使用代数几何的方法来研究交换代数中的奇点相关问题。首席研究员还将为高中生举办几个数学夏令营。他将雇用学生帮助开发与测试p 0特性的理想和奇点有关的软件。最后，他将继续组织会议。代数几何是一个非常活跃的数学领域。明确地说，它是对由多项式方程的解组成的几何对象（称为代数变量）的研究（例如，抛物线是y = x^2的解）。特征代数几何是研究这些方程及其在基本算术规则发生变化时的解。例如，在特征p = 5中，不是简单的4 + 4 = 8，而是当我们到达5时，我们再次从1开始计数，这样4 + 4 = 3（就像计算时钟的小时数）。这种类型的算术以及相关的代数和几何在现代密码学和编码理论中被大量使用。营地、会议和软件开发的运行也将有助于培养下一代研究人员。