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

主要研究者将探讨问题的特点p 0代数几何和交换代数。 解释，他建议应用方法交换代数研究伴随线丛的特点p 0，特别是生产全球部分。 当对代数几何的中心对象进行分类时，这种线丛至关重要。 在复数上，研究伴随线丛的标准技术包括Kodaira-type消失定理。 这些定理在特征p 0中是不可用的，因此，本文采用交换代数中的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与奇点有关的问题。 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理想和特征p 0中的奇点相关的软件。 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。 简单地说，它是研究由多项式方程的解组成的几何对象（称为代数簇）（例如，抛物线是y = x^2的解）。特征p 0代数几何是研究这些方程和他们的解决方案时，基本规则的算术已经改变。 例如，在特征p = 5中，我们不是简单地4 + 4 = 8，而是一旦我们到达5，我们就再次从1开始计数，这样4 + 4 = 3（就像计数时钟的小时）。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得到了大量应用。 举办夏令营、会议和开发软件也将有助于培养下一代研究人员。

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.