Algebraic geometry and commutative algebra

纯数学研究的惊人特征之一是，它经常惊人地发现美丽的一般结果，这些结果可以用来帮助回答科学和工程其他部分的重要问题。 这项提案中正在考虑的问题就是这种情况。这项研究是在被称为“代数几何”的领域进行的，这是一个自古希腊人研究圆锥体截面以来就存在的学科。我们在这个提案中考虑的问题是：一组方程的解的集合有多大？知道如何为这项提议中提出的特定方程回答这个问题，将有助于决定：生物学家提出的进化树是否正确；是否有可能使用功能强大的计算机以高效的方式将极大的矩阵相乘(例如，这是经济学家在寻求为我们的经济建模时感兴趣的事情)；医学研究人员如何从数百万患者记录中提取有用数据，而每个患者记录要么不包含任何数据，要么只包含关于一种特定疾病的很小一部分数据。 该提案的目标之一是培训年轻的研究人员成为代数几何方面的专家，并了解生物学、统计学和计算中可以用该提案的方法解决的问题的性质。这在加拿大是一个新的研究领域，有可能提供机会在纯数学、生物学、计算和统计学之间建立新的联系。我的学生还将有机会参加拟议的欧洲代数几何应用研究网络和正在规划的其他倡议(例如，2010年在Mittag-Leffler研究所)，这些网络与本提案中的研究相关联。

One of the amazing features of pure mathematics research is how shockingly often it discovers beautiful general results which can be used to help answer important problems in other parts of science and engineering. This is the case with the problems being considered in this proposal. The research is in the area called "algebraic geometry", a subject that has been around since the ancient Greeks studied the sections of a cone. The problem we consider in this proposal is: how big is the set of solutions to a collection of equations? Knowing how to answer that question for the particular equations that come up in this proposal will help decide: if an evolutionary tree proposed by biologists is correct; if it is possible to use an enormously powerful computer to multiply exceedingly large matrices in an efficient manner (something, for example, that economists are interested in doing when they seek to model our economy); how medical researchers can extract useful data from millions of patient records where each individual record contains either none or only a very little bit of data on one specific disease. One of the goals of the proposal is to train young researchers to become expert in algebraic geometry and to also understand the nature of the problems in biology, statistics and computing which are amenable to solution by the methods of this proposal. This is a new field of research in Canada which has the potential to offer opportunities to forge new connections between pure mathematics, biology, computing and statistics. My students will also have the opportunity to participate in the proposed European-based Research Network on Applications of Algebraic Geometry and other initiatives being planned (e.g. at the Mittag-Leffler Institute in 2010) which are connected to the research in this proposal.