Algebraic geometry and commutative algebra

代数几何和交换代数

• 批准号: 8488-2009
• 负责人: Geramita, Anthony
• 金额: $ 2.04万
• 依托单位: Queen's University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=568595
• 关键词: Algebraic; geometry; commutative; algebra

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.

纯数学研究的一个令人惊讶的特点是，它经常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领域的重要问题。 本提案所考虑的问题就是这种情况。这项研究是在所谓的“代数几何”领域，这是一个自古希腊人研究圆锥截面以来就存在的学科。我们在这个建议中考虑的问题是：一组方程的解的集合有多大？知道如何回答这个问题的具体方程出现在这个建议将有助于决定：如果进化树提出的生物学家是正确的;如果有可能使用一个非常强大的计算机乘以非常大的矩阵在一个有效的方式（例如，经济学家在试图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以百万计的患者记录中提取有用的数据，其中每个单独的记录都不包含或只包含一种特定疾病的非常少的数据。 该建议的目标之一是培养年轻的研究人员成为代数几何专家，并了解生物学，统计学和计算中的问题的性质，这些问题可以通过该建议的方法解决。这是加拿大的一个新的研究领域，有可能为纯数学、生物学、计算和统计学之间建立新的联系提供机会。我的学生也将有机会参加拟议中的欧洲研究网络代数几何的应用和其他计划（例如，在2010年米塔格-莱弗勒研究所），这是连接到本提案中的研究。