Algebra, geometry and applications

代数、几何及其应用

• 批准号: 406709-2011
• 负责人: Shapiro, Ilya
• 金额: $ 1.17万
• 依托单位: University of Windsor
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2013
• 资助国家: 加拿大
• 起止时间: 2013-01-01 至 2014-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=529986
• 关键词: Algebra; geometry; applications

I am interested in the interplay between geometry and algebra and their applications to representation theory, number theory and mathematical physics. In my work I use the language of algebraic geometry and the methods of homological algebra. Modern algebraic geometry is a branch of mathematics which vastly generalizes our geometric intuition to encompass seemingly purely algebraic problems. On the other hand homological algebra can be viewed as a tool designed to extract, in an algebraic form, essential information from geometric and also algebraic mathematical objects in the guise of invariants. Taken together they provide very powerful techniques for dealing with a wide variety of problems. Namely they allow the different branches of mathematics: geometry and algebra, to interact and complement each other. Thus a question in one area may be solved by transforming it into a question in another area where a previously intractable problem may become very accessible to the methods already available.

我感兴趣的是几何和代数之间的相互作用，以及它们在表示论、数论和数学物理中的应用。 在我的工作中，我使用代数几何的语言和同调代数的方法。 现代代数几何是数学的一个分支，它极大地概括了我们的几何直觉，包含了看似纯粹的代数问题。 另一方面，同调代数可以被视为一种工具，旨在提取，在代数形式，基本信息的几何和代数数学对象的伪装不变量。 综合起来，它们为处理各种各样的问题提供了非常强大的技术。 也就是说，它们允许数学的不同分支：几何和代数，相互作用和相互补充。 因此，一个领域的问题可以通过将其转化为另一个领域的问题来解决，在另一个领域，以前难以解决的问题可能变得非常容易使用现有的方法。