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: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=664878
• 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. ****My proposed research consists of several projects that are good examples of the fruitful interaction of algebra and geometry discussed above. Firstly, they involve calculations, development and investigation of homological invariants in algebraic and differential geometry. Secondly, they are concerned with studying algebraic structures geometrically and geometric structures algebraically. ****The anticipated outcome of the line of research described above is the improved understanding of various mathematical objects that occur naturally and are of significant interest to both mathematicians and physicists. ************************

我对几何和代数之间的相互作用及其在表示理论、数论和数学物理中的应用感兴趣。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，以涵盖看似纯粹的代数问题。另一方面，同调代数可以被看作是一种工具，以代数形式从不变量的幌子下的几何和代数数学对象中提取基本信息。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，相互作用和互补。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，在这个领域中，以前难以解决的问题可以用现有的方法来解决。****我提出的研究包括几个项目，这些项目是上面讨论的代数和几何富有成效的相互作用的好例子。首先，它们涉及代数和微分几何中同调不变量的计算、发展和研究。二是从几何角度研究代数结构，从代数角度研究几何结构。****上述研究方向的预期结果是提高对自然发生的各种数学对象的理解，这些对象对数学家和物理学家都有重大的兴趣。************************