Algebra, geometry and applications

我感兴趣的是几何和代数之间的相互作用，以及它们在表象理论、数论和数学物理中的应用。在我的工作中，我使用了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，涵盖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，旨在以不变量的名义，以代数形式从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--相互作用和补充。因此，一个领域的问题可以通过将其转变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会变得非常容易利用现有的方法。我提议的研究由几个项目组成，它们是上面讨论的代数和几何之间卓有成效的互动的很好例子。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。其次，他们致力于从几何上研究代数结构，并从代数上研究几何结构。上述研究路线的预期结果是改进了对自然发生的各种数学对象的理解，并且对数学家和物理学家都非常感兴趣。

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.