Cluster algebras through representation theory

抽象代数是对代数结构的研究。例如，整数与加法和乘法的集合是我们都熟悉的代数结构。它正式地被称为环，因为两个给定的运算，加法和乘法，满足一些非常自然的性质。环不需要由整数构造，所涉及的操作也不需要类似于我们所知道的加法和乘法。抽象代数通常提供一种代数设置，人们可以在其中研究自然界或其他数学领域中出现的问题。

Abstract algebra is the study of algebraic structures. For instance, the set of integers together with addition and multiplication is an algebraic structure that we are all familiar with. It is formally called a ring, because the two given operations, addition and multiplication, satisfy some very natural properties. A ring need not be constructed from integers, and the operations involved need not be similar to addition and multiplication as we know them. Abstract algebra often provides an algebraic setting in which one can study a problem arising in nature, or in another field of mathematics. My research lies in representation theory of algebras, which is a branch of abstract algebra. We are interested in studying some algebraic structures, similar to rings, that are called algebras. Representation theory refers to the idea of representing a complex algebraic object by one that is easier to understand. In representation theory of algebras, we are interested in studying some objects called modules, by using elementary methods from basic linear algebra. Representation theory of algebras arises in many branches of mathematics, and even of physics. For instance, string theory, which is a branch of theoretical physics, has recently been studied by using a class of algebras called cluster algebras. The latter are deeply connected to representation theory of algebras. Indeed, in the recent years, representation theory of algebras has developed powerful tools to better understand these cluster algebras. Being a branch of abstract algebra, representation theory is in close connection to category theory, algebraic geometry and homological algebra. My proposal consists of developing representation theory of algebras, through its interactions with category theory, algebraic geometry and homological algebra. As in most areas of pure mathematics, it is very hard to predict the immediate impacts of this research. Many ideas in mathematics that are crucial today were developed in an abstract setting hundreds years ago. As representation theory is becoming more and more useful as a tool in other research areas, it is important to develop it, in concert with the development of these other fields. As a secondary objective of this proposal, I would like to use the new theoretical methods developed to better understand the cluster algebras. This has the potential to bring new methods for studying problems in physics, including problems in string theory.