Cluster algebras through representation theory

抽象代数是研究代数结构的学科。例如，整数集加上加法和乘法是我们都熟悉的代数结构。它在形式上被称为环，因为加法和乘法这两个给定的运算满足一些非常自然的性质。环不需要由整数构成，所涉及的运算也不需要类似于我们所知道的加法和乘法。抽象代数通常提供了一个代数环境，人们可以在这个环境中研究自然界中出现的问题，或者在另一个数学领域中出现的问题。 我的研究方向是抽象代数的一个分支--代数的表示论。我们感兴趣的是研究一些被称为代数的类似于环的代数结构。表示理论指的是用一个更容易理解的代数对象来表示一个复杂的代数对象。在代数表示论中，我们感兴趣的是利用基本线性代数的初等方法来研究一些称为模的对象。 代数表示论出现在数学的许多分支中，甚至在物理学中也是如此。例如，弦理论是理论物理的一个分支，最近人们利用一类称为簇代数的代数来研究它。后者与代数的表示论有着密切的联系。事实上，近年来，代数的表示理论已经发展出强大的工具来更好地理解这些簇代数。表示论作为抽象代数的一个分支，与范式论、代数几何、同调代数有着密切的联系。 我的建议包括通过它与范式论、代数几何和同调代数的相互作用来发展代数的表示理论。就像在大多数纯数学领域一样，很难预测这项研究的直接影响。今天至关重要的许多数学思想都是在数百年前的抽象环境中发展起来的。随着表征理论作为一种工具在其他研究领域中变得越来越有用，与这些其他领域的发展相协调地发展它是很重要的。作为这个提议的次要目标，我想使用发展起来的新的理论方法来更好地理解簇代数。这有可能为研究物理问题带来新的方法，包括弦理论中的问题。

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.