Cluster algebras through representation theory

抽象代数是研究代数结构的学科。例如，整数集合加上加法和乘法是我们都熟悉的代数结构。它被正式称为环，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。一个环不需要由整数构成，所涉及的运算也不需要与我们所知道的加法和乘法相似。抽象代数通常提供一个代数环境，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。 代数表示论是抽象代数的一个分支。我们感兴趣的是研究一些代数结构，类似于环，被称为代数。表示论是指用一个更容易理解的对象来表示一个复杂的代数对象的思想。在代数的表示论中，我们感兴趣的是使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，甚至出现在物理学中。例如，弦理论是理论物理学的一个分支，最近人们用一类叫做簇代数的代数来研究它。后者与代数的表示论有着深刻的联系。事实上，近年来，代数的表示理论已经开发出了强大的工具来更好地理解这些簇代数。作为抽象代数的一个分支，表示论与范畴论、代数几何、同调代数等学科有着密切的联系。 我的建议包括发展表示理论的代数，通过其与范畴论，代数几何和同调代数的相互作用。正如在纯数学的大多数领域一样，很难预测这项研究的直接影响。数学中许多今天至关重要的思想都是在几百年前的抽象环境中发展起来的。随着表征理论在其他研究领域中作为一种工具变得越来越有用，重要的是要发展它，与这些其他领域的发展相一致。作为这个提议的第二个目标，我想使用新的理论方法来更好地理解簇代数。这有可能为研究物理学问题带来新的方法，包括弦理论问题。

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.