Cluster Algebras in Representation Theory and Symplectic Geometry

表示论是对不同类型的对称性的研究和分类，特别是在线性代数的背景下。从历史上看，它在数学中发挥了重要作用，因为分析一个给定的数学对象，例如方程组，当对象在适当的抽象意义上是对称的时，通常会显着简化。辛几何，另一方面，成长出来的哈密顿力学和今天占据了突出的地方，在数学由于广泛的背景下，例如在拓扑和代数几何，其中底层几何结构的哈密顿力学出现。该研究项目旨在通过利用代数组合学的新思想，特别是最近开发的簇代数理论，促进对这两个主题的理解，并挖掘它们之间的深层新联系。这些联系反过来又都是由数学物理学的最新进展，特别是弦理论和超对称规范理论所提供的信息。在表示论方面，本项目的主要研究对象包括仿射格拉斯曼及其相关范畴，特别是它们的衍生范畴等变相干层。在辛几何方面，主要对象是非紧辛4-流形和6-流形中的拉格朗日膜或微局部层范畴。在前一种情况下，簇结构出现在所涉及的几何对象的等变K-环的水平上，并且描述了由这些环继承的某些组合结构。在后者中，簇结构出现在4维拉格朗日膜的模空间上，或者通过基于6维福谷范畴的Hall代数型构造。通过其组合性质的语言集群代数提供了一种手段，隔离，否则非常丰富和复杂的数学对象，以及发现这些对象之间的新关系的易处理的方面。

Representation theory is the study and classification of different kinds of symmetry, specifically in the context of linear algebra. Historically it has played an important role in mathematics because analyzing a given mathematical object, for example a system of equations, is often dramatically simplified when the object is symmetric in a suitable abstract sense. Symplectic geometry, on the other hand, grew out of Hamiltonian mechanics and today occupies a prominent place in mathematics due to the wide range of contexts, for example in topology and algebraic geometry, where the underlying geometric structures of Hamiltonian mechanics appear. This research project aims to advance understanding of both subjects, as well as unearthing deep new connections between them, by leveraging new ideas from algebraic combinatorics, in particular the recently developed theory of cluster algebras. These connections are in turn all informed by recent advances in mathematical physics, specifically string theory and supersymmetric gauge theory.On the side of representation theory, the main objects of study in this project include the affine Grassmannian and its relatives, in particular their derived categories of equivariant coherent sheaves. On the side of symplectic geometry, the main objects are categories of Lagrangian branes or microlocal sheaves in noncompact symplectic 4- and 6-manifolds. In the former context, cluster structures appear at the level of the equivariant K-rings of the geometric objects involved, and describe certain combinatorial structures inherited by these rings. In the latter, cluster structures appear on moduli spaces of Lagrangian branes in 4 dimensions, or through Hall algebra-type constructions based on Fukaya categories in 6 dimensions. Through its combinatorial nature the language of cluster algebras provides a means of isolating tractable aspects of otherwise very rich and complicated mathematical objects, as well as uncovering new relations between these objects.