Cluster Algebras in Representation Theory and Symplectic Geometry

表征理论是对不同种类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，它在数学中扮演着重要的角色，因为当对象在适当的抽象意义上对称时，分析给定的数学对象，例如方程组，通常会大大简化。另一方面，辛几何是从哈密顿力学中发展出来的，今天在数学中占据着突出的地位，因为它有广泛的背景，例如在拓扑和代数几何中，哈密顿力学的潜在几何结构就出现了。该研究项目旨在通过利用代数组合学的新思想，特别是最近发展的聚类代数理论，促进对这两个主题的理解，并挖掘它们之间的新联系。这些联系都是由数学物理的最新进展，特别是弦理论和超对称规范理论所决定的。在表示理论方面，本项目的主要研究对象包括仿射格拉斯曼及其亲戚，特别是它们的等变相干束的派生范畴。在辛几何方面，主要研究对象是非紧辛4-和6-流形中的拉格朗日膜或微局部轴的范畴。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，并描述了这些环继承的某些组合结构。在后者中，团簇结构出现在4维拉格朗日膜的模空间上，或者在6维中通过基于Fukaya范畴的Hall代数型构造。通过它的组合性质，簇代数语言提供了一种方法，可以从其他非常丰富和复杂的数学对象中分离出易于处理的方面，并揭示这些对象之间的新关系。

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.