Moduli spaces and higher representation theory

表示论是通过线性作用来研究对称性的理论。高级表示理论引入了一种新的范式，其中空间被高级结构（阿贝尔或三角范畴，或高级范畴结构）所取代。在过去的20年里，这种方法一直在提倡，特别是研究量子引力的物理学家，但迄今为止收效甚微。研究范畴之间的函子显得越来越重要，比如将我们感兴趣的范畴与我们更好理解的其他范畴进行比较。我们的主张是，我们应该研究这些函子之间的关系，通过这样做，我们将发现一些基本类型的新对称性，类似于向量空间的经典对称性。这将提供具体的（代数的、数值的）信息，而目前对范畴和函子的研究是在抽象的层次上完成的，具体的数据只能以大量的信息损失为代价来获得。这样的研究在一定程度上与通常的表示论相一致：人们定义了有趣的结构（例如，传统上人们会考虑对称群、简单李代数），并研究了它们可以成为对称的可能对象（例如，传统上人们试图对简单表示进行分类，这是一般表示的基石）。一个重要的新特征是，虽然向量空间是相当基本的结构，但范畴（阿贝尔或三角）不是。一个重要的后果将是更好地理解各种类别的代数或几何起源通过研究他们的更高的对称性。该提案的一个关键方面是提供从其他类别中构造类别的方法。模空间的构造应该被绕过，而相关的范畴结构应该被直接构造。开发一个代数替代模结构是该项目的主要灵感。该项目的目的是开发一种新的方法来计算（交换和非交换）几何中的不变量，基于PI的高级表示理论计划。

Representation theory is the study of symmetries via linear actions. Higher representation theory introduces a new paradigm, wherein spaces are replaced by higher structures (abelian or triangulated categories, or higher categorical structures). Such approaches have been advocated over the last twenty years, in particular by physicists working on quantum gravity, but very little has been achieved so far. It has appeared more and more important to study functors between categories, say to compare a category we are interested in with other categories we understand better. Our claim is that we should study the relations between these functors, and by doing so, we will discover some new symmetries of a fundamental type, analogous to classical symmetries for vector spaces. This would provide concrete (algebraic, numerical) information, while the current study of categories and functors is completed at an abstract level, and concrete data can be obtained only at the expense of a great loss of information. Such a study goes partly in line with usual representation theory: one defines interesting structures (classically one would, for example, consider symmetric groups, simple Lie algebras) and investigates the possible objects they can be symmetries of (classically one tries, for example, to classify simple representations, which are the building bricks for general representations). An important new feature is that, whilst vector spaces are fairly elementary structures, categories (abelian or triangulated) are not. An important consequence would be a better understanding of various categories of algebraic or geometric origin via the study of their higher symmetries. A crucial aspect of the proposal is to provide constructions of categories from other categories. Constructions of moduli spaces should be bypassed and the associated categorical structures should be constructed directly. Developing an algebraic substitute for moduli constructions is the main inspiration for the project. The aim of this project is to develop a new approach to counting invariants in (commutative and non-commutative) geometry, based on the PI's programme of higher representation theory.