Representation theory and homotopical algebra

这项建议围绕着表示理论的两个主要方向，并建议发展同伦方法。在过去的二十年里，有限群的表示理论在很大程度上是由Alperin和Brou‘e的猜想推动的。这一提议将给该地区带来新的拓扑方向和新的猜测。其基本思想是用拓扑空间上的作用量代替向量空间上的作用量。更高表示理论是研究给定的么半范畴在范畴上的作用的理论，是一个较新的研究领域。它的发展主要与Kac-Moody代数有关。最近的一种构造是张量结构。我们的建议将带来理论上的同伦方法。在Khovanov的基础上，它还旨在将更高的表示理论引入到从事计数不变量的几何学家和致力于研究四流形的不变量的拓扑学家领域。该项目将在代数中带来新的拓扑学见解，特别是在表示论中，对称性的研究。一个主要的问题是从局部对称性来理解全局对称性。我们将在更广泛的拓扑环境中接近和重塑这一点。该项目还将产生在代数中建立对象的方法，作为几何中作为族或“模空间”的经典空间构造的对应物。这应该会导致通过代数来理解三维和四维几何的一些性质。

This proposal centers around two main directions in representation theory, and proposes to develop homotopical methods. The representation theory of finite groups has been driven in a large part by the conjectures of Alperin and Brou'e in the last twenty years. The proposal will bring new topological directions in the area, and new conjectures. The basic idea is to replace actions on vector spaces by actions on topological spaces. Higher representation theory, the study of actions of a given monoidal category on categories, is a much more recent area of research. It has been developed mostly in relation with Kac-Moody algebras. A recent construction is that of tensor structures. Our proposal will bring homotopical methods in the theory. It aims also at bringing higher representation theory in the area of geometers working on counting invariants and topologists working on invariants of four-manifolds, in the continuation of Khovanov.The project will bring new topological insight in algebra, and in particular in representation theory, the study of symmetries. A major problem is to understand global symmetries from local symmetries. We will approach and recast this within a wider topological setting. The project will also produce methods to build objects in algebra, as a counterpart of the classical constructions of spaces in geometry as families or "moduli spaces". This should lead to an understanding through algebra of some properties of three and four-dimensional geometry.