Exact Structures in Representation Theory

这个项目的目的是在一般情况下发展稳定条件的理论，从而促进这一领域的知识。这一建议的目的是：1)在新的精确范畴的背景下研究稳定性条件；2)将矩阵约简技术解释为精确结构的改变。在这两种情况下，方法都是使用Quillen在他关于K理论的基础工作中定义的精确结构。它们提供了一个框架，允许使用同调代数中的方法，但严格来说比阿贝尔范畴的设置更具一般性。稳定性条件已在几何学和理论物理学中得到了系统的研究。其动机源于一种名为镜像对称的现象，该现象将两个与理论物理相关的看似无关的几何物体联系在一起。矩阵归约技术是基辅学派用来获得代数表示论基本结果的一个基本工具。矩阵归约的正式设置是使用称为BOCS的特定设置给出的。这种方法的缺点是，矩阵的迭代约简本身并不是一种复杂的技术，它产生了对待研究范畴中对象的越来越复杂的描述。我们在这里提出了一种全新的方法：我们建议保持相同的类别，但改变其确切的结构，而不是改变对象。我们计划应用这种精确结构约简的技术来解决与表象理论和镜像对称性有关的问题。

The aim of this project is to develop a theory of stability conditions in a general setting and thus advance the knowledge in this field. The objectives of this proposal are to: 1) study stability conditions in the new context of exact categories, and  2) interpret the matrix reduction technique as a change of exact structure. The approach is, in both cases, to work with exact structures that have been defined by Quillen in his fundamental work on K-theory. They provide a framework that allows the use of methods from homological algebra, but is strictly more general than the setting of abelian categories. Stability conditions have been studied systematically in geometry and theoretical physics. The motivation stems from a phenomenon called mirror symmetry, relating two seemingly unrelated geometrical objects relevant to theoretical physics. The matrix reduction technique is an elementary tool that has been used by the Kiev school to obtain fundamental results in representation theory of algebras. The formal setup for matrix reduction has been given using a certain setup called called BOCSes. The disadvantage of this approach is that iterated reduction of matrices, per se not a complicated technique, produces more and more complicated descriptions of objects in the categories to be studied. We propose here a fundamentally novel approach: instead of changing the objects, we suggest to keep the same category, but to change its exact structure. We plan to apply this technique of reductions of exact structures to solve questions related to representation theory and mirror symmetry.