Exact Structures in Representation Theory

该项目的目的是在一般情况下发展稳定性条件的理论，从而推进这一领域的知识。这个建议的目的是：1）研究在新的背景下的准确范畴的稳定性条件，和2）解释矩阵约简技术作为一个确切的结构的变化。在这两种情况下，方法都是使用奎伦在他关于K理论的基础工作中定义的精确结构。他们提供了一个框架，允许使用的方法从同调代数，但严格更普遍的设置比阿贝尔范畴。稳定性条件在几何学和理论物理学中都有系统的研究。其动机源于一种称为镜像对称的现象，将两个看似无关的几何对象与理论物理学联系起来。矩阵降阶技术是基辅学派用来获得代数表示论基本结果的一种基本工具。矩阵约简的正式设置已经使用称为BOCSes的特定设置给出。这种方法的缺点是，迭代减少矩阵，本身不是一个复杂的技术，产生越来越复杂的描述对象的类别进行研究。我们在这里提出了一个从根本上新颖的方法：而不是改变对象，我们建议保持相同的类别，但改变其确切的结构。 我们计划应用这种精确结构的简化技术来解决与表示论和镜像对称有关的问题。

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.