Rank functions on triangulated categories, homotopy theory and representations of finite groups

这项拟议的研究是在大约50年前由英国伟大的数学家D·奎伦创立的被称为同伦代数的纯数学领域。同伦代数源于拓扑空间的同伦理论，即研究几何形式和形状在连续变形下不变的性质。同伦理论的相关性质的公理化导致了封闭模型范畴的概念；这个概念在解决拓扑问题方面取得了巨大的成功，但也影响了大量邻近的领域：同调代数、代数和微分几何、表示论，甚至数学物理。最近，由于与微分分次范畴和无穷范畴相关的一个新的强大的思想圈的发展，同伦代数得到了新的推动；这些思想圈是由美国的J.Lurie和他的学派，欧洲的B.Toen和C.-D.西辛斯基等人发展起来的。这一思想圈子隐含地支撑着本项目。更确切地说，后者应该被归类为应用同伦代数，因为它利用了同伦代数的抽象范畴结构，特别是派生的局部化，并将其应用于同伦论、非交换几何和表示论中的各种问题，其中一些是当前非常感兴趣的问题，另一些由于缺乏新的概念而被搁置了多年。推导局部化是最近提出的一个概念，粗略地说，它是一种以同伦不变的方式对非交换环中的元素进行倒置的方法。由于这种同伦不变性，这个过程在许多方面类似于经典的交换局部化过程(这是代数几何的基石之一)。派生局部化在代数拓扑学和(派生)代数几何中已经得到了广泛的应用。该项目的目标是将派生本地化的适用性扩展到新的、可能意想不到的方向。

This proposed research is in the area of pure mathematics called homotopical algebra created around 50 years ago by a great British mathematician D. Quillen. Homotopical algebra takes its motivation from homotopy theory of topological spaces, the study of those properties of geometric forms and shapes that do not change under continuous deformation. Axiomatization of relevant properties of homotopy theory leads to the concept of a closed model category; this concept has been enormously successful for tackling topological problems, but also impacted a vast array of neighbouring fields: homological algebra, algebraic and differential geometry, representation theory and even mathematical physics. Recently, homotopical algebra received a new impetus due to the development of a new powerful circle of ideas related to differential graded and infinity categories; these have been developed by J. Lurie and his school in the USA, B. Toen and C.-D. Cisinski in Europe and others. This circle of ideas implicitly underpins the present project. More precisely, the latter should be classed as belonging to applied homotopical algebra in the sense that it uses the abstract categorical constructions of homotopical algebra, particularly derived localization, and applies it to a wide variety of problems in homotopy theory, noncommutative geometry and representation theory, some of which are of much current interest and others has lain dormant for many years for lack of new ideas.Derived localization is a concept developed in a recent work by proposers; roughly speaking, it is a way to invert elements in non-commutative rings in a homotopy invariant way. Because of this homotopy invariance, this procedure is in many respects similar to the classical procedure of commutative localization (which is one of the cornerstones of algebraic geometry). Derived localization has already found numerous applications in algebraic topology and (derived) algebraic geometry. The goal of this project is to extend applicability of derived localization in new, possibly unexpected, directions.