Homological interactions between singularity theory, representation theory and algebraic geometry

同源的概念出现在一个多世纪前，作为区分几何对象的工具：例如，甜甜圈和球肯定是完全不同的几何对象--最明显的区别是前者中存在一个“洞”。同调理论是一种以严格的数学方式检测这种空洞的方法--反之，也可以检测出空洞的存在。此外，它产生了一个精确的“维度”概念，例如，允许区分一条线和一个平面。其关键思想是以一种“自然”的方式将一个代数结构与一个几何对象相关联，也就是说，如果我们执行一些“可接受的”几何操作--例如拉伸或收缩--那么代数结构应该保持不变。这将我们最初的几何问题--区分几何对象--转化为一个代数问题，它允许计算，通常更容易解决，例如，在直线和平面的情况下，我们最终得到不同的数字1和2。同调理论非常成功：它的产物，同调代数领域，渗透到今天纯数学的许多领域。我们想使用同调技术的力量：这个研究项目的目的是在表示理论和几何之间建立新的关系，特别是奇异空间。我们的研究对象是复杂结构，对复杂结构的深入理解将在这些领域和相关领域有很多应用。让我们用基本术语简要地解释这两个领域。表象理论。对称性是数学中的一个中心概念，它经常导致论证和计算的简化。保持空间对称性的所有变换的集合满足将空间变成一个群的某些公理。相反，给定这样一个群，我们通常可以通过将其认识为空间上的对称性的集合来阐明其结构--这样的认识被称为群的表示。表示论是研究群和更一般的代数结构的表示的理论。几何与奇点理论。多项式属于最简单的数学对象。尽管对几个多项式的(公共)零点集的研究可以追溯到古代，但它在今天仍然具有挑战性。这些消失的集合被称为‘变种’。典型变化上的一个典型点将是很好的：它将局部类似于仿射空间，就像从拓扑观点来看局部看起来像直线的平滑曲线一样。奇点是这种美好的对应关系被打破的地方。他们在数学、物理以及几乎所有应用数学或物理的领域都有丰富的知识。它们是如何联系起来的。给定任何种类(可能带有奇点)，我们可以将同调代数中的一个对象与其联系在一起(“相干层的派生范畴”)。这个对象不允许我们完全重建原来的结构，在转换过程中丢失了一些信息。然而，这是一件好事：我们丢失的一些信息无论如何都是多余的，通过减少到更重要的量，我们的生活变得简单。此外，这使我们能够更清楚地看到以前隐藏的某些对称性。现在我们从表示理论来看对象：给定一个代数，我们可以执行相同的过程并研究其派生范畴。这个派生的范畴通常与变种的范畴重合，揭示了两个对象共享的底层结构的存在。这种重合和相关的结构在两个不同的数学领域之间形成了一座桥梁，这可以用两种方式来增加我们对(奇异)变体和代数的理解。

The notion of homology arose over a century ago as a tool to distinguish geometric objects: e.g. a doughnut and a ball are certainly quite different geometric objects - the most obvious difference beeing the existence of a 'hole' in the former. Homology theory is a way to detect this hole - and conversely, also the absence of a hole - in a mathematically rigorous manner. Moreover, it yields a precise notion of 'dimension', allowing for example, the distinction of a line and a plane.The key idea is to associate an algebraic structure to a geometric object in a 'natural' way, i.e. if we perform some 'admissable' geometric operation - such as stretching or shrinking - then the algebraic structure should stay the same. This translates our original geometric problem - to distinguish geometric objects - into an algebraic one, which allows for computations and is often easier to solve, e.g. in the case of the line and the plane, we end up with the different numbers 1 and 2. Homology theory was very successful: its offspring, the field of Homological algebra, permeates many areas of pure mathematics today.We want to use the power of homological techniques: the aim of this research project is to build new relations between representation theoryand geometry - in particular, of singular spaces. The objects of our study are complicated structures, a deeper understanding of which will have many applications in these and related fields. Let us briefly explain these two areas in elementary terms.1. Representation theory. Symmetry is a central idea in mathematics, which often leads to simplifications of arguments and calculations. The collection of all transformations preserving the symmetry of a space, satisfies certain axioms turning it into a group. Conversely, given such a group, we can often elucidate its structure, by realising it as collection of symmetries on a space - such a realisation is called a representation of the group. Representation theory is the study of representations of groups and more general algebraic structures.2. Geometry & Singularity theory. Polynomials belong to the simplest mathematical objects. Although, the study of (common) zero sets of several polynomials dates back to antiquity, it remains challenging today. These vanishing sets are called 'varieties'. A typical point on a typical variety will be nice: it will locally resemble affine space, just like smooth curves locally look like lines from a topological viewpoint. Singularities are places where this nice correspondence breaks down. They are abundant in mathematics, physics and almost any field in which either mathematics or physics is applied.3. How they are connected. Given any variety (possibly with singularities), we can associate an object from homological algebra to it ('the derived category of coherent sheaves'). This object does not allow us to reconstruct the original structure completely, some information is lost in the transformation process. This, however, is a good thing: some of the information we lose is superfluous anyway and by reducing to more essential quantities, our life simplifies. Moreover, this allows us to see certain symmetries, that were hidden before, more clearly.Now we look at objects from representation theory: given an algebra, we can perform the same process and study its derived category. Often this derived category coincides with that of the variety, revealing the existence of an underlying structure that both objects share. This coincidence and related constructions form a bridge between two different areas of mathematics, which can be exploited in both ways to increase our understanding of (singular) varieties as well as algebras.