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Homological interactions between singularity theory, representation theory and algebraic geometry

奇点理论、表示论和代数几何之间的同调相互作用

基本信息

批准号：
EP/L017962/1
负责人：
Martin Kalck
金额：
$ 31.96万
依托单位：
University of Edinburgh
依托单位国家：
英国
项目类别：
Fellowship
财政年份：
2014
资助国家：
英国
起止时间：
2014 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FL017962%2F1
关键词：
Homological interactions between singularity theory

项目摘要

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.

同调的概念出现在世纪以前，作为区分几何对象的工具：例如，甜甜圈和球当然是完全不同的几何对象-最明显的区别是前者存在一个“洞”。同调理论是一种以数学上严格的方式来检测这个洞的方法--反过来说，也是一种检测洞的缺失的方法。此外，它还产生了“维度”的精确概念，例如，允许区分直线和平面。关键思想是以“自然”的方式将代数结构与几何对象关联起来，即如果我们执行一些“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-转化为代数问题，这允许计算并且通常更容易解决，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：它的后代，同调代数领域，渗透到今天纯数学的许多领域。我们想利用同调技术的力量：这个研究项目的目的是建立表示理论和几何之间的新关系-特别是奇异空间。我们的研究对象是复杂的结构，更深入的了解将在这些领域和相关领域有许多应用。让我们简单地解释一下这两个领域的基本术语。表征理论对称性是数学中的一个中心思想，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，满足某些公理，使其成为一个群。相反，给定这样一个群，我们常常可以通过将它实现为空间上对称性的集合来阐明它的结构--这种实现被称为群的表示。表示论是研究群的表示和更一般的代数结构。2.几何与奇点理论。多项式属于最简单的数学对象。虽然，几个多项式的（公共）零集的研究可以追溯到古代，但今天仍然具有挑战性。这些消失的集合被称为“变种”。一个典型的变种上的一个典型的点将是很好的：它将局部类似于仿射空间，就像光滑曲线局部看起来像从拓扑角度看的线一样。奇点是这种良好的对应关系破裂的地方。他们在数学、物理以及几乎任何数学或物理应用的领域都很丰富。它们是如何连接的。给定任何簇（可能有奇点），我们可以将同调代数中的一个对象与它联系起来（“凝聚性的导出范畴”）。这个对象不允许我们完全重建原始结构，一些信息在转换过程中丢失。然而，这是一件好事：我们失去的一些信息无论如何都是多余的，通过减少到更重要的数量，我们的生活简化了。此外，这使我们能够更清楚地看到某些以前被隐藏的对称性。现在我们从表示论来看对象：给定一个代数，我们可以执行相同的过程并研究它的派生范畴。通常，这个派生的范畴与变种的范畴一致，揭示了两个对象共享的潜在结构的存在。这种巧合和相关的构造形成了两个不同数学领域之间的桥梁，可以以两种方式利用它来增加我们对（奇异）变种以及代数的理解。