Dualities and Correspondences in Algebraic Geometry via Derived Categories and Noncommutative Methods

对由一组多项式的公共零点给出的曲线和曲面的研究是人类几千年来一直在追求的。在现代代数几何学中，我们研究任意维中的这样的集合：这些集合被称为代数簇。对于这样一个品种，人们可以问很多问题：它有多“好”？如果我们站在上面，它看起来像一个弯曲的山还是像一个粗糙的山？如果我们有两个这样的变种，我们能分辨出它们是否相同吗？或者如果它们是相似的，例如，如果站在它们上面的大多数地方，它们看起来都是一样的，只有从某些精确的点看它们时，它们看起来才不同？导出范畴是一种考虑这些几何对象并将有关它们的许多信息转换为代数概念的方法。虽然一个簇的派生范畴保留了我们开始时关于簇的许多信息，但同时它允许我们在代数上下文中工作的额外灵活性。在过去的20年里，由于许多经典的代数几何问题都是通过导出范畴技术来解决的，导出范畴领域经历了大量的活动。关于导出范畴的一个基本问题是关于两个不同几何对象的导出范畴是如何相关的。这些关系中的一些可能来自于两种变体之间的关系和对称性，但它们之间还有其他类型的关系，这些关系更深刻，更难理解：1。首先，理解两个派生范畴之间的映射（函子）是什么样的是很重要的。其中许多-但不是全部，因为人们过去认为！- 有一个非常愉快和有用的几何描述为“傅立叶-Mukai函子”。我的项目的一部分将包括分析和描述不是傅立叶-向井函子的“坏”映射，以及这些映射是如何通过变形我们所知道的“好”映射而自然产生的。2.两个派生范畴之间的另一种关系，是由范畴层次上的“对偶”概念给出的，这将作为我的项目的一部分来研究。描述这种二元性给了我们一种方法来理解尚未发现的派生范畴之间更深层次的关系，这将在派生范畴和几何对象的层次上揭示更多的对称性和行为。3.最后，在某些情况下，导出范畴之间的关系是等价的，因此可以用Fourier-Mukai函子表示，并且在导出范畴的层次上的分析给我们提供了大量的几何信息。我的项目将解决一个这样的例子，即调查的一些商奇点，是一个推广的克莱因奇点，和他们的解决方案的奇点。

The study of curves and surfaces given by the common zeroes of a set of polynomials has been pursued by humanity for thousands of years. In modern algebraic geometry, we study such sets in any dimension: these are called algebraic varieties. There are a number of questions that one can ask about one such variety: how "nice" is it? If we were standing on it, would it look to us like a curvy hill or like a rough mountain? If we are given two such varieties, can one tell if they are the same? Or if they are similar, for example if standing in most places on them they would look the same, and they only look different when looking at them from certain precise spots?Derived categories are a way to consider these geometric objects and translate much of the information about them into algebraic notions. While the derived category of a variety retains much of the information about the variety we started with, at the same time it allows us extra flexibility to work in an algebraic context. In the past two decades the field of derived categories has experienced an outpouring of activity as many classical algebraic geometry problems are solved passing through derived categories techniques.One fundamental question about derived categories is about how the derived categories of two different geometric objects are related. Some of these relations might come from relations and symmetries between the two varieties, but there are also other kinds of relations between them, which are deeper and harder to understand:1. First of all, it is important to understand what the maps (functors) between two derived categories are like. Many of these - but not all, as people used to think! - have a very pleasant and useful geometric description as "Fourier-Mukai functors". Part of my project will consist in analyzing and describing the "bad" maps that are not Fourier-Mukai functors, and how these arise naturally by deforming the "good" maps we know about. 2. Another relation between two derived categories, which will be investigated as part of my project, is given by a concept of "duality" at the categorical level. Describing this duality gives us a way to understand deeper relations between derived categories that haven't yet been discovered, and that will shed more light on the symmetries and behavior both at the level of derived categories and at the level of the geometric objects.3. Finally, in some instances the relations between derived categories turn out to be equivalences and hence representable by Fourier-Mukai functors, and the analysis on the level of derived categories gives us back a big amount of geometric information. My project will tackle one such instance, namely the investigation of some quotient singularities that are a generalization of the Kleinian singularities, and their resolutions of singularities.