Bridgeland stability and the moveable cone

生活中有许多例子，两个数字可能是相同的，但被计数的对象之间没有联系。例如，硬币1 p，2 p，5 p，...，2英镑的英国货币，以及一只健康蜘蛛的腿数。然而，数学中的这种巧合有时只是不同结构之间更深层次上存在的一种更有趣的关系的影子。当这一点得到解释时，我们不仅理解了最初的巧合，而且更好的是，我们可以将我们的理解从一种结构转换到另一种结构。简而言之，我们揭示了两种语言之间的字典，我们可能只是部分理解，一个这样的例子是出现在两个明显不同的数学环境中的正数：一个等于以相当抽象的代数方式定义的某些类型的对称的数量;另一个是通过计算某些几何定义的对象获得的。事实上，这种巧合可以用一种被称为“麦凯对应”的关系来解释。粗略地说，这种对应以一种非常具体的方式描述了两个相当抽象的对象（称为三角范畴），一个是用代数定义的，另一个是用几何定义的，实际上是相同的。每一个这样的类别都编码某些数字，而最初的巧合归结为简单的观察，即两个相同的类别编码完全相同的数字！因此，这就是纯数学家的主要目标之一：研究明显的巧合是否可以用自然的方式来解释，而正是这种对“自然”概念的探索，使纯数学对科学和工程的许多领域都很重要。正如上面所描述的麦凯对应一样，人们知道，即使对于编码同一种类数的不同类型的几何，也确实存在许多这样的对应（称为范畴的等价），其中一些已经在过去15年左右的几位数学家的工作中得到了非常优美的描述。即使是现在，我们也不清楚总体情况，但数学家邦达尔和奥尔洛夫提出了以下问题：“如果我们有两种几何类型，它们虽然不同，但在某种程度上是相似的，是否存在上述对应关系来解释这种相似性？".在这里，我们的目标是通过引入一个特定的地图-一种“机器”-PI已经深入研究的抽象概括，为这个问题奠定一个新的几何方法的基础。至关重要的是，我们相信，我们准确地理解了正确的抽象层次，从而照亮了正确的道路：太少的抽象可能根本无法照亮任何东西;而太多的抽象可能是如此盲目，以至于无法提供任何帮助。虽然我们没有完整的图片，但我们相信我们已经找到了问题的正确基础，并为我们的方法提供“概念证明”，我们将证明它适用于一类有趣的例子。我们相信，这个建议的结果将为几个有趣的问题提供解决方案，这些问题加在一起，为我们理解著名的邦达尔和奥尔洛夫猜想提供了重要的几何步骤。

There are many instances in life when two numbers might be the same and yet there is no relation between the objects that are being counted. For example, the number of coins 1p, 2p, 5p,..., £2 in British currency, and the number of legs on a healthy spider. However, such coincidences in mathematics are sometimes merely the shadow of a much more interesting relation that exists on a deeper level between different structures. When this can be explained, we not only understand the original coincidence but, better yet, we can translate our understanding from one structure to the other. Put simply, we uncover the dictionary between two languages that we might only partially understand.One such example is provided by a positive number that appears in two apparently different mathematical contexts: one is equal to the number of certain types of symmetry that are defined in a rather abstract, algebraic way; and the other is obtained by counting certain geometrically-defined objects. In fact, this coincidence can be explained quite beautifully by a relation known as the "McKay correspondence". Roughly speaking, this correspondence describes in a very concrete way in which two rather abstract objects (called triangulated categories), one defined in terms of algebra and the other defined in terms of geometry, are in fact the same. Every such category encodes certain numbers, and the original coincidence boils down to the simple observation that two identical categories encode precisely the same numbers! This, then, is one of the primary goals of a pure mathematician: to investigate whether apparent coincidences can be explained in a natural way, and it is precisely this search for the "natural" notion that makes pure mathematics important to so many fields of science and engineering.The current proposal aims to do precisely this. As with the McKay correspondence described above, it has been known for some time that many such correspondences (called equivalences of categories) do exist even for rather different types of geometry which encode the same kind of numbers, and some of these have been described very elegantly by the work of several mathematicians over the last fifteen years or so. Even now, the general picture eludes us, but the following question has been posed by mathematicians Bondal and Orlov: "If we have two types of geometry that, while being different are nevertheless similar in a controlled way, does there exist a correspondence as above to explain the similarity?". Here we aim to lay the foundation for a new, geometric approach to this problem by introducing an abstract generalisation of a particular map - a kind of "machine" - that the PI has studied in depth. Crucially, we believe that we understand precisely the right level of abstraction to shed light on the correct path: too little abstraction may illuminate nothing at all; while too much abstraction may be so blinding as to provide no help whatsoever. While we do not have the full picture, we do believe that we have found the correct foundation for the problem, and to provide a "proof of concept" for our approach we will demonstrate that it works for an interesting class of examples. The results that will come from this proposal will, we believe, provide solutions to several interesting problems that, taken together, provide an important, geometric step towards our understanding of the celebrated conjecture of Bondal and Orlov.