Categorifying Turbulence in Borel Reducibility

分类是一个在科学中无处不在的概念。无论是动物还是元素，粒子还是恒星，从一个广泛定义的对象或个体类别中描述可能的亚群的过程对于加深我们的理解非常有效。这在纯数学的抽象世界中也是正确的，在那里被分类的对象是数学结构。退一步说，我们可以用非常一般的术语来理解分类如下。我们有一些数学对象，我们要分类，可能要识别“本质上相同”的对象。我们还有其他的数学对象，称为“不变量”，我们希望用它们来分类（就像颜色用来分类恒星，或者质子数用来分类化学元素一样），同样，对于不变量，可能有一个“本质上相同”的概念。然后，分类是一个合理的可定义的映射，将被分类的对象映射到不变量，尊重这些“本质上相同”的概念。Borel归约的框架封装了这个过程，并使用了这样的观察，即经常被分类的对象和不变量可以被编码成真实的数字，就像信息被编码成计算机上的比特串一样。有了这种编码，我们就可以给出一个“合理定义的映射”的正式特征作为分类。至关重要的是，有时可以表明不存在这样的地图，从而表明不可能找到希望的分类。这对于排除不同数学领域的分类，如遍历理论和C* 代数的研究是很重要的。这些不可能性证明的核心工具是Hjorth的湍流概念，尽管它如此成功，但Borel归约框架在一个重要方面福尔斯欠缺-它忽略了对象之间的映射。在大多数数学领域，对象之间的这些映射被视为与对象本身一样重要，一个好的分类应该尊重它们。然而，Borel归约法传统上并不考虑这一点。这个项目的PI有一个新的Borel归约公式，它确实考虑到了这些映射。为了模仿传统框架的成功，并证明新的不可能性的分类结果，这个新的框架的湍流的模拟是必要的。该项目的目标是找到并证明这种类似湍流的基本性质。

Classification is an idea that is ubiquitous across the sciences. Whether it is animals or elements, particles or stars, the process of characterising possible subgroupings from a broadly defined class of objects or individuals is extremely effective for deepening our understanding. This holds true also in the abstract world of pure mathematics, where the objects being classified are mathematical structures.Taking a step back, we can understand classifications in very general terms as follows. We have some mathematical objects that we which to classify, possibly up to the identification of objects that are "essentially the same". We have other mathematical objects called "invariants" that we wish to use for the classification (much as colours are used to classify stars, or the number of protons classifies chemical elements), and again there might be a notion of "essentially the same" for the invariants. The classification is then a reasonably definable map taking the objects to be classified to the invariants, respecting these notions of "essentially the same".The framework of Borel reducibility encapsulates this process, using as well the observation that frequently the objects to be classified and the invariants can be encoded into real numbers, much as information is coded into strings of bits on a computer. With this encoding in place, we can then give a formal characterisation of a "reasonably definable map" as a classification. Crucially, it is sometimes possible to show that no such map exists, thus showing that a hoped-for classification is impossible to find. This has been important for ruling out classifications in varied areas of mathematics such as ergodic theory and the study of C* algebras. A central tool in these impossibility proofs is Hjorth's notion of turbulence.Although it has been so successful, the Borel reducibility framework falls short in one important regard - it ignores the mappings between objects. In most areas of mathematics, these mappings between the objects are seen as being as important as the objects themselves, and a good classification should respect them. Borel reducibility traditionally takes no account of this however. The PI of this project has a new formulation of Borel reducibility which does take these maps into account. In order to mimic the success of the traditional framework, and prove new impossibility-of-classification results, an analogue of turbulence for this new framework is needed. The goal of this project is to find and prove the fundamental properties of such an analogue of turbulence.