Categorifying Turbulence in Borel Reducibility

分类是一种普遍存在于科学界的观念。无论是动物还是元素、粒子还是恒星，从广泛定义的一类物体或个人中描述可能的分类的过程对于加深我们的理解是极其有效的。这在纯数学的抽象世界中也是如此，在抽象世界中，被分类的对象是数学结构。退后一步，我们可以理解分类的非常一般的术语如下。我们有一些数学对象需要分类，可能直到识别出“本质上相同”的对象。我们还有其他的数学对象，叫做“不变量”，我们希望用来进行分类(就像颜色被用来对恒星进行分类，或者质子的数量被用来对化学元素进行分类一样)，同样，对于不变量，可能会有一个“本质上相同”的概念。然后，分类是一个合理可定义的映射，将待分类的对象归类为不变量，尊重这些“本质相同”的概念。Borel约简的框架封装了这一过程，还利用了这样的观察，即将被分类的对象和不变量经常可以被编码成实数，就像信息被编码成计算机上的位串一样。有了这种编码，我们就可以给出一个“合理定义的地图”的正式特征作为一种分类。至关重要的是，有时有可能表明不存在这样的地图，从而表明不可能找到所希望的分类。这对于排除不同数学领域的分类是很重要的，例如遍历理论和C*代数的研究。在这些不可能性证明中的一个核心工具是赫约思的湍流概念。尽管波雷尔的简约性框架非常成功，但它在一个重要方面存在缺陷--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，波雷尔约化传统上没有考虑到这一点。这个项目的PI有一个新的Borel可归约性公式，它确实考虑了这些映射。为了模仿传统框架的成功，并证明新的不可能分类的结果，需要对这个新框架进行湍流模拟。这个项目的目标是找到并证明这种湍流类比的基本性质。

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.