Categorifying Turbulence in Borel Reducibility

分类是一个在科学领域普遍存在的概念。无论是动物还是元素，粒子还是恒星，从广泛定义的物体或个体类别中描述可能的子组的过程对于加深我们的理解是非常有效的。这在纯数学的抽象世界中也是成立的，在那里被分类的对象是数学结构。退一步看，我们可以从以下非常一般的角度来理解分类。我们有一些数学对象，我们需要分类，可能到识别“本质上相同”的对象。我们还有其他数学对象，叫做“不变量”，我们希望用它们来进行分类（就像用颜色来给恒星分类，或者用质子的数量来给化学元素分类一样）。对于不变量，可能还有一个“本质相同”的概念。然后，分类是一个合理的可定义映射，将要分类的对象映射为不变量，尊重这些“本质相同”的概念。Borel可约性框架封装了这一过程，并使用了这样一种观察，即通常要分类的对象和不变量可以被编码为实数，就像信息在计算机上被编码为比特串一样。有了这种编码，我们就可以将“合理可定义的映射”作为分类给出正式的特征。至关重要的是，有时有可能证明不存在这样的地图，从而表明不可能找到期望的分类。这对于排除各种数学领域的分类非常重要，例如遍历理论和C*代数的研究。这些不可能性证明的一个核心工具是霍约斯的湍流概念。尽管Borel简化框架非常成功，但它在一个重要方面存在不足——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可约性传统上没有考虑到这一点。这个项目的PI有一个新的Borel可约性公式，它确实考虑了这些映射。为了模拟传统框架的成功，并证明新的不可能分类的结果，需要对这种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模拟的基本性质。

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.