# Categorifying Turbulence in Borel Reducibility

Lead Research Organisation:
University of Leeds

Department Name: Pure Mathematics

### Abstract

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.

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.