# 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.

## People |
## ORCID iD |

Andrew Brooke-Taylor (Principal Investigator) |

Description | Recall from the Lay Summary that "Borel reducibility" has been used to great effect in showing that certain hoped-for classifications in other areas of mathematics (such as C*-algebras and ergodic theory) are impossible to achieve, giving a deeper insight into the complexity of these questions, and saving a great deal of effort from mathematicians chasing an impossible goal! The goal of the research funded by this Small Grant was to develop further a refined version of this framework with which we might be able prove more such impossibility-of-classification results. In particular, classifications are often expected to respect mappings between objects, so this work focused on a new "categorified" version of Borel reducibility that takes such mappings into account. The research bolstered our understanding of this new framework as being meaningfully different from the classical framework, which was the high-level goal of the project. In particular, we have shown that numerous classes of structures that were maximal in the old framework - "Borel complete" - are not maximal in our new framework. For example, the classes of linear orders and of quandles are both Borel complete but were shown to not be maximal in the new framework. This clear distinction is promising for the overarching goal of proving new impossibility-of-classification results in cases where the classical version of Borel reducibility is too blunt a tool. The original proposal focused particularly on Hjorth's notion of "turbulence" in classical Borel reducibility as a central tool that we hoped to generalise. We found that this was indeed possible; we were able to sidestep initial concerns about the difficulty of this, related to the formulation of turbulence in terms of group actions, by reformulating turbulence in terms of the relevant equivalence classes instead. This sets the stage for future research developing the theory of this version of turbulence, by analogy with the classical version, with the long-term hope of providing a powerful tool for further impossibility of classification results. |

Exploitation Route | As mentioned above, a clear way forward from the starting point provided by this research is to develop the theory of our generalised form of turbulence, drawing on the classical theory as a guide. Again, the hope is that this notion could then be used by others to prove new impossibility-of-classification results in diverse areas of mathematics, giving a deeper understanding of the complexity of the questions being asked. |

Sectors | Other |

Description | Fields Institute Thematic Program Travel Grant |

Amount | $4,000 (CAD) |

Organisation | Fields Institute for Research in Mathematical Sciences |

Sector | Charity/Non Profit |

Country | Canada |

Start | 04/2023 |

End | 06/2023 |

Description | Calderoni - categorifying turbulence |

Organisation | Rutgers University |

Country | United States |

Sector | Academic/University |

PI Contribution | Working to prove new results on the topic of the grant, presenting the results in various seminars and conferences, and commencing writing up the results. |

Collaborator Contribution | Discussing the work with the PI and making suggestions, including on fruitful further lines of research. Also collaborated in writing up results so far. |

Impact | Although Calderoni and I had worked together previously, the working relationship surely would not have survived the pandemic without the support of the grant, providing impetus to revive this important thread of research - with new, particular focus on "turbulence" - when various circumstances meant it would have been easier for all involved to just drop the topic. We have numerous new results in this area that we have introduced, bringing category theory considerations into descriptive set theory - see the Key Findings section. These new results have been presented in a range of seminars and conferences, and work is underway writing up results for publication. |

Start Year | 2021 |