Applying language models to algebraic topology: generating simplicial cycles using multi-labeling in Wu's formula

TL;DR

This paper explores the use of language models to generate simplicial cycles in algebraic topology, leveraging multi-labeling and machine learning to understand homotopy groups of spheres.

Contribution

It introduces a novel approach of applying multi-label language modeling to generate algebraic topological structures within Wu's formula framework.

Findings

Language models can generate simplicial cycles effectively.

Multi-label input improves the quality of generated cycles.

The approach offers new insights into the structure of homotopy groups.

Abstract

Computing homotopy groups of spheres has long been a fundamental objective in algebraic topology. Various theoretical and algorithmic approaches have been developed to tackle this problem. In this paper we take a step towards the goal of comprehending the group-theoretic structure of the generators of these homotopy groups by leveraging the power of machine learning. Specifically, in the simplicial group setting of Wu's formula, we reformulate the problem of generating simplicial cycles as a problem of sampling from the intersection of algorithmic datasets related to Dyck languages. We present and evaluate language modelling approaches that employ multi-label information for input sequences, along with the necessary group-theoretic toolkit and non-neural baselines.