Machine learning assisted exploration for affine Deligne-Lusztig varieties

TL;DR

This paper introduces a machine learning framework to explore affine Deligne-Lusztig varieties, aiding in understanding their structure, discovering new conjectures, and providing mathematical proofs, thus accelerating research in this complex area.

Contribution

It presents a novel interdisciplinary ML-based approach for studying ADLV, including data generation, pattern analysis, and proof discovery, advancing mathematical understanding and research efficiency.

Findings

Rediscovered the virtual dimension formula

Proved a new lower bound for dimension

Identified new research directions

Abstract

This paper presents a novel, interdisciplinary study that leverages a Machine Learning (ML) assisted framework to explore the geometry of affine Deligne-Lusztig varieties (ADLV). The primary objective is to investigate the nonemptiness pattern, dimension and enumeration of irreducible components of ADLV. Our proposed framework demonstrates a recursive pipeline of data generation, model training, pattern analysis, and human examination, presenting an intricate interplay between ML and pure mathematical research. Notably, our data-generation process is nuanced, emphasizing the selection of meaningful subsets and appropriate feature sets. We demonstrate that this framework has a potential to accelerate pure mathematical research, leading to the discovery of new conjectures and promising research directions that could otherwise take significant time to uncover. We rediscover the virtual dimension formula and provide a full mathematical proof of a newly identified problem concerning a certain lower bound of dimension. Furthermore, we extend an open invitation to the readers by providing the source code for computing ADLV and the ML models, promoting further explorations. This paper concludes by sharing valuable experiences and highlighting lessons learned from this collaboration.