Machine learning assisted exploration for affine Deligne-Lusztig varieties
This work accelerates pure mathematical research in algebraic geometry by enabling faster discovery of conjectures and directions, though it is incremental as it applies existing ML methods to a new domain.
The paper tackles the exploration of affine Deligne-Lusztig varieties by developing a machine learning framework to investigate their nonemptiness patterns, dimensions, and irreducible components, resulting in the rediscovery of a virtual dimension formula and a proof for a new lower bound problem.
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.