Machine learning assisted exploration for affine Deligne-Lusztig
  varieties
Bin Dong Xuhua He Pengfei Jin Felix Schremmer Qingchao Yu
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.