Moduli, Stacks, and Beyond

Algebraic varieties are shapes in space defined by polynomial equations. They play a prominent role across mathematics with important applications to cryptography, computer vision, physics, and artificial intelligence. Among algebraic varieties, moduli spaces stand out as some of the most enchanting varieties, capturing the imagination of mathematicians with their profound elegance and deep connections to other disciplines. This project has two goals: first, to uncover new mathematical insights about moduli spaces, and, second, to use these discoveries to improve our understanding of the interplay between mathematics and artificial intelligence. The project also provides research training opportunities for graduate students. In more detail, over the past decade, the PI has been studying moduli spaces and algebraic stacks. This work has led to several advances, including a local structure theorem for algebraic stacks and an existence theorem for good moduli spaces. So far, these results have only been established in characteristic zero. This project aims to extend these foundational theorems to positive characteristic. Additionally, this project will apply these advances to study specific moduli spaces and to deepen our understanding of modern machine learning. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.