Deep Patterns in Algebraic Structures

Machine Learning Approaches to Quiver Mutation and Exceptional Sequence Classification - Bowdoin College Mathematics Department Independent Study

Overview

This work will apply machine learning techniques to uncover hidden patterns in fundamental mathematical structures called quivers and their associated exceptional sequences. Quivers are directed graphs that encode algebraic relationships. Quivers appear throughout modern mathematics and are a powerful tool used in string theory, particle physics, and machine learning.

Traditional approaches to classifying quiver properties often rely on manual analysis and case-by-case verification. This process becomes intractable for complex structures. Our approach employs machine learning to automatically detect patterns and motifs in exceptional sequences associated with Type A_n quivers.

We aim to develop a comprehensive workflow to:

• Generate complete datasets of exceptional sequences and isomorphism classes of HomExt quivers
• Extract geometric and combinatorial features from quiver transformations using machine learning methods
• Learn generalizable rules across arbitrary sequence lengths
• Develop predictive models that can classify exceptional sequences and HomExt mutation patterns for any n

This research aims to deliver new theoretical insights in representation theory while demonstrating the utility of machine learning to solve problems beyond the reach of existing methods.