RUI: Symmetries, Stability, and Related Topics

In the study of algebraic objects in mathematics, a central problem is to classify all the relevant objects in a given context. A powerful and time-tested approach to this classification problem is to begin with a measure of how "stable" an object is. Since there are different measures of stability, it is important to understand how they are related.

Moreover, when there are inherent symmetries underlying the spaces in which algebraic objects arise, these symmetries induce even richer relations among different measures of stability and the corresponding stable objects. This project aims to discover new relations among different types of stability that arise in the area of algebraic geometry and nearby areas such as representation theory, using symmetries as a main tool.

The project will also train undergraduate and graduate students in research through integrated student projects.

This project will incorporate stability conditions in algebraic geometry and representation theory under a single framework. Different aspects of the project are tied together by the problem of solving the generalized Gepner equation in a variety of situations. The generalized Gepner equation is a relation involving different group actions on stability conditions.

When specialized to different contexts, the generalized Gepner equation is connected to the birational geometry of moduli spaces, modularity of counting invariants, dynamical systems in algebraic geometry, and the existence of new kinds of stability conditions. The main tools will be the notion of weak polynomial stability functions and their associated weight functions.

The actions of exact equivalences on these functions and relevant t-structures will be studied. These results will then be refined to yield concrete equations involving stability conditions.

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.