The overarching goal of this project is to transition the study of friezes from a collection of isolated cases into a comprehensive, unified theoretical framework. Our research will bridge algebraic, geometric, and categorical perspectives to uncover the fundamental principles underlying these structures.
Key Research Directions
Foundational Theory and Generalizations
We aim to establish a rigorous basis for infinite-rank tilings by leveraging advancements in cluster algebra theory. This includes extending classical SL2 results to higher-rank SLn systems and exploring the properties of friezes associated with broader classes of simple algebraic groups.
Quantum and Noncommutative Structures
A primary direction involves the transition from classical to quantum and noncommutative frieze varieties. We will investigate how cluster-theoretic properties manifest in quantum settings and seek to classify the underlying schemes that govern these quantum analogs.
Integrable Systems and Discrete Geometry
The project will explore the deep intersections between friezes and discrete integrable systems. By viewing tilings through the lens of discrete differential geometry, we intend to analyze their moduli spaces and the Poisson structures that emerge from these connections.
Categorical Origins and Growth Phenomena
We seek to identify the categorical roots of friezes within 2-Calabi-Yau categories and Auslander-Reiten theory. Furthermore, the research will investigate the analytic properties of infinite friezes, specifically focusing on the growth phenomena derived from various algebraic varieties and Riemann surfaces.