Circle Patterns Bridging Hyperbolic Geometry and Statistical Mechanics

Objective

In our digital age, the development of discrete models that accurately capture the essential properties of their continuous counterparts is crucial for precise predictions and broad applications, from statistical mechanics to computer science. The CircleBridge project focuses on discrete conformal structures, particularly circle patterns, which serve as effective discretizations of conformal (angle-preserving) geometry — a field deeply rooted in complex analysis, geometry, and differential equations. CircleBridge seeks to expand the concept of circle patterns into the broader realm of complex projective structures, laying the groundwork for new theoretical insights. The project has two primary objectives: First, it aims to deepen our understanding of the space of circle patterns, developing computationally feasible tools to connect discrete conformality theories with their continuous analogs in Teichmüller theory and hyperbolic 3-manifolds. Second, CircleBridge will explore the implications of these findings in statistical mechanics, where circle patterns have recently emerged as a universal framework for the planar bipartite dimer model. By uniting leading experts from these fields in a transatlantic collaboration, CircleBridge will advance both theoretical knowledge and practical methodologies. This interdisciplinary effort aims to create a unified mathematical framework, paving the way for future discoveries across multiple disciplines.