Abstract：
Random curves arise naturally as interfaces in two-dimensional statistical models, from percolation cluster boundaries to domain walls in the Potts model. For nonintersecting loops, the Temperley–Lieb algebra provides a powerful framework connecting lattice models, quantum groups and conformal field theory. Branching interfaces, such as those of the three-state Potts model, lead instead to planar graphs described by $\mathfrak{sl}_3$ webs.

We first recall how to obtain the conformal annulus partition function for the loop model using the Temperley-Lieb algebra. We then present recent results on the cellular structure of the $\mathfrak{sl}_3$ web algebra, along with explicit diagrammatic representations and local generators. These results extend the algebraic approach to branching geometries and give exact annulus partition functions, where the conformal symmetry is described by the $W_3$ algebra, an extension of the Virasoro algebra.

Bio-Sketch：

Jesper Lykke Jacobsen is Professor of Theoretical Physics at Sorbonne Université and the École Normale Supérieure in Paris, France. His research focuses on statistical mechanics, conformal field theory, and exactly solvable models, with particular emphasis on critical phenomena, lattice models, percolation, and disordered systems. His work combines analytical, algebraic, and numerical approaches to understanding the universal properties of complex physical systems. He received the Paul Langevin Prize from the French Physical Society in 2016 and the Grand Prix Ampère of the French Academy of Sciences in 2025. His research spans the interface between theoretical physics and mathematics, including integrable systems and logarithmic conformal field theory.