We discuss recent work that pinpoints the spectral decomposition for the Anderson tight-binding model with an unbounded random potential on the Bethe lattice of sufficiently large degree. We prove that there exist a finite number of mobility edges separating intervals of pure-point spectrum from intervals of absolutely continuous spectrum, confirming a prediction of Abou-Chacra, Thouless, and Anderson. A central component of the proof is a monotonicity result for the leading eigenvalue of a certain transfer operator, which governs the decay rate of fractional moments for the tight-binding model’s off-diagonal resolvent entries. This is joint work with Amol Aggarwal. We will also draw connections to the problem of establishing a mobility edge in the spectrum of heavy-tailed random matrices, which was considered in recent work with Amol Aggarwal and Charles Bordenave.

Abstract: The Kauffman Bracket Skein algebra is a central object in low-dimensional topology. It is an algebra generated by loops on an oriented surface. There have been many attempts to generalize this algebra from various motivations. In this talk, we review one of the generalizations, its motivation, and recent progress on its representation theory. This talk is based on joint work with Hiroaki Karuo and Helen Wong.