学术报告（Sergey Galkin 2025.1.10）
Can one hear the shape of a random walk
Abstract: In a forthcoming joint work with Pieter Belmans and Swarnava Mukhopadhyay we construct many zero mean high-dimensional lattice random walks of pairwise-distinct shapes with pairwise-equal return probabilities for all numbers of steps. The same examples provide pairwise-distinct shapes of discretizations of the standard Laplacian with pairwise-equal density state functions. We discovered them in our study of mirror symmetry for moduli spaces of vector bundles, and of the related Laurent phenomenon for mutations of graph potentials.
The main technical ingredient for distinguishing shapes is the following construction and result relating combinatorics and convex geometry. To a coloured trivalent graph one associates a convex lattice polytope, the quantum Clebsch--Gordan polytope. We prove that this association is a full functor, in particular that from the quantum Clebsch--Gordan polytope one can recover its graph, and that any isomorphism between such polytopes is induced by a unique isomorphism of the underlying colored graphs.
Quantum Clebsch--Gordan polytopes are moment polytopes of toric degenerations of odd character varieties (or moduli spaces of rank-2 bundles on a curve). Thus, the reconstruction can be interpreted algebro-geometrically as a combinatorial non-abelian Torelli theorem. In symplectic geometry it also implies that monotone Lagrangian tori on an odd character variety, accociated with these degenerations are pairwise non-Hamiltonian isotopic.
