Bipartite Graphs in Tn and Toric Mirror Symmetry

Explore a 44-minute lecture on bipartite graphs in Tn and toric mirror symmetry presented by Harold Williams from the University of Southern California at IPAM's Statistical Mechanics and Discrete Geometry Workshop. Delve into joint work with Chris Kuo and earlier collaborations with David Treumann and Eric Zaslow, examining how homological mirror symmetry affects one-dimensional coherent sheaves on toric surfaces. Discover how these sheaves correspond to Lagrangians derived from perturbations of conormals to specific bipartite graphs in T2, with the moduli map implemented through the graph's Kasteleyn operator. Investigate the extension of this combinatorial description to coherent sheaves of codimension one on toric varieties of dimension n≥2, involving bipartite graphs in Tn. Gain insights into the intersection of algebraic geometry, symplectic geometry, and combinatorics in this advanced mathematical exploration.