# Introduction to subfactors and quantum symmetries of Hadamard matrices

** Speaker:** Michael Montgomery, Dartmouth College

** Date:** October 6 and 13, 2022

** Abstract:** The study of subfactors was initiated by Vaughan Jones in the 1980’s. Subfactors are inclusions of tracial von Neumann algebras with trivial centers and led to new results in low dimensional topology, knot theory, and mathematical physics. Hyperfinite subfactors can be built from combinatorial objects called biunitaries. Examples of biunitaries include complex Hadamard matrices and quantum Latin squares. Quantum symmetries of the corresponding subfactor can be captured by its standard invariant. Due to Sorin Popa and Jones the standard invariant can then be axiomatized by a graphical calculus called a planar algebra. In this two part talk we will introduce subfactors, their standard invariants, and planar algebras. Using planar algebraic techniques, we prove a criterion that implies the standard invariant has infinitely many simple objects. We apply this criterion to type II Paley Hadamard matrices and a continuous family of complex Hadamard matrices found by Petrescu.