Sequential Deformations of Hadamard Matrices and Commuting Squares

In this dissertation, we study analytic and sequential deformations of commuting squares of finite dimensional von Neumann algebras, with applications to the theory of complex Hadamard matrices. The main goal is to shed some light on the structure of the algebraic manifold of spin model commuting squares (i.e., commuting squares based on complex Hadamard matrices), in the neighborhood of the standard commuting square (i.e., the commuting square corresponding to the Fourier matrix). We prove two types of results: Non-existence results for deformations in certain directions in the tangent space to the algebraic manifold of commuting squares (chapters 3 and 4), and finiteness results for commuting squares based on Hadamard matrices with certain symmetries (chapter 5).