Universal flops and noncommutative algebras

A classification of simple flops on smooth threefolds in terms of the length invariant was given by Katz and Morrison, who showed that the length must take the value 1,2,3,4,5, or 6. This classification was produced by understanding simultaneous (partial) resolutions that occur in the deformation theory of A, D, E Kleinian surface singularities. An outcome of this construction is that all simple threefold flops of length l occur by pullback from a "universal flop" of length l. Curto and Morrison understood the universal flops of length 1 and 2 using matrix factorisations. I aim to describe how these universal flops can understood for lengths >2 via noncommutative algebra.