Prof. Gavin Brown - Noncommutative singularity theory and flops

(Joint with Michael Wemyss.)

A (simple) flop arises when a smooth complex 3-fold X admits a morphism that is an isomorphism away from a single rational curve C, and C is contracted to a hypersurface singular point in the image Y. In the typical case, the image is the node xy=zt, and the morphism is a (small) resolution of singularities. In fact there is a choice of two such resolutions - that is, two such morphisms from smooth complex 3-folds containing a contracted curve to Y - and the flop is the birational map between them.

On the other hand, in this set up the noncommutative deformation theory of C in X is controlled by a finite-dimensional associative algebra, the Donovan-Wemyss contraction algebra. Conjecturally, in fact, such algebras classify flops. I discuss our project to classify such algebras, which boils down to understanding normal forms of noncommutative power series f(x,y) up to a certain equivalence. This may be viewed as a noncommutative analogue of classical singularity theory, in which germs of (commutative) functions are arranged according to invariants and then assigned normal forms. Indeed, in the case of finite-dimensional algebras, we recover an ADE classification in very explicit terms, and this instantly teaches us infinitely many new things about flops.