Colin Ingalls

Noncrossing partitions and representations of quivers

We situate the noncrossing partitions associated with a finite Coxeter group within the context of the representation theory of quivers. We describe Reading’s bijection between noncrossing partitions and clusters in this context, and show that it extends to the extended Dynkin case. Our setup also yields a new proof that the noncrossing partitions associated with a finite Coxeter group form a lattice. We also prove some new results within the theory of quiver representations. We show that the finitely generated, exact abelian, and extension-closed subcategories of the representations of a quiver Q without oriented cycles are in natural bijection with the cluster tilting objects in the associated cluster category. We also show that these subcategories are exactly the finitely generated categories that can be obtained as the semistable objects with respect to some stability condition.

On nodal Enriques surfaces and quartic double solids

We consider the class of singular double coverings

Non-Commutative Coordinate Rings and Stacks

X \rightarrow {\mathbb {P}}^3

Unramified Brauer Classes on Cyclic Covers of the Projective Plane

X→P3 ramified in the degeneration locus

Blowing up quantum weighted projective planes

D

The minimal model program for orders over surfaces

D of a family of 2-dimensional quadrics. These are precisely the quartic double solids constructed by Artin and Mumford as examples of unirational but nonrational conic bundles. With such a quartic surface