Research
Modules and singularities
Analysis is the study of estimates; Topology is the study of nearness; Combinatorics is the study of counting; Algebra is the study of tautologies.
My interests are in commutative algebra, with leanings toward the homological. In practice this means thinking about the relationship between the structure of a ring and the structure of the category of maximal Cohen–Macaulay modules over it.
Cohen–Macaulay representation theory
A Cohen–Macaulay local ring has finite CM type when it admits only finitely many indecomposable maximal Cohen–Macaulay modules up to isomorphism — a condition that turns out to be a strong statement about how mild the singularity is. Much of my work has gone into classifying such rings, and into the intermediate territory of countable and bounded CM type, where the Brauer–Thrall conjectures live.
Mixed-characteristic hypersurfaces of finite Cohen-Macaulay type · Local rings of countable Cohen-Macaulay type · Local rings of bounded Cohen-Macaulay type · the book
Noncommutative resolutions of singularities
Van den Bergh’s noncommutative crepant resolutions replace a singular variety by a noncommutative algebra with the derived category a crepant resolution would have had. With Ragnar-Olaf Buchweitz and Michel Van den Bergh I constructed such resolutions for determinantal varieties, first for maximal minors and then for arbitrary ones, which meant working out the derived categories of Grassmannians in arbitrary characteristic along the way.
Determinantal varieties I · II: arbitrary minors · Grassmannians in arbitrary characteristic · Scenes from categorical geometry (expository) · MSRI Emissary article
Matrix factorizations and branched covers
Over a hypersurface singularity, maximal Cohen–Macaulay modules are the same thing as matrix factorizations of the defining equation, and Knörrer periodicity relates a hypersurface to its double branched cover. Work with Tim Tribone and Alex Dugas extends this dictionary to higher branched covers, and to factorizations of an equation into more than two matrices.
Branched covers and matrix factorizations · Extensions of theorems of Knörrer and Herzog-Popescu · Wild hypersurfaces
Homological invariants and conjectures
A related thread measures singularities numerically and homologically: the F-signature and its relation to strong F-regularity in prime characteristic, the growth of Betti sequences of canonical modules, rings with nontrivial semidualizing modules, and the Auslander–Reiten conjecture on modules without self-extensions.
Two theorems about maximal Cohen-Macaulay modules · On a conjecture of Auslander and Reiten · Betti sequence of the canonical module · Endomorphism rings of finite global dimension
Who else does these things?
The people listed at commalg.org/people, that’s who.