Jason Bell, Pure Mathematics, University of Waterloo
"The noncommutative Zariski Cancellation Problem"
The Zariski Cancellation problem asks whether a complex affine variety X with X x C isomorphic to affine (n+1)-space must be isomorphic to affine n-space. Algebraically, this is asking about whether one can find a finitely generated C-algebra A such that A[x] is a polynomial ring but A is not. In this setting, one can formulate a noncommutative version. In the noncommutative setting, the analogue of a polynomial ring is an Artin-Schelter regular algebra. Just as with varieties, one can associate a dimension to AS-regular algebras that behaves like Krull dimension. We prove that the cancellation problem has an affirmative answer in dimension at most two. This is joint with James Zhang.