# Cancellation problems and homotopy theory

SPEAKER: Aravind Asok

TITLE: Cancellation problems and homotopy theory

ABSTRACT The classical cancellation question can be phrased in ring theoretic terms as follows. Suppose k is a field. If A and B are two commutative k-algebras such that A[t] is isomorphic to B[t], under what additional hypotheses can one conclude that A and B are isomorphic as k-algebras? We will begin by discussing a class of counterexamples to this problem due to Danielewski and Fieseler, which arise from certain smooth affine surfaces. We will generalize this class of counterexamples and explain how the Morel-Voevodsky A^1-homotopy theory can be used to understand their source. Concrete examples will be emphasized throughout.