On flat deformations and their applications

Michael Wemyss and Will Donovan have developed a method for characterizing commutative rings appearing in algebraic geometry which uses methods from noncommutative ring theory. They introduced contraction algebras, which provide important insight into resolutions of noncommutative singularities. All local contraction algebras were characterised by Michael Wemyss and Gavin Brown by giving generators and relations. Deformations of these algebras give information about Gopakumar–Vafa (GV) invariants, which are very important in algebraic geometry. This leads to purely ring theoretic questions about deformations of local algebras, for example whether or not a given contraction algebra can be deformed to a given semisimple algebra. Some of these problems can be investigated by applying polynomial identities to both algebra A and N when deforming an algebra $N$ into an algebra A.

In this talk, we provide some methods for solving such questions and mention some open questions from algebraic geometry posed by Arend Bayer and Ben Davidson. The talk will be elementary and aimed at people working in noncommutative ring theory.