Kevin Costello

Abstract:   Deformation quantization casts the problem of quantizing classical mechanical systems as an algebraic problem: given a Poisson algebra, find an associative algebra which is a quantization. I'll explain how the problem of quantizing a classical field theory can be expressed as a similar algebraic problem, using the language of factorization algebras. Then, I'll state a theorem describing the obstruction and deformation groups controlling possible quantizations, to each order in hbar. The proof of the theorem relies on the physicists' techniques of renormalization.