Title: Equivariant Analysis
Abstract:
Many classical constructions in analysis secretly depend on a
choice of origin. The Weierstrass theorem produces an entire
function with prescribed zeros — but move the origin, and the
function changes. Equivariant analysis asks which constructions
survive when no origin is available. I will explain how this
question is modeled on descriptive combinatorics, and describe
joint work with M. Sodin and A. Wennman. Whether an equivariant
construction is possible turns out to depend on the group of
periods. For aperiodic objects, the Weierstrass and Mittag-Leffler
theorems, the ∂̄-equation, and the Poisson and heat equations all
have equivariant counterparts; in other instances, taking
antiderivatives among them, equivariant solutions are impossible
in general.