We prove that every circle domain in the Riemann sphere admits a proper holomorphic embedding into the affine plane C^2. By a circle domain we mean a domain obtained by removing from the Riemann sphere a finite or countable family of pairwise disjoint closed round discs. Our proof also applies to some circle domains with punctures. The uniformization theorem of He and Schramm (1996) says that every domain in the Riemann sphere with at most countably many boundary components is conformally equivalent to a circle domain, so our theorem embeds all such domains properly holomorphically in C^2. (Joint work with Erlend F. Wold.)

Lecture series by Rajesh Gopakumar (Harish-Chandra Research Institute). The lectures will be supplemented by talks by other invited speakers.

Lecture series by Jonathan Rosenberg (University of Maryland). There will be additional talks by other invited speakers.