Speaker: Jeremy Tyson (University of Illinois at Urbana-Champaign)

Title: Quasiconformal mappings and dimension distortion

Abstract: The theory of quasiconformal (QC) mappings generalizes to higher dimensional Euclidean spaces and beyond many aspects of the geometry and analysis of planar conformal mappings. I’ll start with a brief historical survey of Euclidean quasiconformal mapping theory, highlighting a few signature applications in holomorphic dynamics, hyperbolic rigidity, differential geometry, and other areas. An unexpected complication in understanding the equivalence of definitions for quasiconformality in the absence of Euclidean or Riemannian structure led Heinonen and Koskela in the 1990s to develop a synthetic metric approach. This was the origin of the modern subject of analysis in metric spaces. The second part of the talk will focus on the following question: How do quasiconformal (and more generally, Sobolev) mappings distort metric notions of dimension? Specifically, I’ll focus on the effect of such maps on recently defined dimension interpolants (one-parameter families of metrically natural dimensions), with applications to QC classification and to conformal dimension. The new results presented at the end of the talk are based on joint work with Stathis Chrontsios (UTK), and separately with Jonathan Fraser (University of St. Andrews).