Piotr Hajłasz

My research is in the Geometric Function Theory which covers a wide range of topics on the borderline of classical analysis, geometric analysis, theory of Sobolev spaces and analysis on metric spaces. My research involves methods of harmonic analysis and geometric and algebraic topology.

Classical analysis: geometric measure theory; maximal functions; the Whitney extension theorem; the Sard theorem.

Sobolev spaces: Sobolev embedding theorems on domains with irregular boundary, characterization of domains with the extension property, Sobolev spaces on metric spaces, approximation of Sobolev mappings between manifolds with connection to algebraic topology, degree theory for Orlicz-Sobolev mappings, continuity properties for mappings of finite distortion.

Analysis on metric spaces:  Sobolev spaces on metric spaces; spaces supporting Poincaré inequalities; geometry and topology of the Heisenberg groups; structure of Lipschitz mappings into metric spaces (differentiability, implicit function theorem, area and co-area formulas). 

Quantitative topology: Lipschitz and Hölder homotopy groups; homotopy and approximation of mappings with low derivative of low rank; factorization through metric trees.

Convex analysis: Differentiability properties of convex functions.