Speaker(s): Pekka Pankka(University of Helsinki and Finnish Mathematical Society)
Time: 16:00-17:00 January 12, 2024
Venue: Room 77201, Jingchunyuan 78, BICMR
In this minicourse, I will discuss quasiconformal geometry from the point of view of mapping theory, especially of quasiregular mappings. Quasiregular mappings can be viewed as a generalization of planar holomorphic mappings in higher dimensional Riemannian geometry. The prefix 'quasi-' refers to bounded distortion of the conformal structure. The classical Liouville's theorem states that conformal mappings between domains of a Euclidean n-space are restrictions of Möbius transformtion in dimensions n\ge 3. Therefore, in higher Euclidean dimensions, distortion is needed to obtain a non-trivial 'conformal' mapping theory.
On the first lecture, I will discuss how conformal invariants lead to familiar results from complex analysis in this higher dimensional setting. On the second lecture, I will discuss measurable conformal structures and a version of complex dynamics in this context and how these ideas lead to cohomological boundedness of closed manifolds which admit quasiregular mappings from Euclidean spaces. Finally, on third lecture I will discuss non-equidimensional version of the quasiregular theory connecting quasiregular mappings to holomorphic curves and, if time permits, recent developments on Picard theorems for mappings without conformality assuptions.
Lecture I: Quasiregular mappings and conformal invariants
Lecture II: Quasiregular dynamics and cohomological results
Lecture III: Quasiregular curves and Picard theorems beyond conformality
mini-course time & venue：
Jan 11 2:30-3:30pm Hou Zhu Lou 1124, Beijing Normal University
Jan 11 4:00-5:00pm Hou Zhu Lou 1124, Beijing Normal University
Jan 12 4:00-5:00pm Room 77201, Huaixinyuan, BICMR, Peking University
Bio-Sketch: Pekka Pankka received his PhD from the University of Helsinki in 2005, and is now a professor at the University of Helsinki and the Vice President of Finnish Mathematical Society. His research interests include analysis of complex variables, metric geometry, geometric topology, and focus on quasiconformal mappings in particular. In these areas, he has produced more than 40 research articles, publishing on renowed journals including Acta Mathematica and Duke Mathematical Journal.
