The theories of quasiconformal mappings and elliptic partial differential equations have classical connections dating back the work of Vekua, Bers, Bojarski, and others. During the last ten years these connections have been revitalized through new methods and breakthroughs and surprising applications that merge geometric and analytic methods. These include the solution of Calderón’s problem of impedance tomography in the plane by Astala and Paivarinta. Current research suggests that the methods of geometric analysis will also be applicable to problems in materials sciences, such as in elasticity and stochastic homogenization. Another development is the extension of the theory to degenerate elliptic equations, through the work of David, Iwaniec, Koskela, Martin and many others. Here the geometric counterpart is the theory of mappings of finite distortion, which played a vital role in recent work on random geometry. The workshop will also study the conformal geometry related open problems in SLE and quantum gravity.
With support from the Finnish government, part of the workshop will commemorate the 60th birthday of Professor Kari Astala and his many important contributions to its topics.
(University of California, Los Angeles (UCLA))
Tadeusz Iwaniec (Syracuse University)
Steffen Rohde (University of Washington)
Eero Saksman (University of Helsinki)
Tatiana Toro (University of Washington)