Many contemporary investigations in geometry lead to analytic questions on non-smooth and fractal spaces different from the usual Euclidean setting. This can be traced back to Mostow’s influential work on the rigidity of negatively curved rank-one symmetric spaces. Here one needs analytic machinery in a non-classical (sub-Riemannian) setting in order to treat problems of quasiconformal geometry on the boundary at infinity of such spaces. This work inspired many subsequent investigations such as the delevopment of the theory of quasiconformal mappings on Heisenberg or general Carnot groups (Koranyi-Reimann) or on general metric spaces (Heinonen-Koskela). In his seminal work on hyperbolic groups Gromov developed a general theory of spaces that are negatively curved in the large. These spaces have an associated boundary at infinity, and one can study their quasiconformal geometry with the associated analytic problems. This analytic trend culminated in the creation of a new field of mathematics, the Analysis on Metric Spaces, that has found many applications in geometry. For example, the work by Bourdon and Pajot on the rigidity of Fuchsian buildings relies on such tools.
A theory related to this field of quasiconformal analysis but with a different flavor can be loosely described as Lipschitz analysis. Its orgins go back to classical results such as Rademacher’s theorem on the differentiability of Lipschitz functions, Whitney’s geometric integration theory, or the theory of rectifiability and currents. Since the notion of a Lipschitz function is meaningful for arbitrary metric space, it is tempting to base generalizations of classical theories on this concept. For example, when Ambrosio and Kirchheim recently extended the classical Federer-Fleming theory of currents in Rn to general metric spaces they defined a “metric current” to be a certain functional acting on Lipschitz functions. Similarly, a theory of cotangent bundles for general metric spaces has been developed recently by Cheeger and by Weaver using two different approaches, but both of them use Lipschitz functions. In these studies it is often important to investigate the finer properties of Lipschitz functions and maps. Even in Rn many questions here are far from being understood. In this workshop we intend to pursue some of these directions with an emphasis on more geometric aspects (another workshop in this program on “Analysis on Metric Spaces” has a more analytic bias). Topics will include analytic problems that arise in geometric group theory or for expanding dynamical systems, differentiablity properties of Lipschitz functions, currents and isoperimetric problems on metric spaces, quasiconformal geometry of fractals, and sub-Riemannian geometry.
Mario Bonk, Chair
(University of California, Los Angeles (UCLA))
Marianna Csörnyei (University of Chicago)
Bruce Kleiner (New York University)
Jeremy Tyson (University of Illinois at Urbana-Champaign)
Stefan Wenger (Université de Fribourg)