Conformal Geometry and Geometric PDE´s
Sign into July 31, 2013
Introduction
The particular interest of Geometric Analysis seems to lie in a combination of its relation to the physical world and the way it lies at the intersection of so many branches of Mathematics (Riemannian/Conformal/Complex/Algebraic Geometry, Calculus of Variations, and PDE’s), or even Physics.
Conformal geometry is the study of the set of angle-preserving (conformal) transformations on a space. While in two dimensions, this is precisely the geometry of Riemann surfaces, in dimensions three and above the answer opens up many new different subjects, leading to the very wide field that is conformal geometry (Yamabe-type problems, non-local or non-linear conformally covariant operators, Poincaré-Einstein metrics and its relation to the AdS/CFT correspondence in Physics, and many more).
On the other hand, in CR geometry there are formal similarities with conformal geometry. The analysis of these operators is closely connected with the geometry of the pseudoconvex manifolds which they may bound, hence of interest in several complex variables.
Another classical topic in Geometric Analysis is the study of variational problems related to the area functional. In this sense, the global theory of minimal and constant mean curvature surfaces in homogeneous three-manifolds,
and more generally in Riemannian and sub-Riemannian manifolds, represents today a tremendously active field of new discoveries and challenges. Applications of minimal surfaces to other subjects include low dimensional topology, general relativity and materials science. Closely related to this topic appears the isoperimetric problem that connects Geometric Analysis with Geometric Measure Theory.
While all these seem to be almost unrelated topics, it is precisely the interaction between them that nurtures the development of such a fundamental branch of mathematics that is Geometrical analysis.
Organizing Committee
Sun-Yung Alice Chang, Princeton University
Maria del Mar González, Universitat Politècnica de Catalunya
Robin Graham, University of Washington
Francisco Martín, Universidad de Granada
Paul Yang, Princeton University
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