Nonlinear Problems in Convex and Conformal Geometry

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AbstractAward: DMS-0604638Principal Investigator: Wenxiong ChenThe principal investigator will work on a series of nonlinearpartial differential equations, integral equations and systems,as well as other nonlinear problems. Most of the problems arisefrom differential or convex geometry, such as the semi-linearelliptic equations from prescribing Gaussian and scalar curvatureand the Monge-Ampere equations from the well-known Lp- Minkowskiproblem.According to Einstein, the Universe we lived in is a curvedspace, in which gravity is realized as a distortion or bending ofspace-time in the neighborhood of a massive object and thischange of shape is measured by curvature. One of these projects is focused on understanding when a given function canbecome a curvature. This is a challenging problem in globalanalysis and has interesting consequences in Riemannian Geometry.The other of the PI's major projects, the Lp Minkowski problem,is the central question of the Brunn-Minkowski Theory, which isthe very core of Convex Geometric Analysis. Results in this areahave been applied to numerous disciplines, including stereology,stochastic geometry, integral geometry, number theory,combinatorics, probability, statistics, and information theory.The nonlinear partial differential equations and integralequations and systems studied in this project also have variousapplications in physics, chemistry, and biology. One example is amodeling of chemical reaction in rivers or in blood streams,which would provide useful information in controlling pollutionin rivers.