Nonlinear Problems in Convex and Conformal Geometry

AbstractAward：DMS-0604638项目负责人：陈文雄项目负责人将研究一系列非线性偏微分方程、积分方程和系统，以及其他非线性问题。大多数的问题都来自于微分几何或凸几何，例如半线性椭圆方程来自于高斯曲率和标量曲率，蒙格-安培方程来自于著名的Lp-Minkowski问题。根据爱因斯坦的观点，我们所生活的宇宙是一个弯曲的空间，在这个空间中，引力被实现为一个大质量物体附近时空的扭曲或弯曲，这种形状的变化是通过曲率来测量的。其中一个项目的重点是理解一个给定的函数何时可以成为一个曲率。这是一个具有挑战性的问题在全球分析和有趣的后果在黎曼几何。另一个PI的主要项目，Lp闵可夫斯基问题，是中心问题的布伦-闵可夫斯基理论，这是非常核心的凸几何分析。这一领域的研究成果已应用于体视学、随机几何、积分几何、数论、组合学、概率论、统计学和信息论等众多学科，本项目所研究的非线性偏微分方程、积分方程和系统在物理、化学和生物学中也有着广泛的应用。一个例子是模拟河流或血流中的化学反应，这将为控制河流污染提供有用的信息。

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.