Mathematical Sciences: Fourier Analysis and Partial Difference Equations

PI：Fefferman DMS-9540525该PI将研究微分几何中出现的解析问题。其中一个项目是关于欧几里德4-空间中单连通区域上的半线性椭圆型方程。域被看作是完备共形平坦流形的泛覆盖。第二个项目是研究耦合的椭圆-抛物型方程。通过将椭圆方程耦合到哈密尔顿的Ricci流，可以对标量曲率施加控制，因此如果标量曲率从一开始就是负的，那么它至少在短时间内保持不变。偏微分方程式是建立物理世界数学模型的基础。数学分析的作用与其说是创建方程，不如说是提供有关解的定性和定量信息。这可能包括回答有关唯一性、平稳性和成长性的问题。此外，分析经常开发出近似解的方法和对这些近似的精度的估计。

PI: Fefferman DMS-9540525 The PI will investigate analytic problems which arise in differential geometry. One project is concerned with semilinear elliptic equations on simply connected domains in Euclidean 4-space. The domain is viewed as the universal cover of a complete conformally flat manifold. The second project will be the study of coupled elliptic-parabolic equations. By coupling an elliptic equation to Hamilton's Ricci flow, one can impose control on the scalar curvature, so that if the scalar curvature is a negative to begin with, then it remains so at least for short time. Partial differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations.