Mathematical Sciences: Fourier Analysis and Partial Difference Equations

PI: Fefferman DMS-9540525 PI将研究微分几何中的解析问题。一个项目涉及欧几里得4空间单连通域上的半线性椭圆方程。该定义域被看作是一个完全共形平面流形的泛盖。第二个项目将是研究耦合椭圆-抛物方程。通过将椭圆方程与汉密尔顿里奇流耦合，可以对标量曲率施加控制，因此，如果标量曲率一开始是负的，那么它至少在短时间内保持负。偏微分方程是物理世界数学建模的基础。数学分析的作用与其说是创建方程，不如说是提供关于解的定性和定量信息。这可能包括回答关于独特性、平滑性和增长性的问题。此外，分析常常发展出解的近似方法和对这些近似精度的估计。

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.