Mathematical Sciences: "The Wave Map Program: Toward a Theory of Regularity and Break-down in Classical Nonlinear Fields"

[504919] Tahvildar-Zadeh提议的研究是在数学物理领域，专门研究波图的规律性和稳定性理论。波映射是调和映射的双曲模拟，其域是洛伦兹流形而不是黎曼流形。当人们考虑从洛伦兹流形到黎曼流形的映射的能量泛函时，得到的欧拉-拉格朗日方程变成了双曲的——对于两个黎曼流形之间的映射，这些方程是椭圆的。波图被物理学家称为sigma模型，它们以各种形式出现在现代物理学中。例如，爱因斯坦的场方程可以简化为sigma模型方程。由于双曲偏微分方程在数学上比椭圆偏微分方程更难处理，迄今为止还没有多少关于波动图的一般理论得到发展。

9504919 Tahvildar-Zadeh The proposed research is in the area of mathematical physics and deals specifically with regularity and stability theory of wave maps. Wave maps are a hyperbolic analogue of harmonic maps, where the domain is a Lorentzian manifold rather than a Riemannain manifold. When one considers the energy functional for maps from a Lorentzian manifold to a Riemannian manifold, the resulting Euler-Lagrange equations become hyperbolic - for maps between two Riemannian manifolds these equations are elliptic. Wave maps are called sigma models by physicists, and they arise in modern physics in various guises. For example, Einstein's field equations can be reduced to sigma model equations. Not much general theory has been developed for wave maps thus far as hyperbolic partial differential equations are mathematically harder to deal with than the elliptic ones.