Mathematical Sciences: Smoothing of Dispersive Waves

该奖项支持的主要研究领域是偏微分方程，特别是那些用于模拟波动现象的偏微分方程。工作包括研究一类线性和非线性色散演化方程的平滑性质。色散波的特点是群速度以非平凡的方式依赖于波数。Korteweg deVries方程就是一个例子。平滑现象是指观察到的某些方程的解的行为——它们比它们的初始条件更平滑。这项工作的方法是利用换向子方法和散射理论。人们需要Sobolev加权规范的微地方版本来取代不能随时间持续的标准规范。其他研究方向包括正则行列式：二维黎曼流形上椭圆算子正则行列式的变分公式研究；完全可积哈密顿系统的全局方面以及环面流形和海森堡流形上的薛定谔算子。偏微分方程是物理科学中数学建模的基础。涉及连续变化的现象，例如在运动、材料和能量中看到的现象，都遵循某些一般规律，这些规律可以用偏导数之间的相互作用和关系来表示。数学的关键作用不是说明关系，而是从中提取定性和定量的意义，并验证表达解决方案的方法。

The primary area of research supported by this award is partial differential equations, especially those used to model wave phenomena. Work includes the investigation of smoothing properties for a large class of linear and nonlinear dispersive evolution equations. Dispersive waves are characterized by the property that the group velocity depends in a nontrivial way on the wave number. An example would be Korteweg deVries equation. The smoothing phenomenon refers to the observed behavior of solutions of certain equations - they are smoother than their initial conditions. The approach to this work is the use of the commutator method together with scattering theory. One needs microlocal versions of Sobolev weighted norms to replace the standard norms which do not persist with time. Other research directions include regularized determinants: the study of variational formulas for the regularized determinants of elliptic operators on Riemannian manifolds of dimension two; global aspects of completely integrable Hamiltonian systems and the Schrodinger operator on tori and Heisenberg manifolds. Partial differential equations form the backbone of mathematical modeling in the physical sciences. Phenomena which involve continuous change such as that seen in motion, materials and energy are known to obey certain general laws which are expressible in terms of the interactions and relationships between partial derivatives. The key role of mathematics is not to state the relationships, but rather, to extract qualitative and quantitative meaning from them and validate methods for expressing solutions.