Partial Differential Equations and Several Complex Variables

小行星9703678 支持的项目是在几个复变量的领域，特别是，它涉及到复欧几里德空间中的真实的超曲面的研究。 具体而言，研究者计划在复欧氏空间中找到刚性超曲面定义方程的完全规范形;找到无穷小Cauchy-Riemann自同构解析的充要条件;研究严格凸超曲面的Levi流演化;研究CR-流形上热核的非对角小时间行为;理解在诺依曼问题和相应的热核研究中出现的核的奇异性。 复欧氏空间中的真实的超曲面推广了空间中的曲面：它们是位于具有复杂结构的空间内的曲面的高维类似物。复数结构--局部复数数组--最初是在多项式方程的研究中引入的;它们在数学的各个部分都有很多用途，通常会导致概念和计算的简化。

9703678 Stanton The supported project lies in the area of several complex variables; in particular, it involves the study of real hypersurfaces in complex Euclidean space. Specifically, the investigator plans to find a complete normal form for the defining equation of a rigid hypersurface in complex Euclidean space; find a necessary and sufficient condition for analyticity of infinitesimal Cauchy-Riemann automorphisms; study the evolution of a strictly convex hypersurface by the Levi flow; study the off-diagonal small time behavior of the heat kernel on a CR-manifold; understand the singularities of kernels arising in the study of the Neumann problem and the corresponding heat kernel. Real hypersurfaces in complex Euclidean space generalize surfaces in space: they are higher dimensional analogs of surfaces sitting inside a space equipped with a complex structure. Complex structures - locally, arrays of complex numbers - were first introduced in the study of polynomial equations; they continue to find many uses in various parts of mathematics, often leading to conceptual as well as computational simplifications.