Mathematical Sciences: Partial Differential Equations and Several Complex Variables

首席研究员将研究由几个复杂变量中的类似物引起的几个问题：你能听到鼓的形状吗？一个问题是对于CR流形是否有类似的黎曼-洛克定理。另一个问题是CR流形上热核的小时间非对角行为。对刚性超曲面的正规形式的研究将继续进行。她将在有限型CR流形上寻找包含一个参数的伪微分算子代数。你能听到鼓的形状吗？它是圆形的还是方形的？这个问题最初是由V. Kac在数学背景下提出的，并继续激发着激烈的研究。信号处理理论的分支是多方面的。首席研究员将继续研究这个问题的复杂类似物，在这个研究领域，她已经是最权威的。

The principal investigator will study several questions arising from the analogues in several complex variables of the problem: can you hear the shape of a drum? One question is whether there is an analogue of the Riemann-Roch Theorem for CR manifolds. Another concerns the small-time off-diagonal behavior of the heat kernel on CR manifolds. Investigations of normal forms for rigid hypersurfaces will continue. And she will search for an algebra of pseudodifferential operators which contains a parametrix for the Kohn Laplacian on CR manifolds of finite type. Can you hear the shape of a drum? Was it circular or square? This question was first addressed in a mathematical context by V. Kac and continues to inspire intense study. Ramifications to signal processing theory are manifold. The principal investigator will continue her study of the complex analogues to this question, a field of study in which she already is the foremost authority.