Analysis and Dynamics in Several Complex Variables

该奖项支持在几个复变量、微分几何和动力系统的界面上的研究。复数分析研究定义在复数空间上并在复数空间取值的函数的行为和规律。在科学、工程和经济的许多领域，它仍然是不可或缺的工具。这个项目考虑了当区域变形时，由复值函数定义的区域上的变换的光滑性。使用积分公式，PI将研究当域的底层结构改变时，域的不变量如何变化。该项目的另一个组成部分涉及对共振的研究。PI将使用测量非共振的小除数来分类在曲线流形的线性近似中出现的复杂结构的奇点。该项目将在职业生涯的早期阶段与研究人员合作，并将支持研究生的培训。最近的反例表明，光滑的域族可能由不连续的双全射族等价，PI将使用双全纯群和其他分析工具(如Bergman度量)来研究域族之间的双全射族的存在性。PI将构造一个全局同伦公式，该公式对复杂流形中的适当区域具有良好的估计。其中一个目标是在局部同伦公式不存在的情况下构造一个全局公式。PI将使用这样的全局同伦公式来研究当区域上的复结构变形时，具有强伪凸或凹边界的区域在复流形中的全纯嵌入的稳定性。PI将使用这种方法来研究高余维柯西-黎曼流形上整体柯西-黎曼结构的稳定性。该项目通过法丛的Levi形式和曲率，寻求紧致复流形的嵌入邻域的全纯分类。此外，PI将使用动力系统中的几个复变量和小因子条件的方法来研究实流形的柯西-黎曼奇点的分类。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This award supports research at the interface of several complex variables, differential geometry, and dynamical systems. Complex analysis studies the behavior and regularity of functions defined on and taking values in spaces of complex numbers. It remains an indispensable tool across many domains in the sciences, engineering, and economics. This project considers the smoothness of transformations on a domain defined by complex valued functions when the domain is deformed. Using integral formulas, the PI will study how invariants of a domain vary when the underlying structure of the domain changes. Another component of the project involves the study of resonance. The PI will use small divisors that measure non-resonance to classify singularities of the complex structure arising in linear approximations of curved manifolds. The project will involve collaboration with researchers in an early career stage and will support the training of graduate students.Motivated by recent counterexamples showing that smooth families of domains may be equivalent by a discontinuous family of biholomorphisms, the PI will study the existence of families of biholomorphisms between families of domains using biholomorphism groups and other analytic tools such as Bergman metrics. The PI will construct a global homotopy formula with good estimates for suitable domains in a complex manifold. One of the goals is to construct a global formula in cases when a local homotopy formula fails to exist. The PI will use such global homotopy formulas to investigate the stability of holomorphic embeddings of domains with strongly pseudoconvex or concave boundary in a complex manifold, when the complex structure on the domains is deformed. The PI will use this approach to investigate stability of global Cauchy-Riemann structures on Cauchy-Riemann manifolds of higher codimension. The project seeks a holomorphic classification of neighborhoods of embeddings of a compact complex manifold in complex manifolds via the Levi-form and curvature of the normal bundle. In addition, the PI will study the classification of Cauchy-Riemann singularities for real manifolds using methods from several complex variables and small-divisor conditions in dynamical systems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.