Analysis and Dynamics in Several Complex Variables

该奖项支持在几个复杂变量、微分几何和动力系统的界面上的研究。复变分析研究在复数空间上定义并取值的函数的行为和规律性。在科学、工程和经济的许多领域，它仍然是不可或缺的工具。本文研究了由复值函数定义的域在变形时变换的平滑性。使用积分公式，PI将研究当域的底层结构改变时，域的不变量是如何变化的。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对曲线流形线性近似中产生的复杂结构的奇点进行分类。该项目将涉及与早期职业阶段的研究人员合作，并将支持研究生的培训。由于最近的反例表明光滑域族可以由不连续的生物全纯族等效，PI将使用生物全纯群和其他分析工具（如Bergman度量）来研究域族之间生物全纯族的存在性。该PI将构造一个具有良好估计的复流形中合适域的全局同伦公式。目标之一是在局部同伦公式不存在的情况下构造一个全局公式。PI将使用这种全局同伦公式来研究复杂流形中具有强伪凸或凹边界的域的全纯嵌入，当域上的复杂结构变形时的稳定性。PI将利用这种方法研究高余维柯西-黎曼流形上全局柯西-黎曼结构的稳定性。该项目通过列维形式和正常束的曲率寻求复杂流形中紧致复杂流形嵌入的邻域的全纯分类。此外，PI将研究实数流形的Cauchy-Riemann奇点的分类，使用从几个复杂变量和动力系统中的小因子条件的方法。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.