Analysis and Dynamics in Several Complex Variables

该奖项支持在几个复杂的变量，微分几何和动力系统的接口的研究。复分析研究在复数空间中定义并取值的函数的行为和规律性。它仍然是科学，工程和经济学许多领域不可或缺的工具。本文研究复值函数定义的域在变形时变换的光滑性。使用积分公式，PI将研究当域的底层结构发生变化时，域的不变量如何变化。该项目的另一个组成部分涉及共振的研究。PI将使用测量非共振的小因子来对弯曲流形的线性近似中产生的复杂结构的奇异性进行分类。该项目将涉及在职业生涯的早期阶段与研究人员的合作，并将支持研究生的培训。最近的反例表明，光滑的域族可能等价于一个不连续的家庭的双全纯，PI将研究家庭的存在性的双全纯域之间使用双全纯群和其他分析工具，如伯格曼度量。PI将构造一个全局同伦公式，并对复流形中的合适域进行良好的估计。目标之一是在局部同伦公式不存在的情况下构造全局公式。PI将使用这样的整体同伦公式来研究复流形中具有强伪凸或凹边界的域的全纯嵌入的稳定性，当域上的复结构变形时。PI将使用这种方法来研究高余维Cauchy-Riemann流形上的全局Cauchy-Riemann结构的稳定性。该项目寻求通过列维形式和法丛的曲率对紧致复流形在复流形中的嵌入邻域进行全纯分类。此外，PI将使用动力系统中的多复变量和小因子条件的方法研究真实的流形的Cauchy-Riemann奇点的分类。该奖项反映了NSF的法定使命，并通过使用基金会的智力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.