Complexity in Cauchy-Riemann Geometry

复杂分析的数学领域在工程和理论物理等应用科学和纯科学中得到了广泛的应用；复分析在数学的某些分支领域中也具有重要的基础意义。例如，物理现象通常可以用微分方程来建模，而其中某些方程最好是通过研究复数的技术来理解的。另一个例子是理论物理中的弦理论。高能物理模型的各个方面需要对复数几何有深刻的理解。这个项目的重点是复杂变量和真实变量相互作用的空间几何。用这种几何来研究的空间具有一定的复杂性，这种复杂性尚未得到很好的理解，但在数学界引起了极大的兴趣。PI将研究这种复杂性，并努力为该领域建立适当的基础。PI将研究全纯映射和实子流形或实奇异子变种的CR复杂度。关键的方法是研究定义方程的系数厄米矩阵。这种方法将实子流形与CR映射到超二次曲面的复杂性联系起来，超二次曲面是由厄米形式定义的超曲面。超二次曲面是CR几何中的模型子流形。子流形的复杂性越高，目标超二次曲面必须存在的维度就越大，所涉及的CR映射的复杂性也就越大。特别地，PI感兴趣的是进一步改进在列维平面超曲面的奇异集上的结果，即在某种意义上具有最小CR复杂度的集合。所提出的方法是改进在PI之前关于该主题的工作中成功使用的Segre多样性方法。为了研究所涉及的映射集的复杂性，也有必要了解超二次曲面本身之间映射的复杂性。PI将通过应用交换代数和分析的混合技术来改进以前工作中的定量结果，这些技术之前已经被证明是成功的。然后，这项工作将应用于更一般的领域。

The mathematical field of complex analysis has found diverse application in applied and pure sciences such as engineering and theoretical physics; complex analysis is also of foundational importance in certain subfields of mathematics. For example, physical phenomena can often be modeled with differential equations, and certain of these equations are best understood using techniques issuing from the study of complex numbers. Another example is provided by string theory in theoretical physics. Aspects of this model of high energy physics require a deep understanding of the geometry of complex numbers. This project is focussed on the geometry of spaces where complex and real variables interact. The spaces studied with this geometry have a certain complexity that is not yet well understood but is of great interest in the mathematical community. The PI will study this complexity and work towards building proper foundations for the field.The PI will study the CR complexity of holomorphic mappings and real submanifolds or real singular subvarieties. The key method is to study the Hermitian matrix of coefficients of the defining equations. This approach links the complexity of a real submanifold to a CR mapping to a hyperquadric, that is a hypersurface defined by a Hermitian form. Hyperquadrics are the model submanifolds in CR geometry. The greater the complexity of a submanifold, the greater the dimension that the target hyperquadric must live in, and the greater the complexity of the CR mapping involved. In particular the PI is interested in further refining the results on the singular set of Levi-flat hypersurfaces, that is sets which have in some sense the smallest CR complexity. The proposed method is to refine the Segre variety approach that was successfully used in the PI's previous work on the subject. To study the complexity of the set of the mappings involved, it is necessary to also understand the complexity of mappings between hyperquadrics themselves. The PI will work to refine the quantitative results in previous work by applying a mix of techniques from commutative algebra and analysis that has proved successful before. This work will then be applied to more general domains.