Complexity in Cauchy-Riemann Geometry

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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.