Hermitian Forms and CR Geometry

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John D'Angelo will continue his study of Hermitian forms and their applications to several complex variables and CR geometry. This work lies at the foundation of a developing area in mathematics, complexity theory in CR Geometry, while also enabling connections to other branches of mathematics. The starting point concerns CR mappings between spheres and hyperquadrics; it goes on to study various notions of complexity for CR mappings, the signature pair of a Hermitian symmetric function, and representation theory for finite unitary groups. Applications include Hermitian analogues of Hilbert's 17-th problem, revolving around new matrix positivity conditions for Hermitian polynomials on algebraic sets. This work is closely related to a non-linear form of the Cauchy-Schwartz inequality and to the notion of Hermitian nullity of a set with respect to a real ideal. These results will lead to striking links between the geometry of a real hypersurface in complex Euclidean space and basic questions in algebra. Mapping theorems in one complex dimension have long played a central role in mathematics, physics, and engineering. The proposed work can be regarded as developing the fundamental ideas in higher dimensions, where the situation becomes much more subtle and new phenomena arise. The resulting work leads to a promising and unusual combination of analysis, geometry, and algebra. In 1900 the mathematician David Hilbert set out a list of 23 problems. These problems continue to drive much of current day mathematical research. The famous 17th problem was solved in the 1920s. D'Angelo's work has led to Hermitian analogues of this problem, where new difficulties appear and connections to many areas of mathematics arise. D'Angelo will continue to mentor young mathematicians (including graduate students) in these topics and to organize and attend conferences. He will also continue teaching a new complex analysis course he has developed for Honors freshmen.