David Barrett

Much of my recent research activity is connected with complex-analytic aspects of projective duality. Specifically given a (suitably convex) real hypersurface S in complex projective space, the locus of affine complex hyperplanestangent to S form a dual hypersurface S* in the dual projective space. The interaction between (boundary values of) holomorphic functions on the two hypersurfacesis regulated by an explicit singular integral operator, the Leray transform. The norm of this operator is significant for the function theory;in several situations it has been established that the norm (or at the least theessential norm) is tightly connected with the projective geometry of S. (With postdoc Luke Edholm, I am pursuing a program to establish a general conjecture along these lines.) Along more purely geometric lines, along with Dusty Grundmeier, I have shown that that the boundary of any bounded strongly pseudoconvex complete circular domain in ℂ2 must contain points that are exceptionally tangent to a projective image of the unit sphere.