Nicholas Hanges

Non-analytic hypoellipticity in the presence of symplecticity

Here we construct non-analytic solutions to a class of hypoelliptic operators with symplectic characteristic set and in the form of a sum of squares of real analytic vector fields.

Explicit formulas for the Szegö kernel for some domains in C2

We study the Szego kernel for a class of strictly pseudoconvex domains in C2. An explicit algorithm is given to compute the complete asymptotic expansion for the symbol of the Szego kernel for these domains. It is then easy to compute the first three terms explicitly in terms of the defining function and its derivatives. We give an example where the first three terms (including the logarithmic term) are all non zero. Finally, we show that if the second term vanishes identically, then the boundary is locally biholomorphic to the surface Im w = ¦z¦2.

Singular solutions for a class of Grusin type operators

We construct singular solutions for a one-parameter family of partial differential equations with double characteristics and with complex lower order terms. The parameter belongs to a discrete set which is described in terms of the spectrum of a related differential operator.

Analytic singularities of the Bergman kernel for tubes

Let ⊂ R be a bounded, convex, and open set with real analytic boundary. Let T ⊂ C be the tube with base , and let B be the Bergman kernel of T . If is strongly convex, then B is analytic away from the boundary diagonal. In the weakly convex case this is no longer true. In this situation we relate the off-diagonal points where analyticity fails to the characteristic lines. These lines are contained in the boundary of T , and they are projections of the Treves curves. These curves are symplectic invariants that are determined by the CR (Cauchy-Riemann) structure of the boundary of T . Note that Treves curves exist only when has at least one weakly convex boundary point.