# William Alexandre

- FACULTE DES SCIENCES ET TECHNOLOGIES
- DEPARTEMENT MATHEMATIQUES

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I work in complex analysis of several variables and in quasi complex geometry. In complex analysis I use integral fomulae and residu currents which give explicit solutions of d-bar equation, of holomorphic extensions of functions defined on analytic varieties or division formulae. I established :

- optimal regularity of solutions of the Cauchy-Riemann equation d-bar u = f on a bounded domain D, of finite type, when the data f has certains type of regularity over D,
- necessary and sufficient or nearly sufficient conditions under which functions defined over the intersection of a domain D with a manifold ora singular varity has a holomorphic extension to D in L^p(D),
- caracterization of zero sets of H^p (D) functions when D is a bounded convex domain of finite type,
- a proof of the H^p-corona theorem for 2 generators in convex domains of finite type,
- necessary and sufficient or nearly sufficient conditions for the existence of solutions to division problems of holomorphic functions in strictly pseudoconvex domains.

In quasi complex geometry, I prooved that there exsits hypersurfaces with constant rank which do not contain J-holomorphic disc with prescribed derivative at the origin. I also gave conditions under which such discs exist.