Giuseppe Zampieri, professor of Mathematics at University of Padova since 1984 - Head of the nationwide PRIN research program "Microlocal Analysis" - Coordinator of the research unit "Algebaric and Complex Analysis" at University of Padua - Member of the scientific board of CIME foundation at Florence - Member of "Istituto Veneto di Scienze Lettere ed Arti" at Venice - Author of about one hundred publications in Algebraic Analysis and Complex Geometry

Research interests

**Microlocal equations in real analytic and Gevrey category** - Fundamental
principle of Ehrenpreis - Theory of lacunas - Global existence of analytic solutions.
This has been an active field of research since the 70's and early 80's which has
attracted the interest of Garding, Hörmander, Bott, Sato, De Giorgi, Cattabriga to
mention only a few. The new contribution consists in the geometric interpretation
of the "Phragmén-Lindelöf principle" for existence of real analytic solutions in the
formulation given by Hörmander. The geometry of the domain is completely
described in terms of the propagation cones of the differential operator: bicharacteristic
lines or light cones (for the cases of simple and multiple characteristics
respectively). A further contribution consists in proving that the analytic solvability
"propagates" to Gevrey indices *s < m/(m - 1)* where m is the multiplicity.
Leaders in this line of research were Cattabriga, Andreotti, Kaneko.

**Algebraic Analysis** - Microlocal theory of boundary
value problems - Light reflection. In this field merge the contribution
of various theories such as the symplectic
and differential geometry, the analytic theory of PDE's, the theory of
sheaves
and derived categories. Among the leading figures are Sato, Kashiwara,
Leray,
Schapira. The new contribution is the theory of the singular
Lagrangians which
are naturally associated to the manifolds with boundary. The theory of
the second
microlocalization at the boundary and the propagation of the
singularity for
the non-microcharacteristic equations has also been settled. There have
been as
coauthors in this research: Schapira, Kataoka, Tose, D'Agnolo.

**Complex and CR Geometry **- Analytic discs in complex spaces - Integral representation
of the solutions to the tangential CR system - Uniform L^2 estimates -
Kobayashi metric and extremal discs - Holomorphic extension from convex hypersurfaces
- CR envelopes. The new contribution are a theory of analytic discs with
singular boundary, the discs in symplectic spaces and their transformations, the
precise description of the forced CR extension, the uniform estimates for weakly q-
pseudoconvex domains of C^n. The techniques are various: Hölder
and L^2 estimates, integral representation by a kernel, systems of real/complex vector
fields, stationary/extremal discs. Some reference names are those of Henkin,
Tumanov, Kohn, Nirenberg. There have been as coauthors: Tumanov, Zaitsev,
Baracco, Ahn.

Direction of thesis: A. D'Agnolo, C. Marastoni, A. Scalari, A. Giacobbe, F. Tonin,
L. Baracco, P.N. Bettiol, A. Siano, R. Mascolo, R. Marigo.