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This quantity contains a suite of articles for the court cases of the fortieth Taniguchi Symposium research and Geometry in different advanced Variables held in Katata, Japan, on June 23-28, 1997. because the inhomogeneous Cauchy-Riemann equation used to be brought within the research of advanced research of a number of Variables, there was powerful interplay among advanced research and actual research, particularly, the speculation of Partial Differential Equations. difficulties in complicated Anal­ ysis stimulate the improvement of the PDE concept which hence could be utilized to advanced research. This interplay includes Differen­ tial Geometry, for example, through the CR constitution modeled at the triggered constitution at the boundary of a posh manifold. Such buildings are clearly with regards to the PDE idea. Differential Geometric formalisms are successfully utilized in settling difficulties in complicated research and the consequences enhance the idea of Differential Geometry. This quantity makes a speciality of the newest advancements during this inter­ motion, together with hyperlinks with different fields corresponding to Algebraic Geometry and Theoretical Physics. Written by means of contributors within the Symposium, this vol­ ume treats a variety of facets of CR geometry and the Bergman kernel/ professional­ jection, including different significant matters in sleek advanced research. we are hoping that this quantity will function a source for all who're attracted to the recent traits during this region. we wish to specific our gratitude to the Taniguchi origin for beneficiant monetary help and hospitality. we'd additionally prefer to thank Professor Kiyosi Ito who coordinated the association of the symposium.

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