By Gen Komatsu,Masatake Kuranishi

This quantity involves a set of articles for the lawsuits of the fortieth Taniguchi Symposium research and Geometry in different complicated Variables held in Katata, Japan, on June 23-28, 1997. because the inhomogeneous Cauchy-Riemann equation used to be brought within the research of complicated research of a number of Variables, there was powerful interplay among complicated research and actual research, particularly, the idea of Partial Differential Equations. difficulties in advanced Anal­ ysis stimulate the advance of the PDE idea which as a result should be utilized to advanced research. This interplay consists of Differen­ tial Geometry, for example, through the CR constitution modeled at the precipitated constitution at the boundary of a posh manifold. Such buildings are obviously regarding the PDE conception. Differential Geometric formalisms are successfully utilized in settling difficulties in advanced research and the consequences increase the idea of Differential Geometry. This quantity makes a speciality of the newest advancements during this inter­ motion, together with hyperlinks with different fields equivalent to Algebraic Geometry and Theoretical Physics. Written by means of contributors within the Symposium, this vol­ ume treats a number of elements of CR geometry and the Bergman kernel/ professional­ jection, including different significant matters in smooth complicated research. we are hoping that this quantity will function a source for all who're attracted to the recent traits during this quarter. we want to specific our gratitude to the Taniguchi origin for beneficiant monetary aid and hospitality. we might additionally wish to thank Professor Kiyosi Ito who coordinated the association of the symposium.

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