Complex Analysis: Several Complex Variables and Connections with PDE Theory and Geometry (Trends in Mathematics)

Several Complex Variables is a beautiful example of a ?eld requiring a wide rangeoftechniquescoming fromdiverseareasin Mathematics.Inthe lastdecades, many major breakthroughs depended in particular on methods coming from P- tial Di?erential Equations and Di?erential and Algebraic Geometry. In turn, S- eralComplexVariablesprovidedresultsandinsightswhichhavebeenoffundam- tal importance to these ?elds. This is in particular exempli?ed by the subject of Cauchy-Riemanngeometry,whichconcernsitselfbothwiththetangentialCauchy- Riemannequationsandtheuniquemixtureofrealandcomplexgeometrythatreal objects in a complex space enjoy. CR geometry blends techniques from algebraic geometry, contact geometry, complex analysis and PDEs; as a unique meeting point for some of these subjects, it shows evidence of the possible synergies of a fusion of the techniques from these ?elds. The interplay between PDE and Complex Analysis has its roots in Hans Lewy's famous example of a locally non solvable PDE. More recent work on PDE has been similarly inspired by examples from CR geometry. The application of analytic techniques in algebraic geometry has a long history; especially in recent - years, the analysis of the ?-operator has been a crucial tool in this ?eld. The - ?-operator remains one of the most important examples of a partial di?erential operator for which regularity of solutions under boundary constraints have been extensively studied. In that respect, CR geometry as well as algebraic geometry have helped to understand the subtle aspects of the problem, which is still at the heart of current research.