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Texas A&M University

Several Complex Variables

Several complex variables makes contact with many areas of mathematics: function theory, partial differential equations, geometry, functional analysis, algebra. Applications arise not only in pure mathematics but also in mathematical physics and control theory.

Fascinating new phenomena appear in multi-dimensional complex analysis that are not seen in the one-variable theory. From the point of view of function theory, for example, a notable difference from the theory of one complex variable is that zeroes of holomorphic functions of several variables are never isolated. From the point of view of partial differential equations, a notable difference in the multi-dimensional theory is that the Cauchy-Riemann equations in several variables form an over-determined system. From the point of view of geometry, a notable difference in dimensions higher than one is that complex analysis can take place on submanifolds.

The theory of functions of several complex variables presents both a surprising rigidity and a remarkable richness. The rigidity is evident, for example, in the lack of a higher-dimensional Riemann mapping theorem: even such simple domains as a two-dimensional ball and the Cartesian product of two discs are not holomorphically equivalent. The richness is already apparent in the observation that convergence domains for multi-variable power series have infinitely many different shapes, in dramatic contrast to the situation in one variable.

Harold P. Boas and Emil J. Straube (currently the Head of the Department of Mathematics) are particularly noted for their research on the theory of the Bergman kernel function and on the the boundary regularity theory of the inhomogeneous Cauchy-Riemann equations in pseudoconvex domains. They received the Stefan Bergman Prize from the American Mathematical Society in 1995. Emil J.Straube is also the author of the book Lectures on the L2-Sobolev Theory of the D-bar-Neumann Problem. The group hosts a working seminar on complex analysis. Regular course offerings for graduate students include Math 617, Math 618, and Math 650.