Wolfgang Ebeling starts Functions of Several Complex Variables and Their Singularities off with the following paragraph (p. xi):
The study of singularities of analytic functions can be considered as a sub-area of the theory of functions of several complex variables and of algebraic/analytic geometry. It has in the meantime, together with the theory of singularities of differentiable mappings, developed into an independent subject, singularity theory. Through its connections with very many other mathematical areas and applications to natural and economic sciences and in technology (for example, under the heading, ‘catastrophe theory’) this theory has aroused great interest. The particular appeal, but also its particular difficulty, lies in the fact that deep results and methods from various branches of mathematics come into play here.
If anything, these remarks understate the reality that the important subject of singularity theory is dependent on a number of other mathematical fields, each of which must be covered in some detail before the results of singularity theory proper can come into reach. Naturally, this is a condition that affects all of mathematics except for those areas which can be classed as foundational in some obvious sense: group theory (even after the classification of finite simple groups), ring theory (especially the non-commutative sort), hard analysis, much of complex analysis and complex function theory, and just about all of set theory and logic come to mind right away. But autonomy in this regard is increasingly an illusion, what with the ubiquity of cross-fertilization among mathematical disciplines, and, for instance, the current resurgence of dialogues between theoretical physics and, e.g., algebraic geometry and number theory.
This pervasive reality implies a proportionately growing need for the student or the novice to master more and more mathematics, of very recent vintage and ever greater variety, in order to begin to do original work at the frontier. This certainly applies to singularity theory, and Ebeling’s remarks regarding several complex variables, algebraic geometry, and analytic geometry are particularly apt.
Additionally, in his “Foreword to the English Translation.” Ebeling discloses that the original German title of the work is Funktionentheorie, Differentialtopologie, und Singularitäten, which gives us an even broader context for the subject of singularity theory. This said, the book under review constitutes an offering along the lines of an autonomous preparation for original research on singularities by taking care of the assorted prerequisites in one place, assuming only the standard beginning graduate courses in algebra and complex variables.
Ebeling starts off with a tight treatment of Riemann surfaces, followed by a more ramified but still compact discussion of several complex variables. Immediately after this he tackles the subject of isolated singularities of holomorphic functions with appropriate thoroughness and in sufficient depth to allow for the material’s ready application. So it is that Morse theory makes its appearance in this context and is followed in short order by a discussion of the Milnor number of a singularity.
Ebeling’s style is concise and effective, and Functions of Several Complex Variables and Their Singularities is filled to overflowing with elegant results, presented rigorously and compactly. The result on p. 165 is a case in point: “Let f : (C2,0) → (C,0) be a simple holomorphic function germ with an isolated singularity at 0 and grad f(0) = 0. Then f is right equivalent to one of the following simple function germs: (a) xk+1 + y2 with k≥1 (Ak), (b) x2y + yk-1 with k≥4 (Dk), (c) x3 + y4 (E6), (d) x3 + xy3 (E7), (e) x3 + y5 (E8) …” (Yes, Ak, Dk, E6, E7, E8 do indeed come from Lie theory.)
Ebeling goes on to cover differential topology in some depth and closes his admirable book with a treatment of the topology of singularities focused on the deep theme of monodromy.
Manifestly this is a marvelous source whereby to gain relatively rapid access to contemporary work in singularity theory, modulo an appropriate investment of commitment and intensity on the reader’s part. The book is well worth it.
Michael Berg is professor of mathematics at Loyola Marymount University in Los Angeles, CA.