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For conditions, please view our photographs. A rare book from the library gathered by the famous Cambridge Don, computer scientist, food and wine connoisseur, Jack Arnold LANG.

                           A rare book collected by Jack’s father.

Forward by Theodor Brikker

In the summer semester of 1972 I gave a course of lectures on the local theory of differentiable maps at the University of Freiburg. These lectures have formed the basis for the first thirteen chapters of the book, the next three chapters having been written for a summer school organised by the Studienstiftung des deutschen Volkes. My students were responsible for removing many mistakes from the original manuscript which has now been translated into English by L. Lander. He has also made a number of improvements and corrections and provided the last chapter together with its pictures and list of publications. The later chapters discuss a subject which has been the real motivation for writing the book: classical catastrophe theory. We have both profited greatly from a lecture course on catastrophe theory by K. J!inich, given in Regensburg during the winter semester 1971/72, which contained most of the information and pictures presented in chapter 1 7. A small number of copies of the German text of the present book were printed for our students under the title: Der Regensburger Trichter, Band 3, Differenzierbare Abbildungen. On the pages that follow, the reader will not find any new results or methods. Our purpose is to make it easier for those students, who have properly understood the basic lecture courses on analysis and possess a basic knowledge of algebra, to learn about recent work on differentiable maps, in particular, the mysteries of catastrophe theory. What are the following pages about? Let f : Rn … Rk be a differentiable map. What can be said in general about f- 1 { 0 } , that is, about the solution set of a system of non-linear equations? To start with one refers to a theorem of Whitney and Sard’s theorem, given in 2. 1 and 3. 3, in particular one discovers that interesting structure can only be found for ‘generic’ sets of maps. v Of special interest are the stable differentiable maps, where f is called stable if for a ‘small perturbation’ li : Rn -Rk there are invertible transformations such that the diagram commutes. In fact, one expects that natural forms must be described by stable maps because everything in nature is subject to small disturbances. Is ‘almost every’ map stable? How is the concept of stability to be interpreted? Any introduction to analysis explains that a differentiable germ f : (It, 0) – (R, 0) with non-vanishing Taylor expansion at the origin can be transformed into the first non-vanishing term by a suitable coordinate change. In higher dimensions, when is a germ determined by a finite part of its Taylor expansion (up to equivalence under coordinate transformations)? Those are a few of the questions which are discussed below. Perhaps the reader will thereby be encouraged to join in the task of clarifying and understanding some of the ideas of R. Thorn. Regensburg, Spring 1974 Theodor Brikker

These notes give a fairly elementary introduction to the local theory of differentiable mappings. Sard’s Theorem and the Preparation Theorem of Malgrange and Mather are the basic tools and these are proved first. There follows a number of illustrations including: the local part of Whitney’s Theorem on mappings of the plane into the plane, quadratic differentials, the Instability Theorem of Thom, one of Mather’s theorems on finite determinacy and a glimpse of the theory of Toujeron. The later part of the book develops Mather’s theory of unfoldings of singularities. Its application to Catastrophe theory is explained and the Elementary Catastrophes are illustrated by many pictures. The book is suitable as a text for courses to graduates and advanced undergraduates but may also be of interest to mathematical biologists and economists. This book gives a fairly elementary introduction to the local theory of differentiable mappings and is suitable as a text for courses to graduates and advanced undergraduates.

• Theodor Bröcker: A German mathematician specializing in differential topology, known for his contributions to the study of differentiable mappings and catastrophes.

• Key Works: Authored or co-authored books such as Differentiable Germs and Catastrophes (with L. Lander) and Introduction to Differential Topology (with K. Jänich).