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It is close enough to the end of the century to make a guess as to what the Encyclopedia Britannica article on the history of mathematics will report in 2582: "We have said that the dominating theme of the Nineteenth Century was the development and application of the theory of functions of one variable. At the beginning of the Twentieth Century, mathematicians turned optimistically to the study off unctions of several variables. But wholly unexpected difficulties were met, new phenomena were discovered, and new fields of mathematics sprung up to study and master them. As a result, except where development of methods from earlier centuries continued, there was a recoil from applications. Most of the best mathematicians of the first two-thirds of the century devoted their efforts entirely to pure mathe­ matics. In the last third, however, the powerful methods devised by then for higher-dimensional problems were turned onto applications, and the tools of applied mathematics were drastically changed. By the end of the century, the temporary overemphasis on pure mathematics was completely gone and the traditional interconnections between pure mathematics and applications restored. "This century also saw the first primitive beginnings of the electronic calculator, whose development in the next century led to our modern methods of handling mathematics.



Combinatorics: Trends and Examples

Combinatorics is currently a very active branch of mathematics, and there is good reason to believe that it will become even more active in the future. One trend in combinatorics is the increasing use of methods from other branches of mathematics to solve purely combinatorial questions. A wellknown example of this situation is the field of algebraic topology where topological questions are converted into algebraic ones and vice versa, an idea which has been used quite effectively in combinatorics, and which we will illustrate with two examples.
Kenneth Baclawski

Control Theory and Singular Riemannian Geometry

This paper discusses the qualitative and quantitative aspects of the solution of a class of optimal control problems, together with related questions concerning a corresponding stochastic differential equation. The class has been chosen to reveal what one may expect for the structure of the set of conjugate points for smooth problems in which existence of optimal trajectories is not an issue but for which Lie bracketing is necessary to reveal the reachable set. It is, perhaps, not too surprising that in thinking about this problem various geometrical analogies are useful and, in the final analysis, provide a convenient language to express the results. Indeed, the geodesic problem of Riemannian geometry is commonly taken to be the paradigm in the calculus of variations; a point of view which is supported by a variety of variational principles such as the theorem of Euler which identifies the path of a freely moving particle on a manifold with a geodesic and the whole theory of general relativity. Nonetheless, the class of variational problems considered here can only be thought of as geodesic problems in some limiting sense in which the metric tends to infinity. For this reason the geodesic analogy has to be developed rather carefully.
R. W. Brockett

On Certain Topological Invariants Arising in System Theory

This paper is an extension of the lecture presented by the first author at the conference, New Directions in Applied Mathematics, held at The Cleveland Museum of Art and at The Case Western Reserve University on the occasion of that university’s centennial anniversary. This author would like to thank the organizers of the conference, Professors Peter Hilton and Gail Young, for their cordial invitation to join in CWRU’s celebration as well as for the opportunity to give such a lecture at a time when the mathematics of control systems is expanding quite rapidly on several exciting frontiers. The lecture itself covered, roughly speaking, the material treated in sections 1, 3, and 5 of the present paper and was intended to give an indication of the kind of research which is currently going on in the application of geometry and topology to the problems and theory of linear systems. These topics included a survey of known results and of joint work with the second author, and with R. W. Brockett (this has been reported in more detail elsewhere [10])
Christopher I. Byrnes, Tyrone E. Duncan

Operations Research and Discrete Applied Mathematics

Operations research is an area of great potential for growth. Managing of any kind of system consists to a large extent of diagnosing, recommending, and supervising change. As more and more aspects of society become computerized, such management requires understanding what programs and machines can do and how one changes them to do whatever is best in a new situation.
D. Kleitman

Symplectic Projective Orbits

Let H be a complex Hilbert space and P(H) the corresponding projective space, i.e., the space of all one dimensional subspaces of H. If v is a non-zero vector in H we shall denote the corresponding point in P(H), i.e., the line throughv by [v]. Let G be a compact Lie group which is unitarily represented on H so that we may consider the corresponding action of G on P(H). In conjunction with the Hartree-Fock and other approximations, a number of physicists, [2], [4], and [5], have become interested in the following problem: For which smooth vectorsv is the orbit G [v] symplectic? Since G is compact, by projecting onto components we may reduce the problem to the case where the representation is irreducible and hence H is finite dimensional. We restate the question in this case: The unitary structure on H makesP(H) into a Kaehler manifold and thus, in particular, into a symplectic manifold. Each G orbit in P(H) is a submanifold. The question is: for which G orbits is the restriction of the symplectic form of P(H) nondegenerate so that the orbit becomes a symplectic manifold? We shall show that the only symplectic orbits are orbits through projectivized weight vectors. But not all such orbits are symplectic; there is a further restriction on the weight vector that we shall describe. The orbit through the projectivized maximal weight vector is not only symplectic but is also Kaehler, i.e., is a complex submanifold of P(H) and is the only orbit with this property.
Bertram Kostant, Shlomo Sternberg

Four Applications of Nonlinear Analysis to Physics and Engineering

My goal is to describe, in as accessible terms as possible, four separate applications of nonlinear analysis to relativity, elasticity, chaotic dynamics and control theory that I have recently been involved with. The descriptions are in some sense superficial since many interesting technical points are glossed over. However, this is necessary to efficiently convey the flavor of the methods.
Jerrold E. Marsden

Bifurcation, Catastrophe, and Turbulence

Bifurcation occurs in a parametrised dynamical system when a change in a parameter causes an equilibrium to split into two. Catastrophe occurs when the stability of an equilibrium breaks down, causing the system to jump into another state. The elementary theory concerns dynamical systems with steady state equilibria (point attractors), and the non-elementary theory concerns systems with dynamic equilibria (periodic attractors and strange attractors). In the elementary case Thom [72] has used singularities to classify both bifurcation and catastrophe, and this has led to a great variety of applications [22]. We illustrate the contrasting styles of application in biology and physics by describing two recent examples. The first is a model by Seif [59] concerning hyperthyroidism, and the second is a model by Schaeffer [58] concerning Taylor cells in fluid flow.
E. C. Zeeman

The Emphasis on Applied Mathematics Today and Its Implications for the Mathematics Curriculum

1. Across the country, and beyond the borders of the United States, the cry is being heard that we mathematicians should be concerning ourselves more, both in our research and in our teaching, with applications of mathematics. It is being argued that we have been overemphasizing mathematics itself, the autonomous discipline of mathematics, at the expense of due attention to its usefulness, to its role in science, in engineering, in the conduct of modern society. Some put it crudely—there is too much “ pure mathematics,” too little “ applied mathematics.”
Peter J. Hilton
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