Applied Mathematics at M.I.T.

Harvey P Greenspan worked hard to establish Applied Mathematics as a subject in its own right at the Massachusetts Institute of Technology. Having achieved a successful outcome, he wrote the paper 'Applied Mathematics at M.I.T.', Amer. Math. Monthly 80 (1) (1973), 67-72. we present a version of this paper below.

Applied Mathematics at M.I.T., by H P Greenspan, Massachusetts Institute of Technology.

The use of mathematics pervades modern society and its impact, already immense, is growing rapidly. This development, reflected in a rising and often insistent demand for a relevant mathematics curriculum, has placed great pressure on the existing structure of American mathematics. It is now clear that fundamental changes are necessary to make the system function properly in the required capacity. Consequently, serious consideration must be directed to a new program of education and research in mathematics which will fulfil the assumed obligations, and all expectations, in the most natural and spontaneous manner. At M.I.T., the response has been to take the obvious but decidedly non-trivial course, namely, the establishment of Applied Mathematics as a separate discipline and its federation with Pure Mathematics in a single, balanced, and assuredly unique Department of Mathematics. The success of this effort warrants elaboration and it is my purpose here to describe, first, the field of Applied Mathematics and then, more briefly, departmental organisation.

Despite a long, rich history and a current state of great vitality, Applied Mathematics still suffers from a rather amorphous image, due mainly, I believe, to the lack of a precise and accepted statement of its basic philosophy and distinctive content. In many respects then, the development of Applied Mathematics amounts to the creation of a new discipline, whose existence has to be secured by unique, indispensable subject matter, research of high intellectual standards, and a stimulating interaction with other scientific fields. The more practical aspects of such an endeavour are the development of a comprehensive curriculum, the appointment of a superior faculty, good students, and sufficient funding; I will comment on each of these later.

A careful delineation of the philosophy of Applied Mathematics constitutes a blue print for development and a consistent means of formulating policy and making decisions. General agreement among various sub-groups about the nature of Applied Mathematics is possible because common attributes are easily recognised and form a genuine bond.

An acceptable definition has several practical advantages for it eliminates old frictions without creating new ones, and in general provides a measure of internal stability which is essential during the formative stage. It proclaims a unique identity, but need be no more precise, and should not be, than that given for any other subject and a little thought shows this to be a fairly weak condition.

Applied Mathematics at M.I.T. means the mathematical study of general scientific concepts, principles and phenomena. Emphasis should be placed on the interpretation of the subject as a science, because the primary goal is knowledge about the real world. Progress is measured by this standard and, of course, results are subjected to experimental test and verification. In this respect, the rigour and elegance of analysis are subordinated as ends to the truth and validity of the knowledge attained, although one expects, as a matter of faith, that a close correlation between these goals exists.

To enumerate the specific content of Applied Mathematics, it is necessary to identify the concepts, principles, and phenomena which, because of their widespread occurrence and application, relate or unify various fields of study. Although the list of topics can be amended easily, we have taken the following scientific principles and their mathematical formulations to form the core of our program:
propagation, equilibrium, stability, optimisation, cybernetics, statistics and random processes.
These receive explicit elucidation in such courses as: mechanics, fluid dynamics, electromagnetics, elasticity, astrophysics, geophysics, wave motion, turbulence, thermodynamics, kinetic theory, statistical mechanics, modern physics, probability, statistics, combinatorics, artificial intelligence, automata, computation, control, learning theory and analysis. There are several complementary but less tangible qualities which are essential to any complete description of Applied Mathematics. These are associated with a very versatile and distinctive approach, which manifests itself in the ability to formulate relevant mathematical models made solvable by the art of judicious approximation. Indeed, "approximation" is a key word, a preeminent and ever present characteristic of this subject, as it is of most other sciences. Obviously, implied in all of this is the creation, study and application of advanced mathematical methods relevant to scientific problems.

An Applied Mathematics curriculum constructed on this basis requires a faculty organised along similar lines and this determines a logical policy for growth and expansion. Furthermore, the educational objective implied is simply that every student acquire basic familiarity with each of the topics listed in addition to having specialised in at least one among them.

The problem of identity is difficult because in a sense all theoretical scientists are engaged in applying mathematics. A typical question might ask how the Applied Mathematician investigating wave propagation in the atmosphere differs from a meteorologist doing the same. The answer is that in fact they do not differ, nor should they, during this phase of activity. However, a particular physical problem will occupy only a comparatively short time in the career of the Applied Mathematician and, ideally, on a longer time scale, he will pursue the general concepts discovered in his researches. This can lead far afield from the point of origin. For example, many of the wave phenomena involved in the atmospheric circulation are general dynamical features which are also apparent in vehicular traffic or galactic structure. Thus, the study of specific problems can culminate in theoretical generalisations of broad application. Indeed, the ability to draw relevant analogies in seemingly different situations is one source of tremendous strength and vitality for Applied Mathematics.

Applied Mathematics, by virtue of its emphasis on breadth, versatility, innovation and approximation and its concern with fundamental concepts of interdisciplinary importance, can effectively counteract the trend towards narrow specialisation. An academic program that stresses the recognition and exploitation of important analogies and the transference of methods and techniques from one field to another will undoubtedly stimulate an effective and productive dissemination of mathematics. Furthermore, Applied Mathematics provides, in a completely natural manner, the best and most relevant mathematics education for the entire scientific and technological community.

The Applied Mathematics curriculum at M.I.T. has three essential features. The first is that it begins in the freshman year with calculus taught as Applied Mathematics. The second, and really the core of the undergraduate program, is a full year course for Sophomores and Juniors entitled "Principles of Applied Mathematics". This is designed as a broad survey of the entire field, a general introduction to the fundamental principles listed earlier and their related mathematics. The main objectives are to establish immediate recognition of Applied Mathematics as a unified discipline, and to instil the proper attitudes at the outset. In effect, this course occupies a position roughly equivalent to traditional introductory courses in the departments of physics or chemistry.

The third feature is the in-depth study of several of the principles cited with required courses on physical mathematics, statistics, computing science, and probability. This assures a uniform background for all students in Applied Mathematics and fosters the development of a community of common interest. However, great flexibility is permitted in the program of specialised training.

A special route into Applied Mathematics for the best theoretical students in science and engineering has been devised and this welcomed source of talent compensates for a natural outflow of our own students who wish greater specialisation in fields that have captured their interests.

Study plans for graduate students are handled on an individual basis but the general rule applies - specialisation with adequate breadth. A graduate student, with approval, could receive a degree in Applied Mathematics not having taken any course labelled as such. Of course, the graduate program covers the fundamental topics listed and the entire spectrum of mathematical methods; at this advanced stage, the research interests of the faculty are much in evidence. Students are definitely encouraged to enrol in courses given by other departments. It is contrary to our purpose and, in fact, impossible and undesirable for Applied Mathematics to be complete in itself.

Since Applied Mathematics is a and three are in operation at present. mainly fluid dynamics; the other and includes access to a large computer for more conventional numerical analysis. Experiments maintain the vital connection with reality and are to the theoretician, a source of information, intuition, and sometimes, inspiration. Personally, I limit my use of the laboratory to diagnostic or exploratory studies in the same spirit that I might perform numerical experiments on an analog or digital computer. However, specific problems often necessitate detailed experimental work and the facilities will support this if the enthusiasm and interest exist.

The laboratories are also utilised for educational purposes to help shape desired attitudes and otherwise to contribute to the scientific environment. Students who are brought face to face with nature's difficulties learn a great deal and do not soon forget their experience.

Faculty growth, ideally, should be controlled by the quality and tempo of research and the demands of the curriculum. Unfortunately, these conditions are sometimes in opposition for it is always easiest to over-develop an area that is already strong. A practical appointments policy is to determine whether new capabilities are offered to the group or whether the talents obtainable exceed those of the resident faculty in the same area. An affirmative answer to either is a call for positive action.

The size of the faculty is governed internally by the growth of the subject and externally by the response it generates. In this regard, it is relevant to consider certain desirable constraints, those of maximising professional interaction and maintaining cohesion.

Eventual recognition of Applied Mathematics will stem from its vitality and influence in the scientific and technological world. In the meantime, there is much work to be done to achieve this objective, all of which requires money in sufficient amounts and on a continuing basis. For example, M.I.T. has made available for Applied Mathematics alone several prestigious graduate fellowships. Unfortunately, institute funds are essential because there is no adequate mechanism as yet by which national agencies can channel funds in this direction.

The new M.I.T. journal "Studies in Applied Mathematics" is a vehicle for the publication of research in the entire field which, as the synthesis of the complete research spectrum, will focus international attention on the intrinsic merits of Applied Mathematics.

However, activity should never obscure a fundamental truth - the only really legitimate basis for growth at a distinguished University is excellence - in students, faculty, research and curriculum.

Applied Mathematics requires departmental status, de facto or de jure to develop as I have described it. Only a department has the vital powers of faculty appointment, promotions, curriculum organisation, and fiscal control, which permit the flexibility and freedom of action necessary for success. The exact form of organisation usually depends on circumstances peculiar to each institution. Since the content, philosophy and criteria of Applied Mathematics are exclusive, it is very desirable, and usually crucial, to have de jure status, but this is not easily attained in a system overwhelmingly controlled by one faction of mathematics. Some compromise seems likely and since Pure and Applied Mathematics do have a common heritage and do share a joint teaching responsibility, it is possible, with tolerance and good will, to construct a federated department. It is not inconceivable that this may even lead, at some future time, to a redefinition of mathematics in which the appellations pure and applied will no longer be necessary.

The Mathematics Department at M.I.T. is organised on this basis, and in essence, two departments co-exist as one. Briefly, the salient features of the arrangement are the following: Two separate and equal policy committees for Pure and Applied Mathematics are in permanent operation, each possessing the essential departmental prerogatives. The chairmen of these committees and the head of the department (a rotating position) together act as a coordinating body. Continuity is thereby secured and the stability of the operation is further enhanced by a liaison Departmental Committee which reviews all tenured appointments. The rare issues which cannot be resolved within this framework are settled by the higher administration and this should be viewed as normal procedure when, for example, different departments compete for a limited number of appointments. The routine chores of university life are handled by a single administrative staff which allows a definite economy of effort.

This federation has proved remarkably adaptable thus far, and has won general faculty acceptance and approval. The fact that Applied Mathematics can prosper in such a union, which also appears to benefit all groups concerned, invites emulation by universities facing similar problems.

There are other interpretations of Applied Mathematics, less viable, I believe, than ours. However, debate is somewhat irrelevant because, in time, the most successful program will have its definition accepted.

Last Updated November 2020