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What is Applied Mathematics?

How is it related to PURE MATH?

Undergraduate and graduate students sometimes wonder what is applied mathematics. Students often form an impression of applied math based on conversations with other students or conversations with and courses taught by particular faculty members. Unfortunately this impression can be wrong in the sense that a subset of applied mathematics may seem to represent all of applied math. It is the intention of this webpage to discuss different mathematicians' views on applied mathematics, offer some examples, discuss some history, etc.

Applied (or perhaps Applicable) Mathematics consists of mathematical techniques and results, including those from "pure" math areas such as (abstract) algebra or algebraic topology, which are used to assist in the investigation of problems or questions originating outside of mathematics.
My view is somewhat similar to some of those expressed in A Mathematician's Apology by G. H. Hardy (study guide, the book (pdf file)). In 1940, Hardy wrote:
" One rather curious conclusion emerges, that pure mathematics is on the whole distinctly more useful than applied. A pure mathematician seems to have the advantage on the practical as well as on the aesthetic side. For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics. " (page 41)
Mathematical and computational techniques from ordinary and partial differential equations are often (justifiably) used as examples of applied mathematics (e.g. APPLIED MATHEMATICS: An Introduction by Harry Pollard, Addison-Wesley 1972). Mathematicians occasionally disagree on the answers to the questions in the title and some may view applied math (outside of probability & statistics) as being largely limited to ordinary and partial differential equations, continuum mechanics, the calculus of variations and numerical computations. In my opinion, this view is wrong. For example, it ignores the contributions of algebra to physics and to the recently completed Human Genome Project.

Various definitions of applied math can be found. A few of these are:

1. Applied Mathematics concerns the application of mathematics in a wide range of disciplines in various areas such as science, technology, business and commerce. Applied mathematicians are engaged in the creation, study and application of advanced mathematical methods relevant to specific problems. Once this referred mainly to the application of mathematics to such disciplines as mechanics and fluid dynamics but currently, applied mathematics has assumed a much broader meaning and embraces such diverse fields as communication theory, theory of optimisation, theory of games and numerical analysis. Indeed, today there is a remarkable range and variety of applications of mathematics in industry and government, involving important real-world problems such as materials processing, design, medical diagnosis, development of financial products, network management and weather prediction. (U.So.Africa);

2. Human beings have innate natural tendencies to count, to quantify, and to apply logic in their attempts at understanding the world. Mathematics is the human endeavor which has come to provide definition and scope to these activities, in terms which employ the utmost precision of thought. A few ancient civilizations developed mathematical systems to some extent, for example some of the Babylonians, but the first great step in the establishment of mathematics was made by the Greeks between 600 and 300 BCE. The contribution of Euclid was to state theorems in geometry and to construct their proofs from a small number of basic statements, called axioms, taken as the starting point of the subject. This showed that new knowledge could be obtained by pure reasoning about the basic axioms. Though geometrical ideas and structures have since been greatly broadened and deepened Euclidean geometry remains an important and useful part of modern mathematics and science.
Attention to geometrical ideas predates Euclid and arose from a desire to quantify physical space. Arithmetic, the subject of operations with various kinds of numbers, arose from the need to count and quantify any objects. The roots of probably all major divisions of mathematics go back to concern about practical matters or knowledge of the natural world. In a few situations serious investigation of the natural world has lead directly to the creation of whole areas of mathematics that not only produced methods for formulating and solving important physical problems but lead, by further development, to new advanced mathematical subjects. The most striking example of this is the invention of calculus in the 17th century to solve the problem of motion, particularly the motion of the planets under their mutual gravitational attractions. The development and extension of calculus, along with the construction of a mathematical foundation for it, lead to the subject called analysis, a major part of mathematics.
But mathematics itself is not physical theory. The growth of mathematical roots, as mathematics, involves abstraction away from concern with particular objects towards an emphasis on relations among abstract objects, axiomatization, and establishment of the basic characteristics and facts of a subject by rigorous proof. A piece of mathematics developed this way may be considered to stand alone as a coherent logical structure independent of any connections to the physical world that may be possible. Its inherent content often suggests to mathematicians ways for further axiomatization and logical development which lead to new interesting structures, often to ones for which no relation to the physical world seems possible. However, experience has shown that areas of mathematics developed from purely mathematical motivation do find, with surprising frequency, significant application in the real world, sometimes many years after their development. Thus there are two intrinsic connections of mathematics with the real world, the direct one, illustrated by the invention of calculus, and what might at first be called the serendipitous. But the latter is so prevalent that it forces the recognition that any area of mathematics may prove useful. In addition to geometry and analysis the other major subject areas of mathematics are algebra (the study of equations and associated abstract structures), discrete mathematics (the study of sets of discrete quantities), topology (the study of continuous transformations of objects), number theory (the study of the properties of numbers), and set theory and logic (the basis for how to make precise arguments starting from the foundations of mathematics). Each of these subjects contains areas of fruitful application to real world problems.
This is the setting in which the meaning of the term "applied mathematics" is to be understood. Modern applied mathematics is two things. It is the attempt to use mathematics to quantify and solve problems which arise in investigation of the physical world and human enterprise. It is also the study and further development of those areas of mathematics that have proven the most useful in solving real world problems or seem to offer promise for present problems. Thus a distinction between applied and pure mathematics is one of the interest and purpose of the practitioner, it is not a fixed dichotomy or division of the subject areas of mathematics. The two different interests, one in applying mathematics and the other in mathematics for its own sake, have always progressed together and probably could not progress without each other. Attempts to solve real world problems have created important and lasting aspects of mathematics, and the purest of mathematical constructs have found application.
Historically, applied mathematics mostly concerned theoretical physics and physical problems. The fundamental laws of physics are formulated as mathematical equations governing the behavior of physical quantities. These provide our deepest understanding of the physical world and our most accurate predictions of physical phenomena. While problems arising from physics and ranging from the study of classical fluids to quantum systems remain a significant part, applied mathematics has grown to include a wide variety of other areas such as bio-mathematics, cryptography, scientific computation, mathematical modeling, economics, financial mathematics, operations research, and engineering problems.
( U. Calgary);

3. Mathematics is used to help understand the real world and to help to change parts of it for Man's benefit. It is used in such diverse fields as engineering, physics, biology, economics, environmental studies, chemistry, political studies, medicine etc, etc, etc. The first step in this process is often the construction of a mathematical model, i.e. a description of the problem in mathematical terms. This model is then studied by using analytical or numerical methods to obtain exact or approximate solutions. Finally, the conclusions are interpreted in the language of the original problem in terms more familiar to a client or user. Often the model is changed to be more realistic or to include more features of the problem. Thus, the modeling process may involve false starts, modifications, and simplifications.
Mathematics enters primarily in the second stage: the solution of mathematically well-formulated problems and the development and analysis of the underlying theory. This stage may include analytical or numerical methods. The approach can range from specific algorithms and formal methods to abstract, general theories. It is often not clear which mathematical skills will be useful in the study of a new problem; thus, applied mathematicians need to be broadly trained so they will have a wide variety of mathematical tools available.
The mathematical scientist must not only be a competent mathematician but must be knowledgeable in the area to which mathematics is being applied. Thus, the applied mathematician must be concerned with the construction and interpretation of appropriate models; students must communicate with scientists in their language.
The art of formulating models requires that the modeler make choices about which factors to include and which to exclude. The goal is to produce a model that is realistic enough that it reflects the essential aspects of the phenomena being modeled, but simple enough that it can be treated mathematically.
Often the model is constructed to answer a specific question. Sometimes the modeler must either simplify the model so it can be analysed, or devise new mathematical methods that will permit an analysis of the model. Often a combination of analytical and numerical methods are used. The modeling process may involve a sequence of models of increasing complexity. Problems sometimes lead to new mathematical methods, and existing mathematical methods often lead to new insights into the problems. The successful applied mathematical scientist must be comfortable and confident in both mathematics and the field of application.
Applied mathematical and computational sciences is a name used to encompass the many analytical and numerical methods used to solve certain types of scientific problems. This name more accurately reflects the nature of modern "applied mathematics" since it also includes the area of scientific computing, which includes many components of computational processes, such as numerical analysis, algorithms for machines with vector and parallel architectures, visualization, simulation, and computer-aided design. In the Applied Mathematics division of the Department of Mathematical Sciences at Liverpool , we study the behaviour of materials, the environmental sciences, mathemaitcal bioloy, etc. Why not look at the entries under the research heading for more detail?
(U. Liverpool);

4. Applied mathematics is a branch of mathematics that concerns itself with the application of mathematical knowledge to other domains. Such applications include numerical analysis, mathematics of engineering, linear programming, optimization and operations research, continuous modelling, mathematical biology and bioinformatics, information theory, game theory, probability and statistics, financial mathematics, actuarial science, cryptography and hence combinatorics and even finite geometry to some extent, graph theory as applied to network analysis, and a great deal of what is called computer science.
The question of what is applied mathematics does not answer to logical classification so much as to the sociology of professionals who use mathematics. The mathematical methods are usually applied to the specific problem field by means of a mathematical model of the system.
Engineering mathematics describes physical processes, and so is often indistinguishable from theoretical physics. Important subdivisions include: fluid dynamics, acoustic theory, Maxwell's equations that govern electromagnetism, mechanics, numerical relativity etc.

5. Certainly, in the past, pure and applied mathematicians have argued at length over which endeavor is nobler. I think that there is not so much conflict along those lines today. It certainly is the case that mathematics is much broader in scope than the portion of it which is traditionally of highest interests to physicists. Physics is only one area of applications. There are lots of significant applications of mathematics to fields such as economics, business, biology, computer science, the social sciences, etc. There seems to be both an old and a new meaning to the term "applied mathematics". The Mathematical Experience by Philip J. Davis and Reuben Hersh. Check the table of contents for the appropriate section. This book also contains a bibliography with additional references. (Robert Allan Chaffer);

6. Mathematics is an enigmatic field of research. On the one hand, it is a field of research in its own right in which fundamental mathematical studies are undertaken without particular applications in mind. On the other hand, mathematics is the essential language used in the development of abstract models of reality that underpin R and D in countless other fields, including applications in industry. Applied mathematics covers both the development of new mathematical techniques that can be used for practical applications and the application of such techniques. Industrial mathematics is the extension of applied mathematics to industrial applications (International Congress on Industrial & Applied Mathematics (pdf file) );

7. To see how the interactions of biology and mathematics may proceed in the future, it is helpful to map the present landscapes of biology and applied mathematics.
The biological landscape may be mapped as a rectangular table with different rows for different questions and different columns for different biological domains. Biology asks six kinds of questions. How is it built? How does it work? What goes wrong? How is it fixed? How did it begin? What is it for? These are questions, respectively, about structures, mechanisms, pathologies, repairs, origins, and functions or purposes. The former teleological interpretation of purpose has been replaced by an evolutionary perspective. Biological domains, or levels of organization, include molecules, cells, tissues, organs, individuals, populations, communities, ecosystems or landscapes, and the biosphere. Many biological research problems can be classified as the combination of one or more questions directed to one or more domains.
In addition, biological research questions have important dimensions of time and space. Timescales of importance to biology range from the extremely fast processes of photosynthesis to the billions of years of living evolution on Earth. Relevant spatial scales range from the molecular to the cosmic (cosmic rays may have played a role in evolution on Earth). The questions and the domains of biology behave differently on different temporal and spatial scales. The opportunities and the challenges that biology offers mathematics arise because the units at any given level of biological organization are heterogeneous, and the outcomes of their interactions (sometimes called ?emergent phenomena? or ?ensemble properties?) on any selected temporal and spatial scale may be substantially affected by the heterogeneity and interactions of biological components at lower and higher levels of biological organization and at smaller and larger temporal and spatial scales (Anderson 1972, 1995).
The landscape of applied mathematics is better visualized as a tetrahedron (a pyramid with a triangular base) than as a matrix with temporal and spatial dimensions. (Mathematical imagery, such as a tetrahedron for applied mathematics and a matrix for biology, is useful even in trying to visualize the landscapes of biology and mathematics.) The four main points of the applied mathematical landscape are data structures, algorithms, theories and models (including all pure mathematics), and computers and software. Data structures are ways to organize data, such as the matrix used above to describe the biological landscape. Algorithms are procedures for manipulating symbols. Some algorithms are used to analyze data, others to analyze models. Theories and models, including the theories of pure mathematics, are used to analyze both data and ideas. Mathematics and mathematical theories provide a testing ground for ideas in which the strength of competing theories can be measured. Computers and software are an important, and frequently the most visible, vertex of the applied mathematical landscape. However, cheap, easy computing increases the importance of theoretical understanding of the results of computation. Theoretical understanding is required as a check on the great risk of error in software, and to bridge the enormous gap between computational results and insight or understanding.
The landscape of research in mathematics and biology contains all combinations of one or more biological questions, domains, time scales, and spatial scales with one or more data structures, algorithms, theories or models, and means of computation (typically software and hardware). The following example from cancer biology illustrates such a combination: the question, ?how does it work?? is approached in the domain of cells (specifically, human cancer cells) with algorithms for correlation and hierarchical clustering.
(PLoS Biology);

A Variety of Links

A student's opinion
What is Mathematics?
Steven Strogatz's comments
Wavelets (Postscript file)
How to Study Mathematics
CAREERS in Applied Mathematics & Computational Sciences
Searching For New Mathematics
The Young Mathematicians' Network
Financial mathematics
Eli Sternberg
Math Forum@Drexel
Why do Mathematics?

This page is not complete. If you would like to suggest interesting links or additional definitions of applied math, please write to me (lancaster AT math.wichita.edu). I do not expect this page to be finished until (approximately) August, 2005.

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