## What is Applied Mathematics?## How is it related to |

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

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Mathematical and computational techniques from ordinary and partial differential equations are often (justifiably) used as examples of applied mathematics (e.g.

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 "*
(
U. Calgary);

3.

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?

4.

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.

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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 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.

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.