Applications to the sciences
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Historically, it has been the needs of the physical sciences which have
driven the development of many parts of mathematics, particularly analysis.
The applications are sometimes difficult to classify mathematically,
since tools from several areas of mathematics may be applied. We focus
on these applications not by discussing the nature of their
discipline but rather their interaction with mathematics.
Most of the areas in this group (the blue ones in the picture here) are
collectively known as "mathematical physics". Somewhat more recently,
increasingly sophisticated mathematical tools are used in the
engineering, biology, and the social sciences (the violet areas in the picture).
70: Mechanics of particles and systems studies
dynamics of sets of particles or solid bodies, including rotating and
vibrating bodies. Uses variational principles (energy-minimization) as
well as differential equations.
74: Mechanics of deformable solids considers questions
of elasticity and plasticity, wave propagation, engineering, and topics
in specific solids such as soils and crystals.
76: Fluid mechanics studies air, water, and other
fluids in motion: compression, turbulence, diffusion, wave propagation,
and so on. Mathematically this includes study of solutions of differential
equations, including large-scale numerical methods (e.g the finite-element
78: Optics, electromagnetic theory is the study of
the propagation and evolution of electromagnetic waves, including topics
of interference and diffraction. Besides the usual branches of analysis,
this area includes geometric topics such as the paths of light rays.
80: Classical thermodynamics, heat transfer is the study
of the flow of heat through matter, including phase change and combustion.
Historically, the source of Fourier series.
81: Quantum Theory studies the solutions of the
Schrödinger (differential) equation. Also includes a good deal of
Lie group theory and quantum group theory, theory of distributions and
topics from Functional analysis, Yang-Mills problems, Feynman diagrams,
and so on.
82: Statistical mechanics, structure of matter is the
study of large-scale systems of particles, including stochastic systems
and moving or evolving systems. Specific types of matter studied include
fluids, crystals, metals, and other solids.
83: Relativity and gravitational theory is differential
geometry, analysis, and group theory applied to physics on a grand scale
or in extreme situations (e.g. black holes and cosmology).
85: Astronomy and astrophysics: as celestial mechanics
is, mathematically, part of Mechanics of Particles (!), the principal
applications in this area appear to be concerning the structure, evolution,
and interaction of stars and galaxies.
86: Geophysics applications typically involve
material in Mechanics and Fluid mechanics, as above, but for large-scale
problems (this subject deals with a very big solid and a large pool of
93: Systems theory; control study the evolution
over time of complex systems such as those in engineering. In particular, one
may try to identify the system -- to determine the equations or parameters
which govern its development -- or to control the system -- to select the
parameters (e.g. via feedback loops) to achieve a desired state. Of particular
interest are issues in stability (steady-state configurations) and the
effects of random changes and noise (stochastic systems). While popularly
the domain of "cybernetics" or "robotics", perhaps, this is in practice a
field of application of differential (or difference) equations, functional
analysis, numerical analysis, and global analysis (or differential geometry).
92: Biology and other natural sciences
whose connections merit explicit connection in the MSC scheme include
Chemistry, Biology, Genetics, and Medicine, In chemistry and
biochemistry, it is clear that graph theory, differential geometry,
and differential equations play a role. Medical technology uses
techniques of information transfer and visualization. Biology
(including taxonomy and archaeobiology) use statistical inference and
91: Game theory, economics, social and behavioral
sciences including Psychology, Sociology, and other social
sciences as a group. The more behavioural sciences (including
Linguistics!) use a medley of statistical techniques, including
experimental design and other rather combinatorial topics. Economics
and finance also make use of statistical tools, especially time-series
analysis; some topics, such as voting theory, are more combinatorial.
This category also includes game theory, which is actually not about
games at all but rather about optimization; which combination of
strategies leads to an optimal outcome.
[Area 91 is not shown on the MathMap; it was created in 2000 from
parts of sections 90 and 92.]
Observe that the branches of mathematics most closely allied with
the fields of mathematical physics are the parts of
analysis, particularly those parts related to
differential equations. The other sciences
draw on these as well as probability and
statistics and, increasingly, numerical methods.
In addition to the resources mentioned on the index pages for these
individual disciplines, one might summarize here the many resources which
apply to mathematical physics generally. For example, there is a newsgroup
sci.physics.research (moderated) and unmoderated groups sci.physics and
You might want to continue the tour with a trip through the remaining subject areas of mathematics.
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Last modified 2000/01/25 by Dave Rusin. Mail: