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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 method).
• 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 fluid!)
• 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 other tools.
• 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 sci.physics.particle.

You might want to continue the tour with a trip through the remaining subject areas of mathematics.