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🌐 Mathematical Physics (فیزیک ریاضی)

Mathematical Physics includes the development of new mathematical methods for application to problems in physics, as well as the mathematically rigorous proofs of claims that physicists have only supported with nonrigorous arguments. Mathematics and Physics are traditionally tightly linked subjects, and many historical figures, such as Newton and Gauss, were both physicists and mathematicians. Traditionally Mathematical Physics has been quite closely associated to ideas in calculus, particularly those of differential equations. In recent years however, in part due to the rise of superstring theory, many more branches of mathematics have become major contributors to physics. Mathematical Physics works with topics and concepts such as vector spaces, matrix algebra, differential equations, integral equations, integral transforms, infinite series, and complex variables. It is a very broad topic that, in basic terms, applies mathematical ideas to physics and engineering problems. Mathematical Physics encompasses branches such as statistical mechanics, atomic and molecular physics, electromagnetism, quantum field theory, classical mechanics, and theoretical physics. The formulation of fundamental physical laws has always been tightly interwoven with the deepest mathematics available. This is even more evident today, with geometry providing the foundations of general relativity, gauge theories, string theory, and models of statistical physics. In turn, developments in these fields of physics have been instrumental in driving progress in various areas of mathematics, such as Riemann surface theory, quantum geometry, knot theory, mirror symmetry, representation theory, nonlinear partial differential equations, and differential geometry. Studying Mathematical Physics reveals the hidden intricacies of relativity and quantum theory. The cross-fertilization between mathematics and physics has perhaps never been richer.

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There is a long history of fruitful exchanges between Probability Theory and Statistical Mechanics and Mathematical Physics. Models of physical systems present challenging and relevant problems to probabilists; ideas and techniques flow back and forth. A general problem is to understand how microscopic noise and local interactions generate random structures on the macroscopic scale, and to describe qualitatively and quantitatively these structures. While the formulation of basic models of physical phenomena, such as percolation, the Ising model, random growth models and interacting particle systems, is completely elementary, their analysis involves a wide array of techniques and concepts: combinatorics and graphs, complex analysis and algebraic curves, operators and representation theory, symmetric functions and integrable systems, Brownian motion, Gaussian fields, stochastic partial differential equations. The fact that current progress toward the understanding of these models involves such diverse ideas is a testimony to the vivacity and depth of the field. Mathematical research in physical modeling focuses on the formulation and analysis of mathematical representations of problems motivated by other branches of science and engineering. In addition to generating novel problems with new computational and analytical challenges, constructing accurate models for complex systems may uncover the need for fundamental extensions to the governing equations.

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