Mathematical physics is a research area focused on formulating and analyzing physical theories using rigorous mathematical structures and methods, and conversely, on developing new mathematics motivated by physical problems. It encompasses the study of quantum mechanics, quantum field theory, statistical mechanics, general relativity, and dynamical systems through functional analysis, operator algebras, differential geometry, probability theory, and partial differential equations. Core topics include spectral theory of Hamiltonians, rigorous renormalization, exactly solvable models, phase transitions, integrable systems, and geometric and topological methods in gauge theories and gravitation, with emphasis on precise definitions, theorems, and proofs underpinning fundamental physical concepts.
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