Mathematical physics
Mathematical physics develops and analyzes mathematical structures, methods, and rigorous foundations for formulating and solving physical problems.
Differential equations: Equations relating functions to their derivatives, used to describe changing quantities and spatial fields. Many physical laws become differential equations for motion, waves, or fields.
Quantum mechanics: The physical theory of matter and radiation at atomic and subatomic scales, built around quantum states and probabilistic outcomes. Its formalism draws on Hilbert spaces, operators, and spectral theory.
Henri Poincaré: French mathematician and physicist whose work shaped topology, celestial mechanics, and mathematical physics. His qualitative study of dynamical systems transformed the mathematics of classical mechanics.
Theoretical physics: The branch of physics that develops conceptual and mathematical models to explain physical phenomena. It prioritizes physical explanation, while mathematical physics often emphasizes rigorous structure and proof.
Navier–Stokes existence and smoothness: The open problem of proving whether three-dimensional incompressible Navier–Stokes solutions always exist smoothly for all time. It asks whether a foundational fluid model can develop singularities from smooth initial data.
Linear algebra: The study of vector spaces and linear maps, including matrices, eigenvalues, and eigenvectors. Quantum states, symmetries, and many physical transformations use vector spaces and linear operators.
General relativity: Einstein’s theory of gravity, in which spacetime curvature is related to matter and energy. Differential geometry gives its equations a precise language for curved spacetime.
David Hilbert: German mathematician whose work influenced analysis, geometry, mathematical logic, and theoretical physics. His axiomatic approach and work on integral equations affected the foundations of physical theory.
Applied mathematics: The use of mathematical methods to model and solve problems arising in science, engineering, and other fields. Its scope is broader than physics, though many of its methods overlap with mathematical physics.
Yang–Mills existence and mass gap: The problem of constructing quantum Yang–Mills theory rigorously in four dimensions and proving it has a positive mass gap. It tests whether a central particle-physics theory has a mathematically sound continuum foundation.