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Having ostensibly reduced the Keplerian celestial laws of motion as well as Galilean terrestrial laws of motion to a unifying force, Newton achieved great mathematical rigor, but with theoretical laxity. In the 18th century, the Swiss Daniel Bernoulli — made contributions to fluid dynamics , and vibrating strings. The Swiss Leonhard Euler — did special work in variational calculus , dynamics, fluid dynamics, and other areas.

Also notable was the Italian-born Frenchman, Joseph-Louis Lagrange — for work in analytical mechanics : he formulated Lagrangian mechanics and variational methods. A major contribution to the formulation of Analytical Dynamics called Hamiltonian dynamics was also made by the Irish physicist, astronomer and mathematician, William Rowan Hamilton Hamiltonian dynamics had played an important role in the formulation of modern theories in physics, including field theory and quantum mechanics. The French mathematical physicist Joseph Fourier — introduced the notion of Fourier series to solve the heat equation , giving rise to a new approach to solving partial differential equations by means of integral transforms.

Into the early 19th century, the French Pierre-Simon Laplace — made paramount contributions to mathematical astronomy , potential theory , and probability theory.

In Germany, Carl Friedrich Gauss — made key contributions to the theoretical foundations of electricity , magnetism , mechanics , and fluid dynamics. In England, George Green published An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism in , which in addition to its significant contributions to mathematics made early progress towards laying down the mathematical foundations of electricity and magnetism. A couple of decades ahead of Newton's publication of a particle theory of light, the Dutch Christiaan Huygens — developed the wave theory of light, published in By , Thomas Young 's double-slit experiment revealed an interference pattern, as though light were a wave, and thus Huygens's wave theory of light, as well as Huygens's inference that light waves were vibrations of the luminiferous aether , was accepted.

Jean-Augustin Fresnel modeled hypothetical behavior of the aether.

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Michael Faraday introduced the theoretical concept of a field—not action at a distance. Midth century, the Scottish James Clerk Maxwell — reduced electricity and magnetism to Maxwell's electromagnetic field theory, whittled down by others to the four Maxwell's equations.

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Initially, optics was found consequent of [ clarification needed ] Maxwell's field. Later, radiation and then today's known electromagnetic spectrum were found also consequent of [ clarification needed ] this electromagnetic field.

## Differential Geometry and Mathematical Physics

The English physicist Lord Rayleigh [—] worked on sound. The Irishmen William Rowan Hamilton — , George Gabriel Stokes — and Lord Kelvin — produced several major works: Stokes was a leader in optics and fluid dynamics; Kelvin made substantial discoveries in thermodynamics ; Hamilton did notable work on analytical mechanics , discovering a new and powerful approach nowadays known as Hamiltonian mechanics.

Very relevant contributions to this approach are due to his German colleague Carl Gustav Jacobi — in particular referring to canonical transformations. The German Hermann von Helmholtz — made substantial contributions in the fields of electromagnetism , waves, fluids , and sound. In the United States, the pioneering work of Josiah Willard Gibbs — became the basis for statistical mechanics.

Fundamental theoretical results in this area were achieved by the German Ludwig Boltzmann Together, these individuals laid the foundations of electromagnetic theory, fluid dynamics, and statistical mechanics. By the s, there was a prominent paradox that an observer within Maxwell's electromagnetic field measured it at approximately constant speed, regardless of the observer's speed relative to other objects within the electromagnetic field.

Thus, although the observer's speed was continually lost [ clarification needed ] relative to the electromagnetic field, it was preserved relative to other objects in the electromagnetic field. And yet no violation of Galilean invariance within physical interactions among objects was detected. As Maxwell's electromagnetic field was modeled as oscillations of the aether , physicists inferred that motion within the aether resulted in aether drift , shifting the electromagnetic field, explaining the observer's missing speed relative to it.

The Galilean transformation had been the mathematical process used to translate the positions in one reference frame to predictions of positions in another reference frame, all plotted on Cartesian coordinates , but this process was replaced by Lorentz transformation , modeled by the Dutch Hendrik Lorentz [—]. In , experimentalists Michelson and Morley failed to detect aether drift, however. It was hypothesized that motion into the aether prompted aether's shortening, too, as modeled in the Lorentz contraction.

It was hypothesized that the aether thus kept Maxwell's electromagnetic field aligned with the principle of Galilean invariance across all inertial frames of reference , while Newton's theory of motion was spared. In the 19th century, Gauss 's contributions to non-Euclidean geometry , or geometry on curved surfaces, laid the groundwork for the subsequent development of Riemannian geometry by Bernhard Riemann — Austrian theoretical physicist and philosopher Ernst Mach criticized Newton's postulated absolute space.

In , Pierre Duhem published a devastating criticism of the foundation of Newton's theory of motion. Refuting the framework of Newton's theory— absolute space and absolute time —special relativity refers to relative space and relative time , whereby length contracts and time dilates along the travel pathway of an object. In , Einstein's former professor Hermann Minkowski modeled 3D space together with the 1D axis of time by treating the temporal axis like a fourth spatial dimension—altogether 4D spacetime—and declared the imminent demise of the separation of space and time.

Einstein initially called this "superfluous learnedness", but later used Minkowski spacetime with great elegance in his general theory of relativity , [11] extending invariance to all reference frames—whether perceived as inertial or as accelerated—and credited this to Minkowski, by then deceased. General relativity replaces Cartesian coordinates with Gaussian coordinates , and replaces Newton's claimed empty yet Euclidean space traversed instantly by Newton's vector of hypothetical gravitational force—an instant action at a distance —with a gravitational field.

The gravitational field is Minkowski spacetime itself, the 4D topology of Einstein aether modeled on a Lorentzian manifold that "curves" geometrically, according to the Riemann curvature tensor , in the vicinity of either mass or energy. Under special relativity—a special case of general relativity—even massless energy exerts gravitational effect by its mass equivalence locally "curving" the geometry of the four, unified dimensions of space and time.

Another revolutionary development of the 20th century was quantum theory , which emerged from the seminal contributions of Max Planck — on black-body radiation and Einstein's work on the photoelectric effect. This revolutionary theoretical framework is based on a probabilistic interpretation of states, and evolution and measurements in terms of self-adjoint operators on an infinite dimensional vector space.

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That is called Hilbert space , introduced in its elementary form by David Hilbert — and Frigyes Riesz , and rigorously defined within the axiomatic modern version by John von Neumann in his celebrated book Mathematical Foundations of Quantum Mechanics , where he built up a relevant part of modern functional analysis on Hilbert spaces, the spectral theory in particular.

Paul Dirac used algebraic constructions to produce a relativistic model for the electron , predicting its magnetic moment and the existence of its antiparticle, the positron. From Wikipedia, the free encyclopedia. Application of mathematical methods to problems in physics. Main articles: Lagrangian mechanics and Hamiltonian mechanics. Main article: Partial differential equations. Main article: Quantum mechanics. Main articles: Theory of relativity and Quantum field theory. Main article: Statistical mechanics. Archived from the original on Retrieved Depending on the ratio of these two components, the theorist may be nearer either to the experimentalist or to the mathematician.

In the latter case, he is usually considered as a specialist in mathematical physics.

## PhD Candidate in geometry and mathematical physics

Frenkel, as related in A. Filippov, The Versatile Soliton , pg Birkhauser, Good theory is like a good suit. Thus the theorist is like a tailor. Frenkel, as related in Filippov , pg J De mechanisering van het wereldbeeld. Meulenhoff, Amsterdam. Areas of mathematics. Category theory Information theory Mathematical logic Philosophy of mathematics Set theory. Abstract Elementary Linear Multilinear.

### Categories within Mathematics

Calculus Real analysis Complex analysis Differential equations Functional analysis. Combinatorics Graph theory Order theory Game theory. Arithmetic Algebraic number theory Analytic number theory Diophantine geometry. Algebraic Differential Geometric. Control theory Mathematical biology Mathematical chemistry Mathematical economics Mathematical finance Mathematical physics Mathematical psychology Mathematical statistics Probability Statistics.