In the mathematical fields of differential geometry and geometric analysis, the Ricci flow, sometimes also referred to as Hamilton's Ricci flow, is a certain partial differential equation for a Riemannian metric. It is often said to be analogous to the diffusion of heat and the heat equation, due to formal similarities in the mathematical structure of the equation. However, it is nonlinear and exhibits many phenomena not present in the study of the heat equation.
Jean Louis, baron Bourgain was a Belgian mathematician. He was awarded the Fields Medal in 1994 in recognition of his work on several core topics of mathematical analysis such as the geometry of Banach spaces, harmonic analysis, ergodic theory and nonlinear partial differential equations from mathematical physics.
The Bôcher Memorial Prize was founded by the American Mathematical Society in 1923 in memory of Maxime Bôcher with an initial endowment of $1,450. It is awarded every three years for a notable research work in analysis that has appeared during the past six years. The work must be published in a recognized, peer-reviewed venue. The current award is $5,000.
Terence Chi-Shen Tao is an Australian mathematician who is a professor of mathematics at the University of California, Los Angeles (UCLA), where he holds the James and Carol Collins chair. His research includes topics in harmonic analysis, partial differential equations, algebraic combinatorics, arithmetic combinatorics, geometric combinatorics, probability theory, compressed sensing and analytic number theory.
In theoretical physics, the (one-dimensional) nonlinear Schrödinger equation (NLSE) is a nonlinear variation of the Schrödinger equation. It is a classical field equation whose principal applications are to the propagation of light in nonlinear optical fibers and planar waveguides and to Bose–Einstein condensates confined to highly anisotropic, cigar-shaped traps, in the mean-field regime. Additionally, the equation appears in the studies of small-amplitude gravity waves on the surface of deep inviscid (zero-viscosity) water; the Langmuir waves in hot plasmas; the propagation of plane-diffracted wave beams in the focusing regions of the ionosphere; the propagation of Davydov's alpha-helix solitons, which are responsible for energy transport along molecular chains; and many others. More generally, the NLSE appears as one of universal equations that describe the evolution of slowly varying packets of quasi-monochromatic waves in weakly nonlinear media that have dispersion. Unlike the linear Schrödinger equation, the NLSE never describes the time evolution of a quantum state. The 1D NLSE is an example of an integrable model.
The Schrödinger–Newton equation, sometimes referred to as the Newton–Schrödinger or Schrödinger–Poisson equation, is a nonlinear modification of the Schrödinger equation with a Newtonian gravitational potential, where the gravitational potential emerges from the treatment of the wave function as a mass density, including a term that represents interaction of a particle with its own gravitational field. The inclusion of a self-interaction term represents a fundamental alteration of quantum mechanics. It can be written either as a single integro-differential equation or as a coupled system of a Schrödinger and a Poisson equation. In the latter case it is also referred to in the plural form.
In mathematics, in the theory of integrable systems, a Lax pair is a pair of time-dependent matrices or operators that satisfy a corresponding differential equation, called the Lax equation. Lax pairs were introduced by Peter Lax to discuss solitons in continuous media. The inverse scattering transform makes use of the Lax equations to solve such systems.
In mathematical physics and the theory of partial differential equations, the solitary wave solution of the form is said to be orbitally stable if any solution with the initial data sufficiently close to forever remains in a given small neighborhood of the trajectory of
Derrick's theorem is an argument by physicist G. H. Derrick which shows that stationary localized solutions to a nonlinear wave equation or nonlinear Klein–Gordon equation in spatial dimensions three and higher are unstable.
In mathematical analysis, Strichartz estimates are a family of inequalities for linear dispersive partial differential equations. These inequalities establish size and decay of solutions in mixed norm Lebesgue spaces. They were first noted by Robert Strichartz and arose out of connections to the Fourier restriction problem.
Gigliola Staffilani is an Italian-American mathematician who works as the Abby Rockefeller Mauze Professor of Mathematics at the Massachusetts Institute of Technology. Her research concerns harmonic analysis and partial differential equations, including the Korteweg–de Vries equation and Schrödinger equation.
Andrea Rica Nahmod is a mathematician at University of Massachusetts, Amherst. She is known for her work in nonlinear partial differential equations and other areas of nonlinear analysis.
In mathematics, cocompact embeddings are embeddings of normed vector spaces possessing a certain property similar to but weaker than compactness. Cocompactness has been in use in mathematical analysis since the 1980s, without being referred to by any name ,(Lemma 2.5),(Theorem 1), or by ad-hoc monikers such as vanishing lemma or inverse embedding.
Francis Michael Christ is an American mathematician and professor at University of California, Berkeley, specializing in harmonic analysis, partial differential equations, and several complex variables. He is known for the Christ–Kiselev maximal inequality.
Monica Vișan is a Romanian mathematician at the University of California, Los Angeles who specializes in partial differential equations and is well known for her work on the nonlinear Schrödinger equation.
Walter Alexander Strauss is an American applied mathematician, specializing in partial differential equations and nonlinear waves. His research interests include partial differential equations, mathematical physics, stability theory, solitary waves, kinetic theory of plasmas, scattering theory, water waves, and dispersive waves.
Gustavo A. Ponce is a Venezuelan mathematician.
Yvan Martel is a French mathematician.
Pokhozhaev's identity is an integral relation satisfied by stationary localized solutions to a nonlinear Schrödinger equation or nonlinear Klein–Gordon equation. It was obtained by S.I. Pokhozhaev and is similar to the virial theorem. This relation is also known as G.H. Derrick's theorem. Similar identities can be derived for other equations of mathematical physics.
Rowan Killip is an American–New Zealand mathematician at the University of California, Los Angeles whose work focuses on mathematical physics, particularly partial differential equations. He won a Sloan Research Fellowship in 2004 and a Simons Fellowship in Mathematics in 2015. In 2023, he won, along with Monica Vișan, the Frontiers of Science Award at the International Congress for Basic Science in Beijing, China for proving the global well-posedness of the Korteweg–De Vries equation in the Sobolev space H-1.
