Richard Samuel Ward FRS (born 6 September 1951) is a British mathematical physicist. He is a Professor of Theoretical Physics at the University of Durham.^{ [4] }
Ward earned his Ph.D. from the University of Oxford in 1977, under the supervision of Roger Penrose. He is most famous for his extension of Penrose's twistor theory to nonlinear cases, which he with Michael Atiyah used to describe instantons by vector bundles on the three-dimensional complex projective space. He has related interests in the theory of monopoles, topological solitons and skyrmions.
Ward was awarded the Whitehead Prize in 1989 for his work in mathematical physics.^{ [5] } He was elected as a fellow of the Royal Society of London in 2005.^{ [6] } His certificate of election reads:
Richard Ward is distinguished for pioneering and elegant research in mathematical physics. He adapted the twistor transform to the self-dual Yang-Mills (SDYM) equation, and with Atiyah constructed general multi-instanton solutions. His discovery of the toroidal BPS two-monopole was a breakthrough in soliton theory. He showed that virtually all known integrable equations arise from SDYM by dimensional and algebraic reductions, allowing a unified solution method. Ward's twistor transform of SDYM, applied to string theory, is leading to striking progress in quantum Yang-Mills theory.^{ [7] }
Sir Michael Francis Atiyah was a British-Lebanese mathematician specialising in geometry.
Edward Witten is an American mathematical and theoretical physicist. He is currently the Charles Simonyi Professor in the School of Natural Sciences at the Institute for Advanced Study. Witten is a researcher in string theory, quantum gravity, supersymmetric quantum field theories, and other areas of mathematical physics. In addition to his contributions to physics, Witten's work has significantly impacted pure mathematics. In 1990, he became the first physicist to be awarded a Fields Medal by the International Mathematical Union, awarded for his 1981 proof of the positive energy theorem in general relativity. He is considered to be the practical founder of M-theory.
In theoretical physics, twistor theory was proposed by Roger Penrose in 1967 as a possible path to quantum gravity and has evolved into a branch of theoretical and mathematical physics. Penrose proposed that twistor space should be the basic arena for physics from which space-time itself should emerge. It leads to a powerful set of mathematical tools that have applications to differential and integral geometry, nonlinear differential equations and representation theory and in physics to general relativity and quantum field theory, in particular to scattering amplitudes.
Nigel James Hitchin FRS is a British mathematician working in the fields of differential geometry, gauge theory, algebraic geometry, and mathematical physics. He is a Professor Emeritus of Mathematics at the University of Oxford.
Vladimir Gershonovich Drinfeld, surname also romanized as Drinfel'd, is a renowned mathematician from the former USSR, who emigrated to the United States and is currently working at the University of Chicago.
Alexander Markovich Polyakov is a Russian theoretical physicist, formerly at the Landau Institute in Moscow and, since 1990, at Princeton University, where he is the Joseph Henry Professor of Physics.
In mathematics, a monopole is a connection over a principal bundle G with a section of the associated adjoint bundle.
Montonen–Olive duality or electric–magnetic duality is the oldest known example of strong–weak duality or S-duality according to current terminology. It generalizes the electro-magnetic symmetry of Maxwell's equations by stating that magnetic monopoles, which are usually viewed as emergent quasiparticles that are "composite", can in fact be viewed as "elementary" quantized particles with electrons playing the reverse role of "composite" topological solitons; the viewpoints are equivalent and the situation dependent on the duality. It was later proven to hold true when dealing with a N = 4 supersymmetric Yang–Mills theory. It is named after Finnish physicist Claus Montonen and British physicist David Olive after they proposed the idea in their academic paper Magnetic monopoles as gauge particles? where they state:
There should be two "dual equivalent" field formulations of the same theory in which electric (Noether) and magnetic (topological) quantum numbers exchange roles.
In theoretical physics, Seiberg–Witten theory is a theory that determines an exact low-energy effective action of a supersymmetric gauge theory—namely the metric of the moduli space of vacua.
In string theory, K-theory classification refers to a conjectured application of K-theory to superstrings, to classify the allowed Ramond–Ramond field strengths as well as the charges of stable D-branes.
Wolf Paul Barth was a German mathematician who discovered Barth surfaces and whose work on vector bundles has been important for the ADHM construction. Until 2011 Barth was working in the Department of Mathematics at the University of Erlangen-Nuremberg in Germany.
In mathematical physics and gauge theory, the ADHM construction or monad construction is the construction of all instantons using methods of linear algebra by Michael Atiyah, Vladimir Drinfeld, Nigel Hitchin, Yuri I. Manin in their paper "Construction of Instantons."
In differential geometry and gauge theory, the Nahm equations are a system of ordinary differential equations introduced by Werner Nahm in the context of the Nahm transform – an alternative to Ward's twistor construction of monopoles. The Nahm equations are formally analogous to the algebraic equations in the ADHM construction of instantons, where finite order matrices are replaced by differential operators.
In theoretical physics, the Penrose transform, introduced by Roger Penrose, is a complex analogue of the Radon transform that relates massless fields on spacetime to cohomology of sheaves on complex projective space. The projective space in question is the twistor space, a geometrical space naturally associated to the original spacetime, and the twistor transform is also geometrically natural in the sense of integral geometry. The Penrose transform is a major component of classical twistor theory.
David Ian Olive was a British theoretical physicist. Olive made fundamental contributions to string theory and duality theory, he is particularly known for his work on the GSO projection and Montonen–Olive duality.
The Journal of Nonlinear Mathematical Physics (JNMP) is a mathematical journal published by Atlantis Press. It covers nonlinear problems in physics and mathematics, include applications, with topics such as quantum algebras and integrability; non-commutative geometry; spectral theory; and instanton, monopoles and gauge theory.
Nikita Alexandrovich Nekrasov is a mathematical and theoretical physicist at the Simons Center for Geometry and Physics and C.N.Yang Institute for Theoretical Physics at Stony Brook University in New York, and a Professor of the Russian Academy of Sciences.
Periodic instantons are finite energy solutions of Euclidean-time field equations which communicate between two turning points in the barrier of a potential and are therefore also known as bounces. Vacuum instantons, normally simply called instantons, are the corresponding zero energy configurations in the limit of infinite Euclidean time. For completeness we add that ``sphalerons´´ are the field configurations at the very top of a potential barrier. Vacuum instantons carry a winding number, the other configurations do not. Periodic instantons werde discovered with the explicit solution of Euclidean-time field equations for double-well potentials and the cosine potential with non-vanishing energy and are explicitly expressible in terms of Jacobian elliptic functions. Periodic instantons describe the oscillations between two endpoints of a potential barrier between two potential wells. The frequency of these oscillations or the tunneling between the two wells is related to the bifurcation or level splitting of the energies of states or wave functions related to the wells on either side of the barrier, i.e. . One can also interpret this energy change as the energy contribution to the well energy on either side originating from the integral describing the overlap of the wave functions on either side in the domain of the barrier.
Integrable algorithms are numerical algorithms that rely on basic ideas from the mathematical theory of integrable systems.
Olaf Lechtenfeld is a German mathematical physicist, academic and researcher. He is a full professor at the Institute of Theoretical Physics at Leibniz University, where he founded the Riemann Center for Geometry and Physics.