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Wu–Yang Dictionary
In topology and high energy physics, the Wu–Yang dictionary refers to the mathematical identification that allows back-and-forth translation between the concepts of gauge theory and those of differential geometry. The dictionary appeared in 1975 in an article by Tai Tsun Wu and C. N. Yang comparing electromagnetism and fiber bundle theory. This dictionary has been credited as bringing mathematics and theoretical physics closer together. A crucial example of the success of the dictionary is that it allowed the understanding of monopole quantization in terms of Hopf fibrations. History Equivalences between fiber bundle theory and gauge theory were hinted at the end of the 1960s. In 1967, mathematician Andrzej Trautman started a series of lectures aimed at physicists and mathematicians at King's College London regarding these connections. Theoretical physicists Tai Tsun Wu and C. N. Yang working in Stony Brook University, published a paper in 1975 on the mathematical fram ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such as Stretch factor, stretching, Torsion (mechanics), twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a Set (mathematics), set endowed with a structure, called a ''Topology (structure), topology'', which allows defining continuous deformation of subspaces, and, more generally, all kinds of List of continuity-related mathematical topics, continuity. Euclidean spaces, and, more generally, metric spaces are examples of topological spaces, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and Homotopy, homotopies. A property that is invariant under such deformations is a to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michael Atiyah
Sir Michael Francis Atiyah (; 22 April 1929 – 11 January 2019) was a British-Lebanese mathematician specialising in geometry. His contributions include the Atiyah–Singer index theorem and co-founding topological K-theory. He was awarded the Fields Medal in 1966 and the Abel Prize in 2004. Early life and education Atiyah was born on 22 April 1929 in Hampstead, London, England, the son of Jean (née Levens) and Edward Atiyah. His mother was Scottish and his father was a Lebanese Orthodox Christian. He had two brothers, Patrick (deceased) and Joe, and a sister, Selma (deceased). Atiyah went to primary school at the Diocesan school in Khartoum, Sudan (1934–1941), and to secondary school at Victoria College in Cairo and Alexandria (1941–1945); the school was also attended by European nobility displaced by the Second World War and some future leaders of Arab nations. He returned to England and Manchester Grammar School for his HSC studies (1945–1947) and did his nati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electric Current
An electric current is a flow of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is defined as the net rate of flow of electric charge through a surface. The moving particles are called charge carriers, which may be one of several types of particles, depending on the Electrical conductor, conductor. In electric circuits the charge carriers are often electrons moving through a wire. In semiconductors they can be electrons or Electron hole, holes. In an Electrolyte#Electrochemistry, electrolyte the charge carriers are ions, while in Plasma (physics), plasma, an Ionization, ionized gas, they are ions and electrons. In the International System of Units (SI), electric current is expressed in Unit of measurement, units of ampere (sometimes called an "amp", symbol A), which is equivalent to one coulomb per second. The ampere is an SI base unit and electric current is a ISQ base quantity, base quantity in the International System of Qua ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heinz Hopf
Heinz Hopf (19 November 1894 – 3 June 1971) was a German mathematician who worked on the fields of dynamical systems, topology and geometry. Early life and education Hopf was born in Gräbschen, German Empire (now , part of Wrocław, Poland), the son of Elizabeth (née Kirchner) and Wilhelm Hopf. His father was born Jewish and converted to Protestantism a year after Heinz was born; his mother was from a Protestant family. Hopf attended Karl Mittelhaus higher boys' school from 1901 to 1904, and then entered the König-Wilhelm- Gymnasium in Breslau. He showed mathematical talent from an early age. In 1913 he entered the Silesian Friedrich Wilhelm University where he attended lectures by Ernst Steinitz, Adolf Kneser, Max Dehn, Erhard Schmidt, and Rudolf Sturm. When World War I broke out in 1914, Hopf eagerly enlisted. He was wounded twice and received the iron cross (first class) in 1918. After the war Hopf continued his mathematical education in Heidelberg (winter 1919/2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Dirac
Paul Adrien Maurice Dirac ( ; 8 August 1902 – 20 October 1984) was an English mathematician and Theoretical physics, theoretical physicist who is considered to be one of the founders of quantum mechanics. Dirac laid the foundations for both quantum electrodynamics and quantum field theory. He was the Lucasian Professor of Mathematics at the University of Cambridge and a professor of physics at Florida State University. Dirac shared the 1933 Nobel Prize in Physics with Erwin Schrödinger for "the discovery of new productive forms of atomic theory". Dirac graduated from the University of Bristol with a first class honours Bachelor of Science degree in electrical engineering in 1921, and a first class honours Bachelor of Arts degree in mathematics in 1923. Dirac then graduated from the University of Cambridge with a PhD in physics in 1926, writing the first ever thesis on quantum mechanics. Dirac made fundamental contributions to the early development of both quantum mechanic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chern–Weil Homomorphism
In mathematics, the Chern–Weil homomorphism is a basic construction in Chern–Weil theory that computes topological invariants of vector bundles and principal bundles on a smooth manifold ''M'' in terms of connections and curvature representing classes in the de Rham cohomology rings of ''M''. That is, the theory forms a bridge between the areas of algebraic topology and differential geometry. It was developed in the late 1940s by Shiing-Shen Chern and André Weil, in the wake of proofs of the generalized Gauss–Bonnet theorem. This theory was an important step in the theory of characteristic classes. Let ''G'' be a real or complex Lie group with Lie algebra and let \Complex mathfrak g/math> denote the algebra of \Complex-valued polynomials on \mathfrak g (exactly the same argument works if we used \R instead of Let \Complex mathfrak gG be the subalgebra of fixed points in \Complex mathfrak g/math> under the adjoint action of ''G''; that is, the subalgebra consisting of al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jim Simons
James Harris Simons (April 25, 1938 – May 10, 2024) was an American hedge fund manager, investor, mathematician, and philanthropist. At the time of his death, Simons's net worth was estimated to be $31.4 billion, making him the 55th-richest person in the world. He was the founder of Renaissance Technologies, a quantitative hedge fund based in East Setauket, New York. He and his fund are known to be quantitative investors, using mathematical models and algorithms to make investment gains from market inefficiencies. Due to the long-term aggregate investment returns of Renaissance and its Medallion Fund, Simons was called the "greatest investor on Wall Street" and more specifically "the most successful hedge fund manager of all time". Simons developed the Chern–Simons form (with Shiing-Shen Chern), and contributed to the development of string theory by providing a theoretical framework to combine geometry and topology with quantum field theory. In 1994, Simons and his wif ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chern Class
In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since become fundamental concepts in many branches of mathematics and physics, such as string theory, Chern–Simons theory, knot theory, and Gromov–Witten invariants. Chern classes were introduced by . Geometric approach Basic idea and motivation Chern classes are characteristic classes. They are topological invariants associated with vector bundles on a smooth manifold. The question of whether two ostensibly different vector bundles are the same can be quite hard to answer. The Chern classes provide a simple test: if the Chern classes of a pair of vector bundles do not agree, then the vector bundles are different. The converse, however, is not true. In topology, differential geometry, and algebraic geometry, it is often important to count how many linearly independent sect ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gauss–Bonnet Theorem
In the mathematical field of differential geometry, the Gauss–Bonnet theorem (or Gauss–Bonnet formula) is a fundamental formula which links the curvature of a Surface (topology), surface to its underlying topology. In the simplest application, the case of a triangle Euclidean geometry, on a plane, the Sum of angles of a triangle, sum of its angles is 180 degrees. The Gauss–Bonnet theorem extends this to more complicated shapes and curved surfaces, connecting the local and global geometries. The theorem is named after Carl Friedrich Gauss, who developed a version but never published it, and Pierre Ossian Bonnet, who published a special case in 1848. Statement Suppose is a Compact space, compact two-dimensional Riemannian manifold with boundary . Let be the Gaussian curvature of , and let be the geodesic curvature of . Then :\int_M K\,dA+\int_k_g\,ds=2\pi\chi(M), \, where is the volume element, element of area of the surface, and is the line element along the bound ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
China
China, officially the People's Republic of China (PRC), is a country in East Asia. With population of China, a population exceeding 1.4 billion, it is the list of countries by population (United Nations), second-most populous country after India, representing 17.4% of the world population. China spans the equivalent of five time zones and Borders of China, borders fourteen countries by land across an area of nearly , making it the list of countries and dependencies by area, third-largest country by land area. The country is divided into 33 Province-level divisions of China, province-level divisions: 22 provinces of China, provinces, 5 autonomous regions of China, autonomous regions, 4 direct-administered municipalities of China, municipalities, and 2 semi-autonomous special administrative regions. Beijing is the country's capital, while Shanghai is List of cities in China by population, its most populous city by urban area and largest financial center. Considered one of six ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kunming
Kunming is the capital and largest city of the province of Yunnan in China. The political, economic, communications and cultural centre of the province, Kunming is also the seat of the provincial government. During World War II, Kunming was a Chinese military center and the location of the headquarters for the US Army Forces China-Burma-India. Kunming Wujiaba International Airport, Wujiaba Airport served as the home of the Flying Tigers, First American Volunteer Group (AVG) of the Republic of China Air Force, nicknamed the Flying Tigers. Kunming was also a transport terminus for the Burma Road. Kunming is at an altitude of Above mean sea level, above sea level and a latitude just north of the Tropic of Cancer, and is situated in the middle of the Yunnan–Guizhou Plateau. Kunming is the fourth most populous city in Western China, after Chongqing, Chengdu, and Xi'an, and the third most populous city in Southwestern China after Chongqing and Chengdu. As of the 2020 census, Kunmin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
National Southwest Associated University
The National Southwestern Associated University was a national public university from 1938 to 1946 based in Kunming, Yunnan, China. It was formed by the wartime incorporation of National Peking University, National Tsinghua University, and National Nankai University. When the Second Sino-Japanese War broke out between China and Japan in 1937, Peking University, Tsinghua University and Nankai University merged to form Changsha Temporary University in Changsha and later National Southwestern Associated University in Kunming and Mengzi, in Southwest China's Yunnan Province. After the war, the universities moved back and resumed their operation. What was left behind in Kunming became the National Kunming Normal University which later emerged as the Yunnan Normal University. History By summer 1937, the Imperial Japanese Army had bombed Nankai University to the ground in Tianjin and occupied areas including the campuses of two of the country's leading universities in Beijing: Pe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
