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Richard Thomas (mathematician)
Richard Paul Winsley Thomas is a British mathematician working in several areas of geometry. He is a professor at Imperial College London. He studies moduli problems in algebraic geometry, and ‘mirror symmetry’—a phenomenon in pure mathematics predicted by string theory in theoretical physics. Education Thomas obtained his PhD on gauge theory on Calabi–Yau manifolds in 1997 under the supervision of Simon Donaldson at the University of Oxford. Together with Donaldson, he defined the Donaldson–Thomas invariants of Calabi–Yau 3-folds, now a major topic in geometry and the mathematics of string theory. Career and research Before joining Imperial College, he was member of the Institute for Advanced Study in Princeton, New Jersey, and affiliated with Harvard University and the University of Oxford. He was made professor of pure mathematics in 2005. Thomas has made contributions to algebraic geometry, differential Geometry, and symplectic geometry. His doctoral thes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Imperial College London
Imperial College London (legally Imperial College of Science, Technology and Medicine) is a public research university in London, United Kingdom. Its history began with Prince Albert, consort of Queen Victoria, who developed his vision for a cultural area that included the Royal Albert Hall, Victoria & Albert Museum, Natural History Museum and royal colleges. In 1907, Imperial College was established by a royal charter, which unified the Royal College of Science, Royal School of Mines, and City and Guilds of London Institute. In 1988, the Imperial College School of Medicine was formed by merging with St Mary's Hospital Medical School. In 2004, Queen Elizabeth II opened the Imperial College Business School. Imperial focuses exclusively on science, technology, medicine, and business. The main campus is located in South Kensington, and there is an innovation campus in White City. Facilities also include teaching hospitals throughout London, and with Imperial College Healthcare ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harvard University
Harvard University is a private Ivy League research university in Cambridge, Massachusetts. Founded in 1636 as Harvard College and named for its first benefactor, the Puritan clergyman John Harvard, it is the oldest institution of higher learning in the United States and one of the most prestigious and highly ranked universities in the world. The university is composed of ten academic faculties plus Harvard Radcliffe Institute. The Faculty of Arts and Sciences offers study in a wide range of undergraduate and graduate academic disciplines, and other faculties offer only graduate degrees, including professional degrees. Harvard has three main campuses: the Cambridge campus centered on Harvard Yard; an adjoining campus immediately across Charles River in the Allston neighborhood of Boston; and the medical campus in Boston's Longwood Medical Area. Harvard's endowment is valued at $50.9 billion, making it the wealthiest academic institution in the world. Endowment inco ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lothar Göttsche
Lothar Göttsche (born January 21, 1961 in Sonderburg, Denmark) is a German mathematician, known for his work in algebraic geometry. He is a research scientist at the International Centre for Theoretical Physics in Trieste, Italy. He is also editor for Geometry & Topology. Biography After studying mathematics at the University of Kiel, he received his Dr. rer. nat. under the direction of Friedrich Hirzebruch at the University of Bonn in 1989. Göttsche was invited as speaker to the International Congress of Mathematicians in Beijing in 2002. In 2012 he became a fellow of the American Mathematical Society. Work Göttsche received international acclaim with his formula for the generating function for the Betti numbers of the Hilbert scheme of points on an algebraic surface: :If S is a smooth surface over an algebraically closed field of characteristic 0, then the generating function for the motives of the Hilbert schemes of S can be expressed in terms of the motivic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rahul Pandharipande
Rahul Pandharipande (born 1969) is a mathematician who is currently a professor of mathematics at the Swiss Federal Institute of Technology Zürich (ETH) working in algebraic geometry. His particular interests concern moduli spaces, enumerative invariants associated to moduli spaces, such as Gromov–Witten invariants and Donaldson–Thomas invariants, and the cohomology of the moduli space of curves. His father Vijay Raghunath Pandharipande was a renowned theoretical physicist who worked in the area of nuclear physics. Educational and professional history He received his A.B. from Princeton University in 1990 and his PhD from Harvard University in 1994 with a thesis entitled `''A Compactification over the Moduli Space of Stable Curves of the Universal Moduli Space of Slope-Semistable Vector Bundles. His thesis advisor at Harvard was Joe Harris. After teaching at the University of Chicago and the California Institute of Technology, he joined the faculty as Professor of M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lagrangian Submanifold
In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology. Symplectic manifolds arise naturally in abstract formulations of classical mechanics and analytical mechanics as the cotangent bundles of manifolds. For example, in the Hamiltonian formulation of classical mechanics, which provides one of the major motivations for the field, the set of all possible configurations of a system is modeled as a manifold, and this manifold's cotangent bundle describes the phase space of the system. Motivation Symplectic manifolds arise from classical mechanics; in particular, they are a generalization of the phase space of a closed system. In the same way the Hamilton equations allow one to derive the time evolution of a system from a set of differential equati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shing-Tung Yau
Shing-Tung Yau (; ; born April 4, 1949) is a Chinese-American mathematician and the William Caspar Graustein Professor of Mathematics at Harvard University. In April 2022, Yau announced retirement from Harvard to become Chair Professor of mathematics at Tsinghua University. Yau was born in Shantou, China, moved to Hong Kong at a young age, and to the United States in 1969. He was awarded the Fields Medal in 1982, in recognition of his contributions to partial differential equations, the Calabi conjecture, the positive energy theorem, and the Monge–Ampère equation. Yau is considered one of the major contributors to the development of modern differential geometry and geometric analysis. The impact of Yau's work can be seen in the mathematical and physical fields of differential geometry, partial differential equations, convex geometry, algebraic geometry, enumerative geometry, mirror symmetry, general relativity, and string theory, while his work has also touched upon applied ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Seidel
Paul Seidel (born December 30, 1970) is a Swiss-Italian mathematician. He is a faculty member at the Massachusetts Institute of Technology. Career Seidel attended Heidelberg University, where he received his Diplom under supervision of Albrecht Dold in 1994. He then pursued his Ph.D. studies at the University of Oxford under supervision of Simon Donaldson (Thesis: ''Floer Homology and the Symplectic Isotopy Problem'') in 1998. He was a chargé de recherche at the CNRS from 1999 to 2002, a professor at Imperial College London from 2002 to 2003, a professor at the University of Chicago from 2003 to 2007, and then a professor at the Massachusetts Institute of Technology from 2007 onwards. Awards In 2000, Seidel was awarded the EMS Prize. In 2010, he was awarded the Oswald Veblen Prize in Geometry "for his fundamental contributions to symplectic geometry and, in particular, for his development of advanced algebraic methods for computation of symplectic invariants." In 2012, he becam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coherent Sheaves
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with reference to a sheaf of rings that codifies this geometric information. Coherent sheaves can be seen as a generalization of vector bundles. Unlike vector bundles, they form an abelian category, and so they are closed under operations such as taking kernels, images, and cokernels. The quasi-coherent sheaves are a generalization of coherent sheaves and include the locally free sheaves of infinite rank. Coherent sheaf cohomology is a powerful technique, in particular for studying the sections of a given coherent sheaf. Definitions A quasi-coherent sheaf on a ringed space (X, \mathcal O_X) is a sheaf \mathcal F of \mathcal O_X-modules which has a local presentation, that is, every point in X has an open neighborhood U in which there is an exac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derived Categories
In mathematics, the derived category ''D''(''A'') of an abelian category ''A'' is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on ''A''. The construction proceeds on the basis that the objects of ''D''(''A'') should be chain complexes in ''A'', with two such chain complexes considered isomorphic when there is a chain map that induces an isomorphism on the level of homology of the chain complexes. Derived functors can then be defined for chain complexes, refining the concept of hypercohomology. The definitions lead to a significant simplification of formulas otherwise described (not completely faithfully) by complicated spectral sequences. The development of the derived category, by Alexander Grothendieck and his student Jean-Louis Verdier shortly after 1960, now appears as one terminal point in the explosive development of homological algebra in the 1950s, a decade in which it had made remarkab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Braid Group
A braid (also referred to as a plait) is a complex structure or pattern formed by interlacing two or more strands of flexible material such as textile yarns, wire, or hair. The simplest and most common version is a flat, solid, three-stranded structure. More complex patterns can be constructed from an arbitrary number of strands to create a wider range of structures (such as a fishtail braid, a five-stranded braid, rope braid, a French braid and a waterfall braid). The structure is usually long and narrow with each component strand functionally equivalent in zigzagging forward through the overlapping mass of the others. It can be compared with the process of weaving, which usually involves two separate perpendicular groups of strands (warp and weft). Historically, the materials used have depended on the indigenous plants and animals available in the local area. During the Industrial Revolution, mechanized braiding equipment was invented to increase production. The braiding te ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homological Mirror Symmetry
Homological mirror symmetry is a mathematical conjecture made by Maxim Kontsevich. It seeks a systematic mathematical explanation for a phenomenon called mirror symmetry first observed by physicists studying string theory. History In an address to the 1994 International Congress of Mathematicians in Zürich, speculated that mirror symmetry for a pair of Calabi–Yau manifolds ''X'' and ''Y'' could be explained as an equivalence of a triangulated category constructed from the algebraic geometry of ''X'' (the derived category of coherent sheaves on ''X'') and another triangulated category constructed from the symplectic geometry of ''Y'' (the derived Fukaya category). Edward Witten originally described the topological twisting of the N=(2,2) supersymmetric field theory into what he called the A and B model topological string theories. These models concern maps from Riemann surfaces into a fixed target—usually a Calabi–Yau manifold. Most of the mathematical predictions of mi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
