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Cartan–Hadamard Conjecture
In mathematics, the Cartan–Hadamard conjecture is a fundamental problem in Riemannian geometry and Geometric measure theory which states that the classical isoperimetric inequality may be generalized to spaces of nonpositive sectional curvature, known as Cartan–Hadamard manifolds. The conjecture, which is named after French mathematicians Élie Cartan and Jacques Hadamard, may be traced back to work of André Weil in 1926. Informally, the conjecture states that negative curvature allows regions with a given perimeter to hold more volume. This phenomenon manifests itself in nature through corrugations on coral reefs, or ripples on a petunia flower, which form some of the simplest examples of non-positively curved spaces. History The conjecture, in all dimensions, was first stated explicitly in 1976 by Thierry Aubin, and a few years later by Misha Gromov, Yuri Burago and Viktor Zalgaller. In dimension 2 this fact had already been established in 1926 by André Weil and r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemannian Geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to point. This gives, in particular, local notions of angle, length of curves, surface area and volume. From those, some other global quantities can be derived by integrating local contributions. Riemannian geometry originated with the vision of Bernhard Riemann expressed in his inaugural lecture "''Ueber die Hypothesen, welche der Geometrie zu Grunde liegen''" ("On the Hypotheses on which Geometry is Based.") It is a very broad and abstract generalization of the differential geometry of surfaces in R3. Development of Riemannian geometry resulted in synthesis of diverse results concerning the geometry of surfaces and the behavior of geodesics on them, with techniques that can be applied to the study of differentiable manifolds of higher dimensio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chris Croke
Chris is a short form of various names including Christopher, Christian, Christina, Christine, and Christos. Chris is also used as a name in its own right, however it is not as common. People with the given name * Chris Abani (born 1966), Nigerian author * Chris Abrahams (born 1961), Sydney-based jazz pianist *Chris Adams (other), multiple people * Chris Adcock (born 1989), English internationally elite badminton player * Chris Albright (born 1979), American former soccer player *Chris Alcaide (1923–2004), American actor *Chris Amon (1943–2016), former New Zealand motor racing driver *Chris Andersen (born 1978), American basketball player * Chris Anderson (other), multiple people *Chris Angel (wrestler) (born 1982), Puerto Rican professional wrestler * Chris Anker Sørensen (born 1984), Danish cycler *Chris Anstey (born 1975), Australian basketball player * Chris Anthony, American voice actress *Chris Antley (1966–2000), champion American jockey *Chri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frank Morgan (mathematician)
Frank Morgan is an American mathematician and the Webster Atwell '21 Professor of Mathematics, Emeritus, at Williams College. He is known for contributions to geometric measure theory, minimal surfaces, and differential geometry, including the resolution of the double bubble conjecture. He was vice-president of the American Mathematical Society and the Mathematical Association of America. Morgan studied at the Massachusetts Institute of Technology and Princeton University, and received his Ph.D. from Princeton in 1977, under the supervision of Frederick J. Almgren Jr. He taught at MIT for ten years before joining the Williams faculty. Morgan is the founder of SMALL, one of the largest and best known summer undergraduate Mathematics research programs. In 2012 he became a fellow of the American Mathematical Society. Frank Morgan is also an avid dancer. He gained temporary fame for his work "Dancing the Parkway". Mathematical work He is known for proving, in collaboration wit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gerrit Bol
Gerrit Bol (May 29, 1906 in Amsterdam – February 21, 1989 in Freiburg) was a Dutch mathematician who specialized in geometry. He is known for introducing Bol loops in 1937, and Bol’s conjecture on sextactic points. Life Bol earned his PhD in 1928 at Leiden University under Willem van der Woude. In the 1930s, he worked at the University of Hamburg on the geometry of webs under Wilhelm Blaschke and later projective differential geometry. In 1931 he earned a habilitation. In 1933 Bol signed the '' Loyalty Oath of German Professors to Adolf Hitler and the National Socialist State''. In 1942–1945 during World War II, Bol fought on the Dutch side, and was taken prisoner. On the authority of Blaschke, he was released. After the war, Bol became professor at the Albert-Ludwigs-University of Freiburg The University of Freiburg (colloquially german: Uni Freiburg), officially the Albert Ludwig University of Freiburg (german: Albert-Ludwigs-Universität Freiburg), is a publi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joel Spruck
Joel Spruck (born 1946) is a mathematician, J. J. Sylvester Professor of Mathematics at Johns Hopkins University, whose research concerns geometric analysis and elliptic partial differential equations. He obtained his PhD from Stanford University with the supervision of Robert S. Finn in 1971. Mathematical contributions Spruck is well known in the field of elliptic partial differential equations for his series of papers "The Dirichlet problem for nonlinear second-order elliptic equations," written in collaboration with Luis Caffarelli, Joseph J. Kohn, and Louis Nirenberg. These papers were among the first to develop a general theory of second-order elliptic differential equations which are fully nonlinear, with a regularity theory that extends to the boundary. Caffarelli, Nirenberg & Spruck (1985) has been particularly influential in the field of geometric analysis since many geometric partial differential equations are amenable to its methods. With Basilis Gidas, Spruck studie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mohammad Ghomi
Muhammad ( ar, مُحَمَّد; 570 – 8 June 632 CE) was an Arab religious, social, and political leader and the founder of Islam. According to Islamic doctrine, he was a prophet divinely inspired to preach and confirm the monotheistic teachings of Adam, Abraham, Moses, Jesus, and other prophets. He is believed to be the Seal of the Prophets within Islam. Muhammad united Arabia into a single Muslim polity, with the Quran as well as his teachings and practices forming the basis of Islamic religious belief. Muhammad was born approximately 570CE in Mecca. He was the son of Abdullah ibn Abd al-Muttalib and Amina bint Wahb. His father Abdullah was the son of Quraysh tribal leader Abd al-Muttalib ibn Hashim, and he died a few months before Muhammad's birth. His mother Amina died when he was six, leaving Muhammad an orphan. He was raised under the care of his grandfather, Abd al-Muttalib, and paternal uncle, Abu Talib. In later years, he would periodically seclude hi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Total Curvature
In mathematical study of the differential geometry of curves, the total curvature of an immersed plane curve is the integral of curvature along a curve taken with respect to arc length: :\int_a^b k(s)\,ds. The total curvature of a closed curve is always an integer multiple of 2, called the index of the curve, or turning number – it is the winding number of the unit tangent vector about the origin, or equivalently the degree of the map to the unit circle assigning to each point of the curve, the unit velocity vector at that point. This map is similar to the Gauss map for surfaces. Comparison to surfaces This relationship between a local geometric invariant, the curvature, and a global topological invariant, the index, is characteristic of results in higher-dimensional Riemannian geometry such as the Gauss–Bonnet theorem. Invariance According to the Whitney–Graustein theorem, the total curvature is invariant under a regular homotopy of a curve: it is the degree of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Calculus Of Variations
The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers. Functionals are often expressed as definite integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations. A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is a straight line between the points. However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist. Such solutions are known as '' geodesics''. A related problem is posed by Fermat's principle: light follows the path of shortest optical length connecting two points, which depend ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Geometry
In mathematics, integral geometry is the theory of measures on a geometrical space invariant under the symmetry group of that space. In more recent times, the meaning has been broadened to include a view of invariant (or equivariant) transformations from the space of functions on one geometrical space to the space of functions on another geometrical space. Such transformations often take the form of integral transforms such as the Radon transform and its generalizations. Classical context Integral geometry as such first emerged as an attempt to refine certain statements of geometric probability theory. The early work of Luis Santaló and Wilhelm Blaschke was in this connection. It follows from the classic theorem of Crofton expressing the length of a plane curve as an expectation of the number of intersections with a random line. Here the word 'random' must be interpreted as subject to correct symmetry considerations. There is a sample space of lines, one on which the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Luis Santaló
Luís Antoni Santaló Sors (October 9, 1911 – November 22, 2001) was a Spanish mathematician. He graduated from the University of Madrid and he studied at the University of Hamburg, where he received his Ph.D. in 1936. His advisor was Wilhelm Blaschke. Because of the Spanish Civil War, he moved to Argentina as a professor in the National University of the Littoral, National University of La Plata and University of Buenos Aires. His work with Blaschke on convex sets is now cited in its connection with Mahler volume. Blaschke and Santaló also collaborated on integral geometry. Santaló wrote textbooks in Spanish on non-Euclidean geometry, projective geometry, and tensors. Works Luis Santaló published in both English and Spanish: ''Introduction to Integral Geometry'' (1953) Chapter I. Metric integral geometry of the plane including densities and the isoperimetric inequality. Ch. II. Integral geometry on surfaces including Blaschke's formula and the isoperimetric inequ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmonic Analysis
Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an extended form of Fourier analysis). In the past two centuries, it has become a vast subject with applications in areas as diverse as number theory, representation theory, signal processing, quantum mechanics, tidal analysis and neuroscience. The term "harmonics" originated as the Ancient Greek word ''harmonikos'', meaning "skilled in music". In physical eigenvalue problems, it began to mean waves whose frequencies are integer multiples of one another, as are the frequencies of the harmonics of music notes, but the term has been generalized beyond its original meaning. The classical Fourier transform on R''n'' is still an area of ongoing research, particularly concerning Fourier transformation on more general objects such as tempered di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conformal Map
In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths. More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-preserving) at a point u_0\in U if it preserves angles between directed curves through u_0, as well as preserving orientation. Conformal maps preserve both angles and the shapes of infinitesimally small figures, but not necessarily their size or curvature. The conformal property may be described in terms of the Jacobian derivative matrix of a coordinate transformation. The transformation is conformal whenever the Jacobian at each point is a positive scalar times a rotation matrix (orthogonal with determinant one). Some authors define conformality to include orientation-reversing mappings whose Jacobians can be written as any scalar times any orthogonal matrix. For mappings in two dimensions, the (orientation-preserving) conformal mappings are precisely the locally ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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