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Chern's Conjecture For Hypersurfaces In Spheres
Chern's conjecture for hypersurfaces in spheres, unsolved as of 2018, is a conjecture proposed by Chern in the field of differential geometry. It originates from the Chern's unanswered question: Consider closed minimal submanifolds M^n immersed in the unit sphere S^ with second fundamental form of constant length whose square is denoted by \sigma. Is the set of values for \sigma discrete? What is the infimum of these values of \sigma > \frac? The first question, i.e., whether the set of values for ''σ'' is discrete, can be reformulated as follows: Let M^n be a closed minimal submanifold in \mathbb^ with the second fundamental form of constant length, denote by \mathcal_n the set of all the possible values for the squared length of the second fundamental form of M^n, is \mathcal_n a discrete? Its affirmative hand, more general than the Chern's conjecture for hypersurfaces, sometimes also referred to as the Chern's conjecture and is still, as of 2018, unanswered even with ''M'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky. The simplest examples of smooth spaces are the plane and space curves and surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries. Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable manifolds. A geometric structure is one which defines some notion of size, distance, shape, volume, or other rigidifying structu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yau's Conjecture On The First Eigenvalue
In mathematics, Yau's conjecture on the first eigenvalue is, as of 2018, an unsolved conjecture proposed by Shing-Tung Yau in 1982. It asks: Is it true that the first eigenvalue for the Laplace–Beltrami operator on an embedded minimal hypersurface of S^ is n? If true, it will imply that the area of embedded minimal hypersurfaces in S^3 will have an upper bound depending only on the genus. Some possible reformulations are as follows: * The first eigenvalue of every closed embedded minimal hypersurface M^n in the unit sphere S^(1) is n * The first eigenvalue of an embedded compact minimal hypersurface M^n of the standard (''n'' + 1)-sphere with sectional curvature 1 is n * If S^ is the unit (''n'' + 1)-sphere with its standard round metric, then the first Laplacian eigenvalue on a closed embedded minimal hypersurface ^n \subset S^ is n The Yau's conjecture is verified for several special cases, but still open in general. Shiing-Shen Chern conjectured that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjectures
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 1995 by Andrew Wiles), have shaped much of mathematical history as new areas of mathematics are developed in order to prove them. Important examples Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, ''b'', and ''c'' can satisfy the equation ''a^n + b^n = c^n'' for any integer value of ''n'' greater than two. This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of '' Arithmetica'', where he claimed that he had a proof that was too large to fit in the margin. The first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995, after 358 years of effort by mathe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Technische Universität Wien
TU Wien (TUW; german: Technische Universität Wien; still known in English as the Vienna University of Technology from 1975–2014) is one of the major universities in Vienna, Austria. The university finds high international and domestic recognition in teaching as well as in research, and it is a highly esteemed partner of innovation-oriented enterprises. It currently has about 28,100 students (29% women), eight faculties and about 5,000 staff members (3,800 academics). The university's teaching and research is focused on engineering, computer science, and natural sciences. History The institution was founded in 1815 by Emperor Francis I of Austria as the '' k.k. Polytechnische Institut'' (Imperial-Royal Polytechnic Institute). The first rector was Johann Joseph von Prechtl. It was renamed the ''Technische Hochschule'' (College of Technology) in 1872. When it began granting doctoral and higher degrees in 1975, it was renamed the ''Technische Universität Wien'' (Vienna Univer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of Southampton
, mottoeng = The Heights Yield to Endeavour , type = Public research university , established = 1862 – Hartley Institution1902 – Hartley University College1913 – Southampton University College1952 – gained university status by royal charter , chancellor = Ruby Wax , vice_chancellor = Mark E. Smith , head_label = Visitor , head = Penny Mordaunt , location = Southampton, Hampshire, England , campus = City Campus , academic_staff = 2,715 (2020) , administrative_staff = 5,001 , students = () , undergrad = () , postgrad = () , colours = Navy blue, light sea green and dark red , endowment = £14.9 million , budget = £578.4 million , affiliations = ACU EUAPort-City University LeagueRussell GroupSES (universities), SESSET ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Princeton University Press
Princeton University Press is an independent publisher with close connections to Princeton University. Its mission is to disseminate scholarship within academia and society at large. The press was founded by Whitney Darrow, with the financial support of Charles Scribner, as a printing press to serve the Princeton community in 1905. Its distinctive building was constructed in 1911 on William Street in Princeton. Its first book was a new 1912 edition of John Witherspoon's ''Lectures on Moral Philosophy.'' History Princeton University Press was founded in 1905 by a recent Princeton graduate, Whitney Darrow, with financial support from another Princetonian, Charles Scribner II. Darrow and Scribner purchased the equipment and assumed the operations of two already existing local publishers, that of the ''Princeton Alumni Weekly'' and the Princeton Press. The new press printed both local newspapers, university documents, ''The Daily Princetonian'', and later added book publishing to it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of Chicago
The University of Chicago (UChicago, Chicago, U of C, or UChi) is a private research university in Chicago, Illinois. Its main campus is located in Chicago's Hyde Park neighborhood. The University of Chicago is consistently ranked among the best universities in the world and it is among the most selective in the United States. The university is composed of an undergraduate college and five graduate research divisions, which contain all of the university's graduate programs and interdisciplinary committees. Chicago has eight professional schools: the Law School, the Booth School of Business, the Pritzker School of Medicine, the Crown Family School of Social Work, Policy, and Practice, the Harris School of Public Policy, the Divinity School, the Graham School of Continuing Liberal and Professional Studies, and the Pritzker School of Molecular Engineering. The university has additional campuses and centers in London, Paris, Beijing, Delhi, and Hong Kong, as well as in downtown ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Marshall Stone
Marshall Harvey Stone (April 8, 1903 – January 9, 1989) was an American mathematician who contributed to real analysis, functional analysis, topology and the study of Boolean algebras. Biography Stone was the son of Harlan Fiske Stone, who was the Chief Justice of the United States in 1941–1946. Marshall Stone's family expected him to become a lawyer like his father, but he became enamored of mathematics while he was a Harvard University undergraduate. He completed a Harvard PhD in 1926, with a thesis on differential equations that was supervised by George David Birkhoff. Between 1925 and 1937, he taught at Harvard, Yale University, and Columbia University. Stone was promoted to a full professor at Harvard in 1937. During World War II, Stone did classified research as part of the "Office of Naval Operations" and the "Office of the Chief of Staff" of the United States Department of War. In 1946, he became the chairman of the Mathematics Department at the University of Chicag ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematisches Forschungsinstitut Oberwolfach
The Oberwolfach Research Institute for Mathematics (german: Mathematisches Forschungsinstitut Oberwolfach) is a center for mathematical research in Oberwolfach, Germany. It was founded by mathematician Wilhelm Süss in 1944. It organizes weekly workshops on diverse topics where mathematicians and scientists from all over the world come to do interdisciplinary, collaborative research. The Institute is a member of the Leibniz Association, funded mainly by the Federal Ministry of Education and Research (Germany), German Federal Ministry of Education and Research and by the state of Baden-Württemberg. It also receives substantial funding from the ''Friends of Oberwolfach'' foundation, from the ''Oberwolfach Foundation'' and from numerous donors. History The Oberwolfach Research Institute for Mathematics (MFO) was founded as the ''Reich Institute of Mathematics'' (German: ''Reichsinstitut für Mathematik'') on 1 September 1944. It was one of several research institutes founded ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of Kansas
The University of Kansas (KU) is a public research university with its main campus in Lawrence, Kansas, United States, and several satellite campuses, research and educational centers, medical centers, and classes across the state of Kansas. Two branch campuses are in the Kansas City metropolitan area on the Kansas side: the university's medical school and hospital in Kansas City, Kansas, the Edwards Campus in Overland Park. There are also educational and research sites in Garden City, Hays, Leavenworth, Parsons, and Topeka, an agricultural education center in rural north Douglas County, and branches of the medical school in Salina and Wichita. The university is a member of the Association of American Universities and is classified among "R1: Doctoral Universities – Very high research activity". Founded March 21, 1865, the university was opened in 1866, under a charter granted by the Kansas State Legislature in 1864 and legislation passed in 1863 under the State Cons ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mimeograph
A mimeograph machine (often abbreviated to mimeo, sometimes called a stencil duplicator) is a low-cost duplicating machine that works by forcing ink through a stencil onto paper. The process is called mimeography, and a copy made by the process is a mimeograph. Mimeographs, along with spirit duplicators and hectographs, were common technologies for printing small quantities of a document, as in office work, classroom materials, and church bulletins. Early fanzines were printed by mimeograph because the machines and supplies were widely available and inexpensive. Beginning in the late 1960s and continuing into the 1970s, photocopying gradually displaced mimeographs, spirit duplicators, and hectographs. For even smaller quantities, up to about five, a typist would use carbon paper. Origins Use of stencils is an ancient art, butthrough chemistry, papers, and pressestechniques advanced rapidly in the late nineteenth century: Papyrograph A description of the Papyrograph meth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
