|
Gromov's Systolic Inequality For Essential Manifolds
In the mathematical field of Riemannian geometry, M. Gromov's systolic inequality bounds the length of the shortest non-contractible loop on a Riemannian manifold in terms of the volume of the manifold. Gromov's systolic inequality was proved in 1983;see it can be viewed as a generalisation, albeit non-optimal, of Loewner's torus inequality and Pu's inequality for the real projective plane. Technically, let ''M'' be an essential Riemannian manifold of dimension ''n''; denote by sys''π''1(''M'') the homotopy 1-systole of ''M'', that is, the least length of a non-contractible loop on ''M''. Then Gromov's inequality takes the form : \left(\operatorname_1(M)\right)^n \leq C_n \operatorname(M), where ''C''''n'' is a universal constant only depending on the dimension of ''M''. Essential manifolds A closed manifold is called ''essential'' if its fundamental class defines a nonzero element in the homology of its fundamental group, or more precisely in the homology of the corre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, due to concerns about competing with the American Journal of Mathematics. The result was the ''Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influential in in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annals Of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the founding editor-in-chief. It was "intended to afford a medium for the presentation and analysis of any and all questions of interest or importance in pure and applied Mathematics, embracing especially all new and interesting discoveries in theoretical and practical astronomy, mechanical philosophy, and engineering". It was published in Des Moines, Iowa, and was the earliest American mathematics journal to be published continuously for more than a year or two. This incarnation of the journal ceased publication after its tenth year, in 1883, giving as an explanation Hendricks' declining health, but Hendricks made arrangements to have it taken over by new management, and it was continued from March 1884 as the ''Annals of Mathematics''. The n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commentarii Mathematici Helvetici
The ''Commentarii Mathematici Helvetici'' is a quarterly peer-reviewed scientific journal in mathematics. The Swiss Mathematical Society started the journal in 1929 after a meeting in May of the previous year. The Swiss Mathematical Society still owns and operates the journal; the publishing is currently handled on its behalf by the European Mathematical Society. The scope of the journal includes research articles in all aspects in mathematics. The editors-in-chief have been Rudolf Fueter (1929–1949), J.J. Burckhardt (1950–1981), P. Gabriel (1982–1989), H. Kraft (1990–2005), and Eva Bayer-Fluckiger (2006–present). Abstracting and indexing The journal is abstracted and indexed in: According to the ''Journal Citation Reports'', the journal has a 2019 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Systolic Geometry
In mathematics, systolic geometry is the study of systolic invariants of manifolds and polyhedra, as initially conceived by Charles Loewner and developed by Mikhail Gromov, Michael Freedman, Peter Sarnak, Mikhail Katz, Larry Guth, and others, in its arithmetical, ergodic, and topological manifestations. See also a slower-paced Introduction to systolic geometry. The notion of systole The ''systole'' of a compact metric space ''X'' is a metric invariant of ''X'', defined to be the least length of a noncontractible loop in ''X'' (i.e. a loop that cannot be contracted to a point in the ambient space ''X''). In more technical language, we minimize length over free loops representing nontrivial conjugacy classes in the fundamental group of ''X''. When ''X'' is a graph, the invariant is usually referred to as the girth, ever since the 1947 article on girth by W. T. Tutte. Possibly inspired by Tutte's article, Loewner started thinking about systolic questions on surfaces in the la ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gromov's Inequality For Complex Projective Space
In Riemannian geometry, Gromov's optimal stable 2-systolic inequality is the inequality : \mathrm_2^n \leq n! \;\mathrm_(\mathbb^n), valid for an arbitrary Riemannian metric on the complex projective space, where the optimal bound is attained by the symmetric Fubini–Study metric, providing a natural geometrisation of quantum mechanics. Here \operatorname is the stable 2-systole, which in this case can be defined as the infimum of the areas of rational 2-cycles representing the class of the complex projective line \mathbb^1 \subset \mathbb^n in 2-dimensional homology. The inequality first appeared in as Theorem 4.36. The proof of Gromov's inequality relies on the Wirtinger inequality for exterior 2-forms. Projective planes over division algebras \mathbb In the special case n=2, Gromov's inequality becomes \mathrm_2^2 \leq 2 \mathrm_4(\mathbb^2). This inequality can be thought of as an analog of Pu's inequality for the real projective plane \mathbb^2. In both cases, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gromov's Inequality (other)
The following pages deal with inequalities due to Mikhail Gromov: * Bishop–Gromov inequality * Gromov's inequality for complex projective space * Gromov's systolic inequality for essential manifolds In the mathematical field of Riemannian geometry, M. Gromov's systolic inequality bounds the length of the shortest non-contractible loop on a Riemannian manifold in terms of the volume of the manifold. Gromov's systolic inequality was proved in 1 ... * Lévy–Gromov inequality {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Filling Area Conjecture
In differential geometry, Mikhail Gromov's filling area conjecture asserts that the hemisphere has minimum area among the orientable surfaces that fill a closed curve of given length without introducing shortcuts between its points. Definitions and statement of the conjecture Every smooth surface or curve in Euclidean space is a metric space, in which the (intrinsic) distance between two points of is defined as the infimum of the lengths of the curves that go from to ''along'' . For example, on a closed curve C of length , for each point of the curve there is a unique other point of the curve (called the antipodal of ) at distance from . A compact surface fills a closed curve if its border (also called boundary, denoted ) is the curve . The filling is said isometric if for any two points of the boundary curve , the distance between them along is the same (not less) than the distance along the boundary. In other words, to fill a curve isometrically is to fill i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grushko Theorem
In the mathematical subject of group theory, the Grushko theorem or the Grushko–Neumann theorem is a theorem stating that the rank (that is, the smallest cardinality of a generating set) of a free product of two groups is equal to the sum of the ranks of the two free factors. The theorem was first obtained in a 1940 article of Grushko and then, independently, in a 1943 article of Neumann. Statement of the theorem Let ''A'' and ''B'' be finitely generated groups and let ''A''∗''B'' be the free product of ''A'' and ''B''. Then :rank(''A''∗''B'') = rank(''A'') + rank(''B''). It is obvious that rank(''A''∗''B'') ≤ rank(''A'') + rank(''B'') since if X is a finite generating set of ''A'' and ''Y'' is a finite generating set of ''B'' then ''X''∪''Y'' is a generating set for ''A''∗''B'' and that , ''X'' ∪ ''Y'', ≤ , ''X'', + , ''Y'', . The opposite inequality, rank(''A''∗''B'') ≥ rank(''A'') + rank(''B''), requires proof. Grushko, but not Neumann, proved a mor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Systoles Of Surfaces
In mathematics, systolic inequalities for curves on surfaces were first studied by Charles Loewner in 1949 (unpublished; see remark at end of P. M. Pu's paper in '52). Given a closed surface, its systole, denoted sys, is defined to be the least length of a loop that cannot be contracted to a point on the surface. The ''systolic area'' of a metric is defined to be the ratio area/sys2. The ''systolic ratio'' SR is the reciprocal quantity sys2/area. See also Introduction to systolic geometry. Torus In 1949 Loewner proved his inequality for metrics on the torus T2, namely that the systolic ratio SR(T2) is bounded above by 2/\sqrt, with equality in the flat (constant curvature) case of the equilateral torus (see hexagonal lattice). Real projective plane A similar result is given by Pu's inequality for the real projective plane from 1952, due to Pao Ming Pu, with an upper bound of ''π''/2 for the systolic ratio SR(RP2), also attained in the constant curvature case. Klein bottl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Filling Radius
In Riemannian geometry, the filling radius of a Riemannian manifold ''X'' is a metric invariant of ''X''. It was originally introduced in 1983 by Mikhail Gromov, who used it to prove his systolic inequality for essential manifolds, vastly generalizing Loewner's torus inequality and Pu's inequality for the real projective plane, and creating systolic geometry in its modern form. The filling radius of a simple loop ''C'' in the plane is defined as the largest radius, ''R'' > 0, of a circle that fits inside ''C'': :\mathrm(C\subset \mathbb^2) = R. Dual definition via neighborhoods There is a kind of a dual point of view that allows one to generalize this notion in an extremely fruitful way, as shown by Gromov. Namely, we consider the \varepsilon-neighborhoods of the loop ''C'', denoted :U_\varepsilon C \subset \mathbb^2. As \varepsilon>0 increases, the \varepsilon-neighborhood U_\varepsilon C swallows up more and more of the interior of the loop. The ''last'' p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
