|
Systolic Category
The systole (or systolic category) is a numerical invariant of a closed manifold ''M'', introduced by Mikhail Katz and Yuli Rudyak in 2006, by analogy with the Lusternik–Schnirelmann category. The invariant is defined in terms of the systoles of ''M'' and its covers, as the largest number of systoles in a product yielding a curvature-free lower bound for the total volume of ''M''. The invariant is intimately related to the Lusternik-Schnirelmann category. Thus, in dimensions 2 and 3, the two invariants coincide. In dimension 4, the systolic category is known to be a lower bound for the Lusternik–Schnirelmann category. Bibliography * Dranishnikov, A.; Rudyak, Y. (2009) Stable systolic category of manifolds and the cup-length. ''Journal of Fixed Point Theory and Applications'' 6, no. 1, 165–177. * Katz, M.; Rudyak, Y. (2008) Bounding volume by systoles of 3-manifolds. ''Journal of the London Mathematical Society'' 78, no 2, 407–417. * Dranishnikov, A.; Kat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Invariant (mathematics)
In mathematics, an invariant is a property of a mathematical object (or a class of mathematical objects) which remains unchanged after operations or transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used. For example, the area of a triangle is an invariant with respect to isometries of the Euclidean plane. The phrases "invariant under" and "invariant to" a transformation are both used. More generally, an invariant with respect to an equivalence relation is a property that is constant on each equivalence class. Invariants are used in diverse areas of mathematics such as geometry, topology, algebra and discrete mathematics. Some important classes of transformations are defined by an invariant they leave unchanged. For example, conformal maps are defined as transformations of the plane that preserve angles. The discovery of invariants is an important ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of n-dimensional Euclidean space. One-dimensional manifolds include lines and circles, but not lemniscates. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane. The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. The concept has applications in computer-graphics given the need to associate pictures with coordinates (e.g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mikhail Katz
Mikhail "Mischa" Gershevich Katz (born 1958, in Chișinău)Curriculum vitae retrieved 2011-05-23. is an Israeli , a professor of mathematics at . His main interests are , and |
Yuli Rudyak
Yuli B. Rudyak is a professor of Mathematics at the University of Florida in Gainesville, Florida. He obtained his doctorate from Moscow State University under the supervision of M. M. Postnikov. His main research interests are geometry and topology and symplectic topology. Books * :Reviewer Donald M. Davis (mathematician) for MathSciNet wrote: "This book provides an excellent and thorough treatment of various topics related to cobordism. It should become an indispensable tool for advanced graduate students and workers in algebraic topology." :The book listed 118 cites at Google Scholar Google Scholar is a freely accessible web search engine that indexes the full text or metadata of scholarly literature across an array of publishing formats and disciplines. Released in beta in November 2004, the Google Scholar index includes p ... in 2011. Personal life Rudyak is the father of Marina Rudyak, who is an Assistant Professor of Chinese Studies at the University of Heidelberg. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lusternik–Schnirelmann Category
In mathematics, the Lyusternik–Schnirelmann category (or, Lusternik–Schnirelmann category, LS-category) of a topological space X is the homotopy invariant defined to be the smallest integer number k such that there is an open covering \_ of X with the property that each inclusion map U_i\hookrightarrow X is nullhomotopic. For example, if X is a sphere, this takes the value two. Sometimes a different normalization of the invariant is adopted, which is one less than the definition above. Such a normalization has been adopted in the definitive monograph by Cornea, Lupton, Oprea, and Tanré (see below). In general it is not easy to compute this invariant, which was initially introduced by Lazar Lyusternik and Lev Schnirelmann in connection with variational problems. It has a close connection with algebraic topology, in particular cup-length. In the modern normalization, the cup-length is a lower bound for the LS-category. It was, as originally defined for the case of X a manifo ... [...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] |
Journal Of The London Mathematical Society
The London Mathematical Society (LMS) is one of the United Kingdom's learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh Mathematical Society and the Operational Research Society (ORS). History The Society was established on 16 January 1865, the first president being Augustus De Morgan. The earliest meetings were held in University College, but the Society soon moved into Burlington House, Piccadilly. The initial activities of the Society included talks and publication of a journal. The LMS was used as a model for the establishment of the American Mathematical Society in 1888. Mary Cartwright was the first woman to be President of the LMS (in 1961–62). The Society was granted a royal charter in 1965, a century after its foundation. In 1998 the Society moved from rooms in Burlington House into De Morgan House (named after the society's first president), at 57–5 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Israel Journal Of Mathematics
'' Israel Journal of Mathematics'' is a peer-reviewed mathematics journal published by the Hebrew University of Jerusalem (Magnes Press). Founded in 1963, as a continuation of the ''Bulletin of the Research Council of Israel'' (Section F), the journal publishes articles on all areas of mathematics. The journal is indexed by ''Mathematical Reviews'' and Zentralblatt MATH. Its 2009 MCQ was 0.70, and its 2009 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 in the last two years in a given journal, as i ... was 0.754. External links * Mathematics journals Publications established in 1963 English-language journals Bimonthly journals Hebrew University of Jerusalem {{math-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematische Annalen
''Mathematische Annalen'' (abbreviated as ''Math. Ann.'' or, formerly, ''Math. Annal.'') is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann. Subsequent managing editors were Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguignon, Wolfgang Lück, and Nigel Hitchin. Currently, the managing editor of Mathematische Annalen is Thomas Schick. Volumes 1–80 (1869–1919) were published by Teubner. Since 1920 (vol. 81), the journal has been published by Springer. In the late 1920s, under the editorship of Hilbert, the journal became embroiled in controversy over the participation of L. E. J. Brouwer on its editorial board, a spillover from the foundational Brouwer–Hilbert controversy. Between 1945 and 1947 the journal briefly ceased publication. References External links''Mathematische Annalen''homepage at Springer''Mathematische Annalen''archive (1869†... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Communications On Pure And Applied Mathematics
''Communications on Pure and Applied Mathematics'' is a monthly peer-reviewed scientific journal which is published by John Wiley & Sons on behalf of the Courant Institute of Mathematical Sciences. It covers research originating from or solicited by the institute, typically in the fields of applied mathematics, mathematical analysis, or mathematical physics. The journal was established in 1948 as the ''Communications on Applied Mathematics'', obtaining its current title the next year. According to the ''Journal Citation Reports'', the journal has a 2020 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 in the last two years in a given journal, as i ... of 3.219. References External links * Mathematics journals Monthly journals Wiley (publisher) academic journals Publications established in 1948 English-lang ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
