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Topic: Homology (mathematics)

Related Topics

Singular homology

CW complex

Abelian group

Adjoint functors

Cohomology

Lie group

Equivalence relation

Euler characteristic

Homological algebra

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	Homology (mathematics) - Wikipedia, the free encyclopedia
		In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homos = identical) is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group).
		For a topological space, the homology groups are generally much easier to compute than the homotopy groups, and consequently one usually will have an easier time working with homology to aid in the classification of spaces.
		In abstract algebra, one uses homology to define derived functors, for example the Tor functors.
	en.wikipedia.org /wiki/Homology_(mathematics) (732 words)

	Mathematician Samuel Eilenberg, 84. Columbia University Record, February 20, 1998
		His work created a new discipline of mathematics, algebraic topology, the use of algebra to describe and understand how certain properties of multidimensional forms—such as the number of holes punched through a surface—remain unchanged even when the forms are twisted, bent or stretched.
		He was a member of the Bourbaki, a French mathematical society dating to 1935 whose members published under the pseudonym Nicolas Bourbaki and undertook, with varying success, to publish treatises that would codify all branches of mathematics.
		The Eilenberg-Steenrod Axioms for Homology clarified the subject by setting forth four simple properties that could be used to discern when a possibly exotic homology theory was in fact the same as the usual one.
	www.columbia.edu /cu/record/23/15/28.html (473 words)

	Amazon.ca: A Combinatorial Introduction to Topology: Books (Site not responding. Last check: 2007-11-06)
		The author treats the case of plane homology (mod 2), which is discussed via the use of polygonal chains on a grating in the plane.
		He proves the invariance theorem for triangulations of surfaces by showing that the homology groups of the triangulation are same as the homology groups of the plane model of the surface.
		Integral homology is also introduced by the author, and he shows the origin of torsion in the consideration of the "twist" in a surface.
	www.amazon.ca /exec/obidos/ASIN/0486679667 (1135 words)

	[No title]
		in Mathematics in 1927 and became adjunct professor of mathematics from 1927 to 1929.
		In 1934, he accepted appointment as professor and chairman of the Department of Mathematics at the University of Virginia, where he stayed for the remainder of his career.
		He researched topology, cyclic elements, the structure of continua, homology theory, and examined different notions of convergents in the space of all subsets of a compact metric space.
	www.lib.utexas.edu /taro/utcah/00199.xml (459 words)

	A Short CV for Lisa Traynor (Site not responding. Last check: 2007-11-06)
		We discuss a different version where the target and domains are required to have first homology group equal to Z^n where 2n is the dimension of the manifolds, and where the embeddings are required to be isomorphisms on the level of first homology.
		The generating function symplectic homology groups are first defined for a symplectomorphism as the relative homology groups of the sublevel sets of the generating function for the lagrangian associated to the diffeomorphism.
		Symplectic homology groups for an open set U are constructed via a limit of homology groups associated to symplectomorphisms supported on U. These groups are invariants of U under the action of symplectic diffeomorphisms of R2n and satisfy isotopy invariance and numerous functorial properties shared by Floer-Hofer homology.
	www.brynmawr.edu /math/people/traynor/cv.html (1982 words)

	PAM Bulletin: Vol. 31, No. 3
		Hamilton was recognized for his introduction of the Ricci flow equation and his development of it into one of the most powerful tools in geometry and topology.
		See the Clay Mathematics Institute website for more information on the award winners, their research, and the Institute, http://www.claymath.org/.
		Of note was a posting to the PAM listserv on Jan. 5, 2004 announcing the forthcoming addition in early 2004 of 8 new journals to the Euclid Prime collection.
	www.sla.org /division/dpam/pam-bulletin/vol31/no3/mathematics.html (578 words)

	PI Summer Program 2006: Topology and its Applications, Mississippi State University, July 10-28, 2006
		During July 10-28, 2006, Mississippi State University, Starkville will be the host of the Institute for Mathematics and its Applications (IMA) summer graduate program in mathematics.
		The goal of this course is to provide an introduction to a recently developed computational version of algebraic topology, called persistent homology, which allows one to infer topological properties of geometric objects from "point clouds" sampled from them.
		We will introduce algebraic topology itself, the theory of persistent homology, software which permits its computation, and demonstrate how it is used in several real world examples.
	www.ima.umn.edu /2005-2006/PISG7.10-28.06 (554 words)

	Sue Geller
		Another and original research hat is in the fields of Algebraic K-Theory and cyclic and Hochschild homologies, where my research has centered on determining the relationships between the K-theory and the homology theories and exploiting these relationships to provide algorthms for computing the K-theory and cyclic homology.
		While I have published no research articles myself, I have been the Ph.D. advisor for two people with Mathematics Education speciality as well as helped design and then "ran", i.e., do the advising for, an MS in Mathematics program for people who want to teach at the secondary and post-secondary levels.
		Starting in January, 1990 and ending in July, 1994, the Mathematical Association of America's Committee on the Participation of Women put on skits at the Winter and Summer Joint Mathematics meetings.
	www.math.tamu.edu /~geller (683 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		His work created a new discipline of mathematics, algebraic topology, the use of algebra to describe and understand how certain properties of multidimensional forms - such as the number of holes punched through the surface - remain unchanged even when the forms are twisted, bent or stretched.
		Professor Eilenberg became interested in art collecting on a trip to Bombay in the mid-1950's and pursued Asian art partly to relieve his mind of the rigors of math, Mr.
		The collection, shown to the public in 1992 as "The Lotus Transcendent," was described by Holland Cotter in the New York Times as "one of New York's hidden treasures." Items from Professor Eilenberg's collection may also be found in the British Museum, the Victoria and Albert Museum and the Brooklyn Museum.
	www.columbia.edu /cu/pr/96_99/19265.html (625 words)

	[No title]
		] A homology for a topological space where the nth group reflects how the space may be filled out by n-dimensional simplicial complexes and detects the presence of analogs of n-dimensional holes.
		A method of approximating a definite integral over an interval which is equivalent to dividing the interval into equal subintervals and applying the formula in the first definition to each subinterval.
		] A mathematical relationship for calculating the oil- or gas-bearing net-pay volume of a reservoir; uses the contour lines from a subsurface geological map of the reservoir, including gas-oil and gas-water contacts.
	www.accessscience.com /Dictionary/S/S27/DictS27.html (2717 words)

	cv (Site not responding. Last check: 2007-11-06)
		Instructor at University of Texas at Austin, Department of Mathematics, 2002~2005.
		Variational problems in the intersection homology theory of singular varities, ANA conference at Carnegie Mellon University, May 2003.
		Intersection homology theory via rectifiable currents, University of Texas at Austin, Fall 2002.
	www.ma.utexas.edu /users/qlxia/cv.html (442 words)

	Clay Mathematics Institute
		The Clay Mathematics Institute announces its 2004 summer school, to be held at the Alfréd Rényi Institute of Mathematics in Budapest.
		These courses will concentrate on recent activity at the crossroads of mathematical disciplines around low-dimensional topology: the theory of holomorphic curves, gauge theory, knot theory, smooth four-manifold topology, and contact geometry.
		The aim of this summer school is to provide a comprehensive introduction to these exciting areas through weeklong courses in Heegaard Floer theory of three- and four-manifolds, Seiberg-Witten theory, contact topology, and knot theory.
	www.claymath.org /programs/summer_school/2004 (337 words)

	Homology, Homotopy and Applications (Site not responding. Last check: 2007-11-06)
		Homology, Homotopy and Applications (HHA) is a fully refereed international journal dealing with homology and homotopy in algebra and topology and their applications to the mathematical sciences.
		Homology, Homotopy and Applications is published by International Press.
		Papers will be published by HHA in all areas of mathematics related to homology and homotopy.
	www.emis.ams.org /journals/HHA (95 words)

	Amazon.com: Combinatorial Foundation of Homology and Homotopy : Applications to Spaces, Diagrams, Transformation ... (Site not responding. Last check: 2007-11-06)
		In this book we consider deep and classical results of homotopy theory like the homological Whitehead theorem, the Hurewicz theorem, the finiteness obstruction theorem of Wall, the theorems on Whitehead torsion and simple homotopy equivalences, and we characterize axiomatically the assumptions under which such results hold.
		This leads to a new combinatorial foundation of homology and homotopy.
		In this chapter we describe the leading examples of combinatorial homology and homotopy theory which are well known fields of algebraic topology.
	www.amazon.com /exec/obidos/tg/detail/-/3540649840?v=glance (631 words)

	Simon K. Donaldson - CIRS (Site not responding. Last check: 2007-11-06)
		His research interests lies in the area of mathematics bordering geometry, topology, and analysis and having substantial connections with mathematical physics.
		Much of his early work hinged on the application of the instanton solutions of the Yang-Mills equations--first introduced in particle physics--as tools to solve purely mathematical problems about the topology of four-dimensional manifolds.
		This has led to novel and wide-ranging results, not obtainable by other methods, that give a glimpse of the special nature of four-dimensional topology and geometry.
	www.cirs.net /researchers/mathematics/DONALDSON.htm (249 words)

	DIMACS Conference on Linking Mathematics and Biology in the High Schools
		To continue to teach biology and mathematics in separate silos is to cut off future progress in biological research at its source.
		In response to new ways of teaching, the presenters developed and implemented a mathematics workshop for secondary mathematics teachers and high school students in Danville, VA. The workshop was designed to introduce mathematical biology, an emerging discipline within the field of mathematics, and to demonstrate innovative ways of teaching mathematics with graphical modeling software.
		Mathematical modeling was the primary topic of the workshop in which the participants explored basic concepts in abstract algebra and graph theory.
	dimacs.rutgers.edu /Workshops/Biomath/abstracts.html (4749 words)

	Facts about homology mathematics (Site not responding. Last check: 2007-11-06)
		In mathematics (especially algebraic topology and abstract algebra), homology is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object.
		A chain complex is said to be exact if the image of the n+1-th map is always equal to the kernel of the n-th map.
		All maps in this long exact sequence are induced by the maps between the chain complexes, except for the maps H
	www.supercrawler.com /Facts/homology__mathematics_.html (697 words)

	Mathematics 261: Algebraic Topology I (Site not responding. Last check: 2007-11-06)
		Algebraic topology studies topological spaces by associating to them algebraic invariants.
		The principal algebraic invariants considered in this course are the fundamental group (also known as the first homotopy group) and the homology groups.
		It is fundamental for students interested in research in Algebraic Geometry, Differential Geometry, Mathematical Physics, and Topology; it is also important for students in Algebra and in Number Theory.
	www.math.duke.edu /graduate/courses/spring04/math261.html (215 words)

	Amazon.com: Homology (Classics in Mathematics): Books: Saunders MacLane (Site not responding. Last check: 2007-11-06)
		This item is not eligible for Amazon Prime, but over a million other items are.
		Differential Forms in Algebraic Topology (Graduate Texts in Mathematics) by Raoul Bott
		Homology theory deals repeatedly with the formal properties of functions and their composites.
	www.amazon.com /exec/obidos/tg/detail/-/3540586628?v=glance (617 words)

	Little Gamers ♥ Hot swedish love
		In mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces.
		It can be broadly defined as the study of homology theories on topological spaces.
		At the intuitive level homology is taken to be an equivalence relation, such that chains C and D are homologous on the space X if the chain C − D is a boundary of a chain of one dimension higher.
	www.little-gamers.com /index.php?id=1016 (492 words)

	[No title]
		This paper is devoted to the study of inverse invariant theory and its relationship with the $\steenrod$--invariant prime spectrum of an unstable algebra over the Steenrod algebra.
		Primary: 13D03, 18G30, 18G55; Secondary: 13D40 James M. Turner 1395 Mathematical Sciences Building Purdue University West Lafayette, IN 47907-1395 jmt@ziplink.net October 1997; Revised: June 12, 1998 In this paper, we study the Andr\'e-Quillen homology of simplicial commutative F-algebras, where F is a field of positive characteristic, with certain vanishing properties.
		We will show, under certain conditions on $\pi_0$ and the vanishing of the homotopy groups, that the vanishing of Andr\'e-Quillen homology implies that the simplicial commutative F-algebra in question is a homology complete intersection.
	www.lehigh.edu /~dmd1/h617 (1576 words)

	Publication-Weiping Li (Site not responding. Last check: 2007-11-06)
		Floer homology for connected sums of homology spheres, J.
		Floer homology of rational homology 3-spheres (with R.
		Instanton Floer homology for connected sums of Poincare spheres PS file, PDf file.
	www.math.okstate.edu /~wli/research/publication.html (591 words)

	Amazon.ca: Algebraic Topology : Homotopy and Homology: Books (Site not responding. Last check: 2007-11-06)
		He assumes only a modest knowledge of algebraic topology on the part of the reader to start with, and he leads the reader systematically to the point at which he can begin to tackle problems in the current areas of research centered around generalized homology theories and their applications.
		After majoring in mathematics at Harvard College, he completed his PhD at Stanford University in 1965.
		In the early 1980s his research concentrated on obstruction theory in connection with holomorphic bundles on projective spaces.
	www.amazon.ca /exec/obidos/ASIN/3540427503 (535 words)

	Math on the Web: Electronic Mathematical Research Journals
		These include electronic-only journals of purely mathematical research.
		Homology, Homotopy and Applications (Razmadze Mathematical Institute, Tbilisi Georgia)
		Journal of Inequalities in Pure and Applied Mathematics (Victoria University of Technology)
	www.ams.org /mathweb/mi-journals2.html (329 words)

	Mathematics Journals
		Journal of the European Mathematical Society [older issues]
		Journal of the Institute of Mathematics of Jussieu
		Journal of Mathematical Sciences, The University of Tokyo
	www.math.neu.edu /~suciu/journals.html (123 words)

	Mathematics: Subject Guide: NCSU Libraries
		This page collates information about professional mathematics for the mathematics community at NCSU.
		Two powerful ways to search are by Library of Congress Subject Headings and by Call Number.
		Math books are located in the QA call number area on the 6th floor of D. Hill.
	www.lib.ncsu.edu /guides/mathematics (141 words)

	Books and proceedings
		Algèbres associatives et de Hopf, homologies et cohomologies usuelles.
		CBMS-NSF Regional Conference Series in Applied Mathematics, 1972.
		All the ressources below are referenced through direct links, for easier use as a library of introductory and comprehensive papers on various subjects.
	nivea.psycho.univ-paris5.fr /~philipona/Biblio/Category/books.html (167 words)

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