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	Homotopy - Wikipedia, the free encyclopedia
		An outstanding use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology.
		Clearly, every homeomorphism is a homotopy equivalence, but the converse is not true: a solid disk is not homeomorphic to a single point.
		In practice homotopy theory is carried out by working with CW complexes, for technical convenience; or in some other reasonable category.
	en.wikipedia.org /wiki/Homotopy (1019 words)

	Homotopy theory (Site not responding. Last check: 2007-10-21)
		Theory of Motives, Homotopy Theory of Varieties, and Dessins d'Enfants AIM Research Conference Center (ARCC), Palo Alto, CA, USA; 12--15 March 2004.
		Homotopy theory, simplicial groups, configuration spaces and braid groups, modular representation theory of symmetric groups.
		Applied Homotopy Theory Fields Institute Program at the University of Western Ontario, Canada; September 2003.
	www.serebella.com /encyclopedia/article-Homotopy_theory.html (319 words)

	Category Theory and Homotopy Theory (Site not responding. Last check: 2007-10-21)
		Category theory was introduced in 1947 to give a richer language than that of set theory, which would be better able to express the structures of homotopy and homology theory then being revealed in the work of Cartan, Eilenberg, Mac Lane, Whitehead and others.
		This language and theory was soon found to have great usefulness in other branches of pure mathematics such as algebra, algebraic geometry, logic and more recently in computer science.
		The basic areas of research in category theory at Bangor are directed towards achieving a greater understanding of the categorical structure and interrelationships between the various objects studied by algebraic topology and homological algebra.
	www.informatics.bangor.ac.uk /public/mathematics/research/cathom/cathom1.html (222 words)

	Books on Douglas C. Ravenel (Site not responding. Last check: 2007-10-21)
		Nilpotence and Periodicity in Stable Homotopy Theory describes some major advances made in algebraic topology in recent years, centering on the nilpotence and periodicity theorems, which were conjectured by the author in 1977 and proved by Devinatz, Hopkins, and Smith in 1985.
		This book is the result of a conference held to examine developments in homotopy theory in honor of Samuel Gitler in August 1993 (Cocoyoc, Mexico).
		Homotopy Theory and Its Applications offers a distinctive account of how homotopy-theoretic methods can be applied to a variety of interesting problems.
	books.bankhacker.com /Douglas+C.+Ravenel (293 words)

	Open problems in homotopy theory of Lie groups
		A major open problem in the homotopy theory of Lie groups is the classification of 2-compact groups, analogous to the classification of compact Lie groups.
		The method of proof employed in [2] for odd primes uses obstruction theory and shows promise of being extendable to p=2, although much work remains to be done....
		The theory of p-local finite groups is a novel theory which aims to formalize the p-local structure found in finite groups, in much the same way as p-compact groups does it for compact (connected) Lie groups.
	www.lehigh.edu /~dlj0/Grodal-problemsession.html (694 words)

	Motivic Homotopy Theory Program (Site not responding. Last check: 2007-10-21)
		The motivic homotopy theory is the homotopy theory for algebraic varieties and, more generally, for Grothendieck's schemes which is based on the analogy between the affine line and the unit interval.
		Eventually, the motivic homotopy theory is expected to provide techniques which may help to solve problems in algebraic geomerty such as various "standard conjectures on algebraic cycles", Beilinson-Soule vanishing and rigidity conjectures, the Bloch-Kato conjecture etc.
		The theory of these categories is closely related to the theory of motivic cohomology pioneered by A.
	www.math.ias.edu /~vladimir/seminar.html (870 words)

	Homotopy Theory I (Site not responding. Last check: 2007-10-21)
		Homotopy theory originated as a branch of topology in which one studied the ways in which geometrical shapes can be deformed.
		As it progressed the methods became quite sophisticated, and general theory was developed to handle the machinery.
		The theory clarified the logical underpinnings, and has led to the expansion of homotopy theoretic thinking to many other areas of mathematics, most notably in algebra and algebraic geometry.
	jdc.math.uwo.ca /hothy-2000 (386 words)

	Parametrized homotopy theory, by J. P. May and J. Sigurdsson
		The essential point is to resolve problems in the homotopy theory of ex-spaces that have no nonparametrized counterparts.
		Instead, a rather intricate blend of model theory and classical homotopy theory is required.
		This allows application of the formal theory of duality in symmetric monoidal categories to the construction and analysis of transfer maps.
	www.mathematik.uni-osnabrueck.de /K-theory/0716 (337 words)

	Amazon.ca: Books: Rational Homotopy Theory (Site not responding. Last check: 2007-10-21)
		As the authors explain eloquently, the (computational) power of rational homotopy theory comes from its algebraic formulation, which was first discussed by Sullivan and the mathematician Daniel Quillen, and involves the use of graded objects with both an algebraic structure and a "differential".
		The rational homotopy types of a space are then in bijective correspondence to isomorphism classes of "minimal" Sullivan algebras, and the homotopy classes of maps between rational spaces are in bijective correspondence to homotopy classes of maps between minimal Sullivan algebras.
		The authors construct the "homotopy Lie algebra" of a simply connected topological space, which is the homotopy group of the loop space tensored with the ground field, and the homotopy Lie algebra of a minimal Sullivan algebra.
	www.amazon.ca /exec/obidos/ASIN/0387950680 (1209 words)

	MH RTN (Site not responding. Last check: 2007-10-21)
		Rational homotopy was always partially motivated by geometric problems; there is a greater realisation that its techniques extend to the "non-rational" world and its geometric applications are again at the forefront of the subject.
		(a) The Goodwillie Calculus of Functors with applications to K-theory, Embedding theory and Homotopy theory, (b) The theory of p-compact groups and its generalisations to the homotopy theory of discrete groups, (c) Stable homotopy theory and its geometric applications.
		BARCELONA (CRM): p-compact groups, homotopy theory of discrete groups, Kac-Moody groups, Steenrod algebra, unstable modules and algebras, H-spaces, homotopical localisation, elliptic genera of manifolds, generalised homology theories, rational homotopy theory, LS-category.
	www.shef.ac.uk /~pm1jg/mhrtn/rtn.html (759 words)

	Algebraic Topology - Cambridge University Press
		This introductory textbook in algebraic topology is suitable for use in a course or for self-study, featuring broad coverage of the subject and a readable exposition, with many examples and exercises.
		The four main chapters present the basic material of the subject: fundamental group and covering spaces, homology and cohomology, higher homotopy groups, and homotopy theory generally.
		Homotopy properties of CW complexes cellular approximation; 63.
	books.cambridge.org /052179160X.htm (335 words)

	Citations: adic homotopy theory - Mandell, -algebras (ResearchIndex) (Site not responding. Last check: 2007-10-21)
		) p adic homotopy theory in terms of cochain algebras, but that is totally unknown territory at present.
		The singular cochain functor with coe cients in Z p induces a contravariant equivalence from the homotopy category of connected nilpotent p complete spaces of nite p type to a full subcategory of the homotopy category of E1 Z p algebras.
		Consequently, a characterization of integral homotopy does not lead to a p local homotopy theory: In killing o all primes other than p, one also kills o crucial information needed to compute the cobar construction of a space.
	citeseer.ist.psu.edu /context/1289109/0 (593 words)

	Algebraic homotopy, Galois theory and Descent
		Algebraic homotopy concerns the use of both algebraic models for homotopy types and homotopy theory applied to algebraic objects.
		As one would expect, mixes of Galois theory and descent are being applied back in algebraic geometric settings with results on ringed topoi, whilst work by Brown continues to look at essential obstructions to extending local to global information measurable by monodromy groupoids generalising the fundamental group.
		The corresponding Galois theory and resulting algebraic homotopy is being studied at Bangor, not only for the potential application within equivariant homotopy and the theory of orbifolds, but also as a test bed for methods relating to Grothendieck's programme.
	www.informatics.bangor.ac.uk /public/mathematics/research/cathom/intas99.html (1107 words)

	INI Programme NST (Site not responding. Last check: 2007-10-21)
		This programme aims to consolidate the advances within homotopy theory itself which have led to these results, to open the way to substantial further developments within the subject, to expose a diversity of new applications and to bring practitioners of these subjects into contact with each other and with practising homotopy theorists.
		Recent developments have culminated in several stable homotopy constructions which are homotopy theoretic enrichments of the category of abelian groups (``spectral algebra'').
		These new stable homotopy categories are useful for studying a wide range of phenomena, from algebraic K-theory and arithmetic to the elliptic cohomology phenomena introduced by Witten.
	www.newton.cam.ac.uk /programmes/NST (262 words)

	Bert Guillou (Site not responding. Last check: 2007-10-21)
		Now usually referred to as motivic homotopy theory, this is a program begun by Fabien Morel and Vladimir Voevodsky (see the webpage for the motivic homotopy theory seminar held a few years ago at the Institute for Advanced Study).
		The idea is to develop a "homotopy theory" for smooth schemes over a nice (Noetherian) base S, in which the affine line A^1 would play the role of the unit interval.
		Once the theory has been set up, one can then use the heavy machinery from homotopy theory to answer questions in algebra and algebraic geometry.
	www.math.uchicago.edu /~bertg/info.html (166 words)

	Amazon.ca: Books: Axiomatic Stable Homotopy Theory (Site not responding. Last check: 2007-10-21)
		Some of these arise in topology (the ordinary stable homotopy category of spectra, categories of equivariant spectra, and Bousfield localizations of these), and others in algebra (coming from the representation theory of groups or of Lie algebras, as well as the derived category of a commutative ring).
		This book presents stable homotopy theory as a branch of mathematics in its own right with applications in other fields of mathematics.
		It is a first step toward making stable homotopy theory a tool useful in many disciplines of mathematics.
	www.amazon.ca /exec/obidos/ASIN/0821806246 (378 words)

	Graduiertenkolleg Homotopy and Cohomology
		The power of Homotopy Theory lies in this invariance up to homotopy: often it allows one to replace complicated objects by simple models of them.
		The strategy of Homotopy Theory is to establish this invariance for as many properties as possible, and then exploit the invariance to obtain a classification.
		The areas of Homotopy Theory in question are Stable Homotopy Theory and Elliptic Cohomology.
	www.math.uni-bonn.de /people/GRK1150 (404 words)

	From Representation Theory to Homotopy Groups (Site not responding. Last check: 2007-10-21)
		Bousfield recently gave a formula for the odd-primary v1-periodic homotopy groups of a finite H-space in terms of its K-theory and Adams operations.
		In this paper, we apply Bousfield's theorem to give explicit determinations of the v1-periodic homotopy groups of (E8,5) and (E8,3), thus completing the determination of all odd-primary v1-periodic homotopy groups of all compact simple Lie groups, a project suggested by Mimura in 1989.
		Its determinant is closely related to the homotopy decomposition of E8 localized at each prime.
	www.lehigh.edu /~dmd1/e8.html (202 words)

	DOCUMENTA MATHEMATICA, Vol. 8 (2003), 409-488 (Site not responding. Last check: 2007-10-21)
		In this paper we employ enriched category theory to construct a convenient model for several stable homotopy categories.
		This is achieved in a three-step process by introducing the pointwise, homotopy functor and stable model category structures for enriched functors.
		The general setup is shown to describe equivariant stable homotopy theory, and we recover Lydakis' model category of simplicial functors as a special case.
	www.mathematik.uni-bielefeld.de /documenta/vol-08/13.html (123 words)

	Fields Institute - Homotopy Theory Program 1995-96
		During 1995-96 the Fields Institute is sponsoring an emphasis year in homotopy theory.
		Course will focus on the use of invariant theory in understanding the cohomology of classifying spaces: invariant theory and pseudo- reflection groups, mod p polynomial cohomology algebras, the Lannes T functor, homotopy fixed point theory, Dwyer-Wilkerson study of p compact groups, homotopy colimit decompositions of classifying spaces
		The major activities will be workshops in unstable homotopy theory during the week of May 27-31 and in rational homotopy theory during the week of June 3-7.
	www.fields.utoronto.ca /programs/scientific/95-96/homotopy (360 words)

	Harvard Gazette: Michael Hopkins, algebraic topologist
		Hopkins has been a major player in recent developments linking algebraic topology to issues in number theory, such as elliptic curves and modular forms, and contemporary physics, such as string theory.
		His work has also brought homotopy theory - specifically, understanding of the homotopy groups of spheres, one of the central problems in topology - into closer contact with other fields of mathematics.
		Born in Alexandria, Va., Hopkins received a B.A. in mathematics from Northwestern University in 1979, a Ph.D. in mathematics from Northwestern in 1984, and a D.Phil.
	www.news.harvard.edu /gazette/2005/04.28/03-hopkins.html (564 words)

	Rational Homotopy Theory (Site not responding. Last check: 2007-10-21)
		Rational homotopy theory is analogous to the study of linear algebra versus general ring and module theory.
		This, along with related methods of localization and completions, marked a change in the approach to the sudy of homotopy types of topological spaces.
		The rational homotopy type of a space X provides the backbone, to which more detailed information concerning various primes is attached to build a picture of the homotopy type of X. The techniques involved in studying rational homotopy theory are simpler and more algebraic than those needed in traditional algebraic topology.
	www.math.purdue.edu /~wilker/rational.html (214 words)

	Problems on axiomatic stable homotopy theory
		In our memoir, we give a conjecture for the thick subcategories in a Noetherian stable homotopy category C--they should be in 1-1 correpondence with subsets of Spec pi_* S closed under specialization.
		This has been done for categories that are homotopy categories of model categories by Schwede, I think, and a lot is known in general.
		This theory takes values in an abelian category that is not modules over a ring--I think it is sheaves over a scheme or something.
	claude.math.wesleyan.edu /~mhovey/problems/axiomatic.html (681 words)

	K-theory Preprint Archives
		450: September 28, 2000, Stacks and the homotopy theory of simplicial sheaves, by J.F. Jardine.
		324: December 18, 1998, Cyclic cohomology of Hopf algebras, and a non-commutative Chern-Weil theory, by Crainic Marius.
		286: June 19, 1998, Hopf algebras, cyclic cohomology and the transverse index theory, by Alain Connes and Henri Moscovici.
	www.mathematik.uni-osnabrueck.de /K-theory (11470 words)

	ARCC Workshop: Theory of motives, homotopy theory of varieties, and dessins d'enfants (Site not responding. Last check: 2007-10-21)
		ARCC Workshop: Theory of motives, homotopy theory of varieties, and dessins d'enfants
		Theory of motives, homotopy theory of varieties, and dessins d'enfants
		The major component of the meeting will be expository lectures on the different areas, at a level which could be appreciated by mathematicians outside the specialized area.
	aimath.org /ARCC/workshops/motivesdessins.html (197 words)

	Arithmetic Invariants and Periodicity in Stable Homotopy Theory (ResearchIndex) (Site not responding. Last check: 2007-10-21)
		Abstract: eories and the Adams spectral sequence A homology theory on finite CW complexes is a covariant functor () for which f g : X Y induce f = g : E and there are Mayer--Vietoris sequences for cofibrations.
		A multiplicative homology theory E () is one where there are natural pairings E (X Y) and E (X) is naturally a module over the graded commutative ring E = E).
		15 Nilpotence and stable homotopy theory (context) - Devinatz, Hopkins et al.
	citeseer.ist.psu.edu /616502.html (336 words)

	A^1-homotopy theory of schemes, by Fabien Morel and Vladimir Voevodsky (Site not responding. Last check: 2007-10-21)
		A^1-homotopy theory of schemes, by Fabien Morel and Vladimir Voevodsky
		In this paper we begin to develope a machinery which we call A^1-homotopy theory of schemes.
		All our constructions are based on the intuitive feeling that if the category of algebraic varieties is in any way similar to the category of topological spaces then there should exist a homotopy theory of algebraic varieties where the affine line plays the role of the unit interval.
	www.math.uiuc.edu /K-theory/0305 (84 words)

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