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Topic: Stable homotopy theory

	Stable homotopy theory - Wikipedia, the free encyclopedia
		In mathematics, stable homotopy theory is a branch of algebraic topology.
		In general, stable homotopy theory tries to isolate the phenomena of homotopy theory that are essentially unchanged after sufficiently many applications of suspension, or then become more perspicuous.
		One of the most important problems in stable homotopy theory is the computation of stable homotopy groups of spheres.
	en.wikipedia.org /wiki/Stable_homotopy_theory (141 words)

	Bott periodicity theorem - Wikipedia, the free encyclopedia
		In mathematics, the Bott periodicity theorem is a result from homotopy theory discovered by Raoul Bott during the latter part of the 1950s, which proved to be of foundational significance for much further research, in particular in K-theory of stable complex vector bundles, as well as the stable homotopy groups of spheres.
		The context of Bott periodicity is that the homotopy groups of spheres, which would be expected to play the basic part in algebraic topology by analogy with homology theory, have proved elusive (and the theory is complicated).
		The subject of stable homotopy theory was conceived as a simplification, by introducing the suspension (smash product with a circle) operation, and seeing what (roughly speaking) remained of homotopy theory once one was allowed to suspend both sides of an equation, as many times as one wished.
	en.wikipedia.org /wiki/Bott_periodicity_theorem (734 words)

	Graduate School : Pure Mathematics (Site not responding. Last check: 2007-10-31)
		The classifying spaces of groups are objects in modern algebraic topology, in particular in the branch of the subject called homotopy theory.
		In fact, many large and important spaces split into pieces for the purposes of stable homotopy theory (the most important family of these decompositions is called the Snaith splittings) and, accordingly, cohomology questions concerning classifying space may be split into smaller pieces too.
		The formula has many applications to topology, algebraic geometry, algebraic K-theory, number theory (where it was used to solve a 1978 conjecture of M J Taylor), to ramification theory, representation theory (where it was used to solve a problem posed by R Brauer in 1946, and to answer a 1960 question of J P Serre).
	www.maths.soton.ac.uk /graduate/pure/topology.shtml (323 words)

	New Contexts for Stable Homotopy Theory
		Stable homotopy theory is the ultimate context in which to perform the type of conversion from geometrical to algebraic data which Poincaré began.
		This exploitation of motivic homotopy theory in number theory and algebraic topology was spurred on by the programme, which also played an important role in spreading the developing body of knowledge.
		The subject of stable homotopy theory has been transformed in the last ten years by key technical advances making distant dreams into reality, but the fact that its methods have also been used in recent spectacular progress in motivic homotopy theory was a quite separate development.
	www.newton.cam.ac.uk /reports/0203/nst.html (2154 words)

	[No title] (Site not responding. Last check: 2007-10-31)
		The objective is to enhance our understanding of fundamental problems in both subjects, including the K-theory of fields and rings of integers, the nature of homotopy inverse limits, and the generating hypothesis in stable homotopy theory.
		Secondary to this main project is a study of certain types of Hopf algebras, and a study of the question of recovering homotopy invariants of a space from its cochains.
		More basically, the projects of Mitchell, Goerss, and Devinatz all involve the relationship between the field of algebra -- especially the theory of numbers -- and the more "geometric" field of homotopy theory, which may be defined as the study of phenomena that remain unchanged under continuous deformations.
	www.cs.utexas.edu /users/yguan/NSFAbstracts/Abstracts/MPS/DMS.MPS.a9504476.txt (425 words)

	INI Programme NST (Site not responding. Last check: 2007-10-31)
		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)

	[No title] (Site not responding. Last check: 2007-10-31)
		Geometrically, this is formulated as an equivalence statement between the algebraic K-theory spectrum of the field and the so-called ``homotopy fixed set'' of the action of the absolute Galois group on the algebraic K-theory spectrum of the algebraic closure.
		In the course of these collaborations, it became clear that one extremely important element in the resolution of these conjectures would be the further development of equivariant stable homotopy theory in some particular directions.
		Equivariant stable homotopy theory has been a key ingredient in the description of various homotopy fixed point spectra under finite group actions, in particular in the Atiyah-Segal completion theorem and in the affirmative resolution of Segal's Burnside ring conjecture.
	www.aimath.org /projects/equivar.html (1330 words)

	[No title] (Site not responding. Last check: 2007-10-31)
		This is a theory for algebraic vareities over an arbitrary base field which is quite analogous to stable homotopy theory for topological spaces.
		That is, the homotopy groups of the model spectrum for algebraic K-theory constructed by Carlsson agree with the homotopy groups coming out of the motivic approach for these absolute Galois groups.
		The development of equivariant stable homotopy theory in the motivic context is also a high priority, and will be addressed at a workshop which will be held at Stanford in August of 2000, with support from AIM, Stanford University, and the NSF.
	www.aimath.org /projects/madsen.html (1084 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.
		Suppose G is a self-equivalence of the stable homotopy category.
	claude.math.wesleyan.edu /~mhovey/problems/axiomatic.html (681 words)

	University of Oregon: Department of Mathematics
		I like to take recent theories developed in algebaic topology, in particular the calculus of functors and equivariant stable homotopy theory, and apply them to answer concrete questions about manifolds, in particular about knots and group actions.
		With the modern development of equivariant stable homotopy theory it is possible to carry this classical approach to group actions much further than in its first development.
		Because transversality does not always hold in the equivariant world, geometric bordism theory differs from homotopical bordism theory and their interplay is a central theme in the subject.
	darkwing.uoregon.edu /~dps/respage.php (1184 words)

	Adams_Frank biography
		His research continued to be of fundamental importance in the homotopy theory of classifying spaces of
		Stable homotopy theory (1964) is a short 74 page book which is based on six lectures Adams gave at the University of California at Berkeley in 1961.
		Stable homotopy theory and generalized homology (1974) comprises of three lecture courses, one on the algebra of stable operations in complex cobordism delivered in 1967, the second on complex cobordism theory delivered in 1970, and the third on stable homotopy and generalized homology theories delivered in 1971.
	www-history.mcs.st-andrews.ac.uk /Biographies/Adams_Frank.html (1450 words)

	Nilpotence and Periodicity in Stable Homotopy Theory
		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.
		The book begins with some elementary concepts of homotopy theory that are needed to state the problem.
		Stable homotopy theory most traditionally concerns itself with the study of groups $\{Y,Z\}$, the group of homotopy classes of stable maps between spaces $Y$ and $Z$, particularly when $Y$ and $Z$ are finite cell complexes.
	www.math.rochester.edu /people/faculty/doug/nilp.html (1015 words)

	[No title]
		Furthermore, this last Lie algebra is given by the homotopy groups modulo torsion of the loop space of the complement of the subspace arrangement.
		Thanks to Hirschhorn's localization technology, we can construct the stable model structure on ordinary spectra with almost no hypotheses on C and G. A new feature of this revision is that we show that, under strong smallness hypotheses on G and C, the stable equivalences coincide with the appropriate generalization of stable homotopy isomorphisms.
		In particular, this is a new construction of the stable model structure on simplicial symmetric spectra.
	www.lehigh.edu /~dmd1/h616 (960 words)

	RESEARCH
		In my thesis (at the University of Washington under the supervision of John Palmieri) I have studied the refinements of chromatic towers and Krull-Schmidt type decompositions for thick subcategories in various stable homotopy categories.
		We begin with an introduction to the stable homotopy category of spectra, and then talk about the celebrated thick subcategory theorem and discuss a few applications to the structure of the Bousfield lattice.
		Abstract: We study the triangulated subcategories of compact objects in stable homotopy categories such as the homotopy category of spectra, the derived categories of rings, and the stable module categories of Hopf algebras.
	www.math.uwo.ca /~schebolu/research.html (1046 words)

	Project Description
		The development of the theory for ``brave new rings'' was historically motivated by concrete applications in nearby areas of mathematics, such as the study of topological and differentiable symmetry groups of manifolds in geometric topology (Waldhausen).
		Such a theory can probably (only) be constructed and analyzed by topological methods, which by now are well developed, and should contain extremely subtle information about ``chromatic'' periodic phenomena in stable homotopy theory that correspond to heights up to 10.
		The theory of brave new rings generalizes the classical theory for rings, in that each ring R in the algebraic sense gives rise to a brave new ring HR (which represents singular cohomology with coefficients in R).
	folk.uio.no /rognes/cas/application.html (4197 words)

	The stable homotopy category at 2 (Site not responding. Last check: 2007-10-31)
		The stable homotopy category has been extensively studied by algebraic topologists for a long time.
		However, passage to the homotopy category looses information and in general the `homotopy theory' can not be recovered from the homotopy category.
		In this paper we show that in contrast to the general case, the stable homotopy category completely determines the stable homotopy theory, at least 2-locally.
	wwwmath.uni-muenster.de /u/stefan.schwede/twolocal.html (343 words)

	Fields Institute - Homotopy Theory Program 1995-96
		During 1995-96 the Fields Institute is sponsoring an emphasis year in homotopy theory.
		What follows is information concerning various aspects of the homotopy program which will be offered after the current January stable homotopy emphasis session ends.
		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)

	CAT'01 abstracts
		We study the homotopy of a quasi-projective variety in a complex projective space following Lefschetz's method, that is, by considering its sections by the hyperplanes of a pencil.
		The theory of toric varieties is a bridge between algebraic geometry and combinatorics.
		The classical braid groups are ubiquitous in modern mathematics, with applications from the theory of operads to the study of the Galois group of the rationals.
	www.mimuw.edu.pl /~cat01/abstracts.html (4304 words)

	Modern Foundations For Stable Homotopy Theory - Elmendorf, Kriz, Mandel, May (ResearchIndex) (Site not responding. Last check: 2007-10-31)
		5 Commutative algebra in stable homotopy theory and a completi..
		4 and algebras in stable homotopy theory (context) - Elmendorf, Kriz et al.
		Stable Splittings For Classifying Spaces Of Alternating, Special..
	citeseer.ist.psu.edu /33579.html (697 words)

	Alan Robinson's bibliography
		Stable homotopy theory over a fixed base space.
		Obstruction theory and the strict associativity of Morava K-theories.
		A.N. Dranishnikov, Spanier-Whitehead duality and the stability of the intersection of compacta (trans.
	www.maths.warwick.ac.uk /~car/biblio.html (319 words)

	Citebase - A uniqueness theorem for stable homotopy theory (Site not responding. Last check: 2007-10-31)
		In this paper we study the global structure of the stable homotopy theory of spectra.
		One sufficient condition is that the associated homotopy category is equivalent to the stable homotopy category as a triangulated category with an action of the ring of stable homotopy groups of spheres.
		In other words, the classical stable homotopy theory, with all of its higher order information, is determined by the homotopy category as a triangulated category with an action of the stable homotopy groups of spheres.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0012021 (204 words)

	[No title]
		This theorem is a 2-local strenghtening of a result with B. Shipley, given in `A uniqueness theorem for stable homotopy theory', in that we use only the triangulated structure of the stable homotopy catgory.
		One sufficient condition is that the associated homotopy category is equivalent to the stable homotopy category as a triangulated category with an action of the ring of stable homotopy groups of spheres, $\pi^s$.
		In other words, the classical stable homotopy theory, with all of its higher order information, is determined by the homotopy category as a triangulated category with an action of $\pi^s$.
	www.lehigh.edu /~dmd1/h119 (1344 words)

	Deformations of formal groups and spectral sheaves in stable homotopy
		Using these results together with Landweber's Exact Functor Theorem we construct an uncountable family of deformations of cohomology theories that might be considered as deformations of complex K-theory.
		The second part which represent joint work with J. Block is concerned with stable homotopy theory of sheaves.
		It is proved that the category of sheaves of spectra and a topological space admits (under some mild restrictions on the space) the structure of a closed model category in the sense of Quillen.
	repository.upenn.edu /dissertations/AAI9800887 (287 words)

	DOCUMENTA MATHEMATICA, Vol. 8 (2003), 409-488 (Site not responding. Last check: 2007-10-31)
		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.univie.ac.at /EMIS/journals/DMJDMV/vol-08/13.html (123 words)

	K-theory Calendar
		Toronto, March 26 - 30, 2007, Workshop on the homotopy theory of schemes.
		Palo Alto, April 23 - 26, 2004, Theory of motives, homotopy theory of varieties, and dessins d'enfants.
		Berkeley, Homotopy theory for algebraic varieties with applications to K-theory and quadratic forms, Mathematical Sciences Research Institute, May 12-16, 1998.
	www.mathematik.uni-osnabrueck.de /K-theory/Calendar (1242 words)

	preprints (Site not responding. Last check: 2007-10-31)
		The second part of this book, `An Introduction to Fibrewise Stable Homotopy Theory', was written specifically with geometric applications in mind.
		Tony Potter and I worked on this theory in the late 1980s, but little of the work was published at that time.
		Methods of equivariant fibrewise stable homotopy theory, and fibrewise homology theory, have also been used in joint work with Sven Bauer (a research student at Aberdeen) and Mauro Spreafico (who spent 18 months at Aberdeen as a Marie Curie Research Fellow in 1998-9 and is now in Milan).
	www.maths.abdn.ac.uk /~crabb/res.htm (273 words)

	[No title]
		I work mostly in stable homotopy theory, which is a part of algebraic topology.
		One area of emphasis is the conversion of problems in homotopy theory to problems in algebraic geometry.
		Earlier, partial versions of these foundations were used in my study of the Morava E-theory of symmetric groups [7,3] using the theory of finite subgroups of formal groups [5].
	neil-strickland.staff.shef.ac.uk /lmscv.html (1085 words)

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