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Topic: Spectrum homotopy theory

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homotopy theory

homotopy groups

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Homotopy lifting property

homotopy class

S duality homotopy theory

stable homotopy theory

scheme homotopy theory

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Homotopy groups of spheres

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	Spectrum (disambiguation) - Wikipedia, the free encyclopedia
		The spectrum of a matrix is a concept used in linear algebra
		The spectrum of an operator is a concept used in functional analysis (a generalisation of the spectrum of a matrix)
		The spectrum of a theory is a concept in mathematical logic
	en.wikipedia.org /wiki/Spectrum_(disambiguation) (265 words)

	Motivic Homotopy Theory Program (Site not responding. Last check: 2007-10-29)
		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)

	[No title]
		It boasts fascinating connections with homotopy theory, string theory, elliptic curves, modular forms, and the mysterious ubiquity of the number 24.
		After all, he became famous for his work on homotopy theory, he invented the axioms of conformal field theory - borrowing lots of ideas from string theory, of course - and I'm sure he mastered the theory of elliptic curves one weekend when he was a kid.
		In the context of homotopy theory, this is almost as good as an abelian group.
	www.math.niu.edu /~rusin/known-math/00_incoming/gen_coho (3543 words)

	[No title] (Site not responding. Last check: 2007-10-29)
		For example, in homotopy theory, which is the study of mathematical phenomena that remain invariant under continuous deformation, one wants to decide if an object has a multiplication that is commutative up to all ``higher homotopies'' -- this is the type of multiplication that is invariant under deformation.
		The broad goal of this project is to further understanding of homotopy theory, the branch of topology concerned with properties of higher-dimensional surfaces and other geometric objects that remain invariant under continuous deformation.
		There are some surprising and rather mysterious connections between homotopy theory and seemingly unrelated fields of mathematics such as number theory and algebraic geometry, and one hopes the project will ultimately illuminate these.
	www.cs.utexas.edu /users/yguan/NSFAbstracts/Abstracts/MPS/DMS.MPS.a9977089.txt (143 words)

	[No title]
		Parametrized homotopy theory is a natural and important part of homotopy theory that is implicitly ce* ntral to all of bundle and fibration theory*.
		Ther* e is a "classical" homotopy* theory based on homotopy equivalences, and there is a mo* re fundamental "derived" homotopy* theory based on a weaker notion of equivalence than that of homotopy equivalence.
		It is perhaps well understood that both homotopy theories ca* n be expressed in terms of model structures on the underlying category, but this asp ect of the classical homotopy* theory has usually been ignored in the model theoreti* *cal literature, a tradition that goes back to Quillen's original paper [83].
	www.math.purdue.edu /research/atopology/May-Sigurdsson/MSMaster.txt (21075 words)

	[No title] (Site not responding. Last check: 2007-10-29)
		The theory of mechanics developed in the ninteenth century was based on 'principles of least action': a ray of light, for example, follows the path which minimizes its time of flight.
		Unfortunately, the theory of such Feynman path integrals has never been made rigorous; indeed, it is now known that no naive generalization of the classical theory of integration can form an adequate basis for the integrals which arise in modern physics.
		In geometry, however, it has become clear recently that ideas from the theory of Feynman integrals can be used to solve classical problems of pure mathematics, and there is evidence that many geometric problems are in some sense 'tame' enough so that an analogue of the theory of Feynman integrals can be established rigorously.
	www.cs.utexas.edu /users/yguan/NSFAbstracts/Abstracts/MPS/DMS.MPS.a9504234.txt (383 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)

	Nick Kuhn, University of Virginia
		This paper is really a paper on representation theory and Mackey functors; thanks to G. Carlsson's work on the Segal Conjecture, it has definitive topological interpretation.
		One novelty was the use of modular representation theory of finite semigroups.
		The three-paper series develops the modular representation theory of the general linear groups over finite fields from a certain categorical 'generic' point of view.
	www.math.virginia.edu /Faculty/Kuhn (1090 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]
		A homology theory on a triangulated category S is an exact functor to an Abelian category which preserves the coproducts that exist in S. Unless we state otherwise, the target category will always be taken to be the category Ab of Abelian groups.
		It is shown in [15, Section 4] that a homology theory defined on F has an essentially unique extension to a homology theory defined on all of S, so the categories of homology theories on F and S are equivalent.
		A cohomology theory with values in an Abelian category B is a homology theory with values in Bop.
	jdc.math.uwo.ca /papers/phantoms.txt (11288 words)

	[No title] (Site not responding. Last check: 2007-10-29)
		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)

	Barnes & Noble.com - Books: Wavelets through a Looking Glass, by Ola E. Bratteli, Hardcover
		The authors have demonstrated further connections with spectral theory, ergodic theory, homotopy theory and the theory of probability---just to name a few of the well-established areas of mathematics which are shown to touch the theory of wavelets.
		On the one hand the goal is to give a modern (but 'timeless') presentation of wavelet theory while on the other hand the goal is to present new results that have not previously been published.
		The latter include material on homotopy of resolutions, approximation theory and results on the spectrum of the associated transfer operators and subdivision operators.
	search.barnesandnoble.com /booksearch/isbnInquiry.asp?isbn=0817642803 (790 words)

	[No title]
		Let G be a closed subgroup of the profinite group Gn, the group of ring spectrum automorphisms of En in the stable homotopy catego* *ry.
		We define homotopy fixed points for towers of discrete G-spectra; we show that these homotopy fixed points are the total right derived functor of fixed points* * in the appropriate sense; and we construct the associated descent spectral sequence.
		Homotopy fixed points for discrete G-spectra are defined in x5, and x6 shows that En is a continuous Gn-spectrum, proving the first half of Theorem 1.3.
	hopf.math.purdue.edu /DavisDaniel/p1v5ams.txt (4968 words)

	[No title]
		The "spectrum" of an observable A is the set of values it's allowed to have, and mathematically this is the set of numbers x such that the operator A - x has no inverse.
		In topology, a "spectrum" is defined to be a sequence of pointed topological spaces, each of which is homeomorphic to the space of all based loops in the next.
		The corresponding spectrum is called the "sphere spectrum" and the corresponding generalized cohomology theory is called "stable homotopy theory".
	math.ucr.edu /home/baez/twf_ascii/week199 (4337 words)

	Research interests of Don Davis (Site not responding. Last check: 2007-10-29)
		I work in homotopy theory, which is a branch of algebraic topology.
		One important advance was a 50-page paper, published in Memoirs of the AMS in 2002 detailing the calculation of the v1-periodic homotopy groups of (E8,5) and (E8,3) by a new method pioneered by Bousfield.
		My Topology paper with Mahowald on the Image of the J homomorphism in the stable homotopy groups of spheres is considered by many to be the definitive work on that topic, which served as a focal point for much of algebraic topology in the 1960s.
	www.lehigh.edu /~dmd1/res.html (452 words)

	theory (Site not responding. Last check: 2007-10-29)
		Einstein's theory of gravitation languished on the sidelines for nearly half a century after the initial flurry of 1919, because relativistic deviations from Newtonian predictions were, in most cases, immeasurably small under all astrophysical conditions then conceivable.
		Today the theory is central to our understanding of some of the most exotic realms of science, from fl holes to the evolution of the cosmos fractions of a second after the big bang.
		In particular, by studying the effective field theory limits of string theories, necessarily result in field theories that are internally consistent: they are free of problematic anomalies.
	www.phys.uvic.ca /research/theory (1378 words)

	Springer Online Reference Works
		More generally, pairings of two generalized cohomology theories into a third may be defined [5].
		, and the universality of cobordisms is reflected in the fact that the formal group of the theory of unitary cobordism is universal (purely algebraically) in the class of all formal groups.
		There is a natural problem of "comparing" different generalized cohomology theories, and, in particular, the problem of expressing one cohomology theory in terms of another.
	eom.springer.de /g/g043780.htm (1103 words)

	Apr 3-7 VU Math Events (Site not responding. Last check: 2007-10-29)
		In generic cases, there is a cononical imbedding of the ordinary spectrum into the functional spectrum.
		We conjecture that the spaces CFT_n of "super symmetric conformal field theories of degree n" fit together in a spectrum homotopy equivalent to the "topological modular form spectrum" TMF of Miller-Hopkins.
		Furthermore, the analogous spaces EFT_n of "supersymmetric 1-dimensional euclidean field theories of degree n" fit together in a spectrum homotopy equivalent to the K-theory spectrum.
	www.math.vanderbilt.edu /~calendar/archive/04_03.html (616 words)

	[No title]
		The basic definitions of homotopy theory have analogs for C* algebras, and it is natural to ask how the homotopy-theoretic properties of X relate to those of C(X).
		A standard procedure in the theory of C* algebras is to tensor a given algebra with the algebra of compact operators on a separable Hilbert space: this process is called ``stabilization'' (and is unrelated to homotopy-theoretic stablization).
		We show that the localization of $BG$ with respect to a multiplicative complex oriented homology theory $h_$ is again a space of type $K(\pi,1)$; in fact, it is the same as the localization of $BG$ with respect to the ordinary homology theory* determined by the ring $h_0$.
	claude.math.wesleyan.edu /~mhovey/archive/letter40 (1445 words)

	[No title]
		Much of homotopy theory can be redone in this spirit, with an arbitrary but fixed space $\M$ and its suspensions replacing the spheres not only in the definition of homotopy groups, but also in that of a $CW$-complex, loop space, and so on.
		Phantom Maps and Homology Theories J. Daniel Christensen and Neil P. Strickland (jdchrist@mit.edu and neil@pmms.cam.ac.uk) Keywords: phantom map, stable homotopy theory, spectrum, triangulated category Abstract: We study phantom maps and homology theories in a stable homotopy category $\cS$ via a certain Abelian category $\cA$.
		Then it is known that $T(j)$ is a retract of a suspension spectrum, is dual to a stable summand of $\Omega^2 S^3$, and that the homotopy colimit of a certain sequence $T(j) \rightarrow T(2j) \rightarrow \ldots$ is a wedge of stable summands of $K(V,1)$'s, where $V$ denotes an elementary abelian 2 group.
	claude.math.wesleyan.edu /~mhovey/archive/letter27 (1919 words)

	Complex Cobordism and Stable Homotopy Groups of Spheres
		The history of computing homotopy groups is illustrated by a brief discussion of the Cartan-Serre method of killing homotopy groups and of its descendent, the classical Adams spectral sequence.
		In this case and in Bott's computation of the homotopy of $b\text{o}$, the $E\sb 2$ term is rather nice and the spectral sequence collapses.
		The computations for the homotopy of spheres are more difficult and useful techniques such as the May spectral sequence and the lambda algebra are introduced.
	www.math.rochester.edu /people/faculty/doug/mu.html (1161 words)

	[No title]
		The periodic homology theories we consider are K(n), the nth Morava K-theory at a fixed prime p and with n > 0, and the `telescopic' variants T (n), where T (n) denotes the telescope of a vn-self map of a finite complex of type n.
		Recall that R is a ring spectrum, and we are assuming that tZ=p(R) is E*-acyclic.
		X2 ____//_X12 is the homotopy fiber of the evident map from X0 to the homotopy pullback of the square with X0 omitted.
	hopf.math.purdue.edu /Kuhn/Tate.txt (6741 words)

	A Quick Tour of Basic Concepts in Homotopy Theory
		Actually homotopy theory is tremendously fun, but it takes quite a bit of persistence to come anywhere close to the coal face.
		In homotopy theory it causes problems to work with spaces equipped with algebraic structures satisfying equational laws, because one cannot transport such structures along homotopy equivalences.
		A homotopy type is roughly a topological space "up to homotopy equivalence", and an omega-groupoid is a kind of limiting case of an n-groupoid as n goes to infinity.
	math.ucr.edu /home/baez/calgary/homotopy.html (10210 words)

	[No title]
		We show that this ring spectrum structure extends to an operad action of the the ``cactus operad", originally defined by Voronov, which is equivalent to the operad of framed disks in R^2.
		Using these, we propose a definition of ``homotopy sheaves'' and show that a twisted diagram is a homotopy sheaf if and only if it gives rise to a ``sheaf in the homotopy category''.
		We predict the unstable Adams filtration of the homotopy elements of SO based on the conjecture, and we give an example of how the chain complex predicts the differentials of the unstable Adams spectral sequence.
	www.lehigh.edu /~dmd1/h86.txt (832 words)

	AMS Summer 1999 Research Conference in Algebraic Topology Abstracts
		Basterra reformulated the foundations of this theory in a convenient form and wrote the first complete arguments in a preprint to appear soon in JPAA.
		With coefficients in an Eilenberg-MacLane spectrum, the Topological Andre-Quillen cohomology of an E-infinity ring spectrum can be calculated as the Quillen cohomology (as defined in the monograph Homotopical Algebra) of a related E-infinity differential graded algebra.
		I will emphasize on the analogy with standard algebraic topology, showing how analogues of some classical results, for instance dealing with rational homotopy theory or Adams spectral sequences, are related to highly non-trivial results or conjectures on motives.
	www.math.wayne.edu /~rrb/Summer99/abstracts4.html (497 words)

	Hecke Algebras Acting on Elliptic Cohomology - Baker (ResearchIndex) (Site not responding. Last check: 2007-10-29)
		Arithmetic Invariants and Periodicity in Stable Homotopy Theory - Baker (1998)
		15 the homotopy type of the spectrum representing elliptic coho..
		On the Homotopy Type of the Spectrum Representing Elliptic..
	citeseer.ist.psu.edu /98498.html (688 words)

	Amazon.com: Wavelets through a Looking Glass (Applied and Numerical Harmonic Analysis): Books: Ola Bratteli,Palle ... (Site not responding. Last check: 2007-10-29)
		Focuses on recent developments in wavelet theory, emphasizing fundamental and relatively timeless techniques that have a geometric and spectral theoretic flavor.
		This book gives a general presentation of some recent developments in wavelet theory with an emphasis on techniques that have a geometric and spectral-theoretic flavor.
		1- The book covers the theory of wavelets from the point of view of operators and functional analysis and will appeal to a growing number of pure as well as applied mathematicians interested in the subject.
	www.amazon.com /exec/obidos/tg/detail/-/0817642803?v=glance (1761 words)

	W. Dwyer: Home Page
		80j:55010 The tame homotopy groups of a suspension, Proceedings of the 1977 Northwestern University Homotopy Theory Conference, Geometric Applications of Homotopy Theory, Lecture Notes in Math.
		81a:55020 Tame homotopy theory, Topology (18), 1979, 321-338.
		87h:55010 (with D. Kan) Reducing equivariant homotopy theory to the theory of fibrations, Contemporary Mathematics Volume 37, American Math.
	www.nd.edu /~wgd (1651 words)

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