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Topic: Cobordism theory

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	PlanetMath: oriented cobordism
		Cobordism is an equivalence relation, and a very coarse invariant of manifolds.
		There is a cobordism category, where the objects are manifolds, and the morphisms are cobordisms between them.
		This is version 4 of oriented cobordism, born on 2003-09-05, modified 2004-10-01.
	www.planetmath.org /encyclopedia/Cobordism.html (110 words)

	Cobordism - Biocrawler
		In mathematics, cobordism is a relation between manifolds, based on the idea of boundary.
		It came to prominence when Thom showed that cobordism groups could be computed by means of homotopy theory (the Thom complex construction).
		Cobordism theory became part of the apparatus of the extraordinary cohomology theory, alongside K-theory.
	www.biocrawler.com /encyclopedia/H-Cobordism (360 words)

	Springer Online Reference Works
		Cobordism theory is dual (in the sense of
		The fundamental problem of the first stage of the development of cobordism theory was the calculation of the cobordism rings of a point.
		The second stage in the development of cobordism theory is the study of cobordisms as specific generalized cohomology theories.
	eom.springer.de /C/c022780.htm (1949 words)

	[No title]
		A number of topics related to cobordism have been developed since the publication of Stong's book and are dealt with thoroughly by Rudyak.
		The expressed purpose of this chapter is to present facts to be used in the final two chapters, which deal with cobordism with singularities.
		One of these chapters concentrates on the geometric theory of manifolds with singularities, while the second concentrates on the various spectra, such as $P(n)$ and $BP\langle n\rangle$, that can be obtained as cobordism of complex manifolds with certain types of singularities.
	www.lehigh.edu /~dmd1/rudyak (982 words)

	IUM10 conference: M.Kazarian
		This problem was studied for last 30 years in the framework of intersection theory.
		The motivation for this formula came from topology, namely, from cobordism theory --- the field that looks rather far from algebraic geometry and intersection theory, perhaps, this was the reason why this formula were not observed before.
		This result demonstrates the unity of mathematics: combinig the methods of completely different domains (intersection theory and cobordism theory in this case) one may obtain new results in both of them.
	www.mccme.ru /ium/ium10/kazarian.html (184 words)

	LMS Proceedings Abstract, paper PLMS 1508 (Site not responding. Last check: 2007-08-12)
		In this paper, we study applications of Atiyah--Hirzebruch spectral sequences for motivic cobordism recently found by Hopkins and Morel.
		The cobordism ring $MGL / 2^{,} (Spec(\mathbb{R}))$ is computed and compared with the results of the real cobordism theory of Hu and Kriz.
		For example, the Chow ring $CH^* (BG_2)_(2)$ for the exceptional Lie group $G_2$ is determined by using motivic cobordism and motivic cohomology.
	www.lms.ac.uk /publications/proceedings/abstracts/p1508a.html (121 words)

	MATH40121 Advanced Algebraic Topology
		Cobordism theory gives an extremely powerful tool for the solution of geometrical problems by the methods of algebraic topology.
		The main goal of this course is to base the construction of cobordism theory on the differential geometry of smooth manifolds and to show some important applications of the theory.
		4.3 Axioms of a cohomology theory for cobordism groups.
	www.maths.manchester.ac.uk /undergraduate/ugstudies/units/level4/MATH40121 (254 words)

	Lessons from Topological Quantum Field Theory
		In quantum field theory on curved spacetime, space and spacetime are not just manifolds: they come with fixed `background metrics' that allow us to measure distances and times.
		The big difference is that in topological quantum field theory we cannot measure time in seconds, because there is no background metric available to let us count the passage of time.
		Thus, we hasten to reassure the reader that this peculiarity of topological quantum field theory is not crucial to our overall point, which is the analogy between categories describing space and spacetime and those describing quantum states and processes.
	math.ucr.edu /home/baez/quantum/node2.html (1456 words)

	Grad course descriptions
		Theory of manifolds: differentiable manifolds, charts, tangent bundles, transversality, Sard's theorem, vector and tensor fields and differential forms: Frobenius' theorem, integration on manifolds, Stokes' theorem in n dimensions, de Rham cohomology.
		Theory of fibre bundles and classifying spaces, fibrations, spectral sequences, obstruction theory, Postnikov towers, transversality, cobordism, index theorems, embedding and immersion theories, homotopy spheres and possibly an introduction to surgery theory and the general classification of manifolds.
		Some topics are: the classical theory of the wave and Laplace equations, general hyperbolic and elliptic equations, theory of equations with constant coefficients, pseudo-differential operators, and nonlinear problems.
	www.math.upenn.edu /grad/courses.html (2588 words)

	[No title]
		Drinfeld's construction of quantum doubles is one of sev- eral recent advances in the theory of Hopf algebras (and their actions on rings) which may be attractively presented within the framework of complex cobordism; these developments were pioneered by S P Novikov and the first author.
		Double complex cobordism DU() is the universal example of a coho- mology theory* D*() equipped with two complex orientations, and we dis- cuss this fundamental property in x3, paying particular attention to the consequences for the D-homology and D-cohomology of complex projective spaces, Grassmannians, and Thom complexes.
		Philosophically, double complex cobordism theory is based on manifolds M whose stable normal bundle M (abbreviated to whenever the context allows) possesses a specific splitting ~=` r into two complex bundles, which we often label the left and right components.
	www.math.purdue.edu /research/atopology/Buchstaber-Ray/dcfmqd.txt (12405 words)

	[No title]
		In the first flowering of stable algebraic topology, with t* he introduction of cobordism* and K-theory, the solidly established theory of fiber bundles was absolutely central to the translation of problems in geometric topo* *logy to problems in stable algebraic topology.
		Defining the category of cohomology theories on spa* *ces in the evident way, we see that it is equivalent to the homotopy category of M- spectra E whose spaces En are homotopy equivalent to CW complexes.
		Combining ideas, they view the algebraic theory as a theory of bundles over a point and generalize it to a theory of bundles over X. Starting from a fixed Euclidean vector bundle V over X, they construct an associated Clifford bundle C(V) over X whose fiber over x is the Clifford algebra C(Vx).
	hopf.math.purdue.edu /May/history.txt (14491 words)

	Graduate Courses
		The centerpiece of the course is the theory of canonical forms, including the Jordan canonical form and the rational canonical form.
		Study of topics selected are from the theory of curves and surfaces in Euclidean space and the theory of manifolds.
		Study of classical cobordism theories; Pontryagin-Thom construction; bordism and cobordism of spaces; K-theory and Bott periodicity; formal groups, and cobordism.
	www.math.virginia.edu /grad/grad6.htm (1548 words)

	Cohomology information - Search.com (Site not responding. Last check: 2007-08-12)
		With hindsight, general homology theory should probably have been given an inclusive meaning covering both homology and cohomology: the direction of the arrows in a chain complex is not much more than a sign convention.
		A cohomology theory is a family of contravariant functors from the category of pairs of topological spaces and continuous functions (or some subcategory thereof such as the category of CW complexes) to the category of Abelian groups and group homomorphisms that satisfies the Eilenberg-Steenrod axioms.
		When one axiom (dimension axiom) is relaxed, one obtains the idea of extraordinary cohomology theory; this allows theories based on K-theory and cobordism theory.
	c10-ss-1-lb.cnet.com /reference/Cohomology?redir=1 (720 words)

	Ph.D. supervision
		Hirsch's theorem claims that the immersion question for a manifold is a problem in homotopy theory: it's just a matter of determining the geometric dimension of the stable normal bundle of the manifold, that is, lifting the corresponding classifying map from BO to a space BO(k) --smallest possible such k.
		The theory of formal group laws has shown to be of special importance in mathematics mainly due to the wide variety of connections it has had with other mathematical branches like geometry, algebraic topology, number theory and combinatorics.
		Number theory is perhaps the area most naturaly linked to the theory of formal groups, and the relations have become abundant over the time.
	chucha.math.cinvestav.mx /proyectos.html (3167 words)

	Topology Seminar, UCSB Mathematics Department
		From the point of view of mapping class groups, the swing presentation can be interpreted as stating that the pure braid group is generated by a finite number of Dehn twists and that the only relations needed are the disjointness relation and the lantern relation.
		I'll emphasise that this cobordism theory is a "categorification" of the usual sl_3 theory, and describe a decategorification theorem making this more precise.
		I will discuss how a technical result in 3-manifold theory can be used to show that the genus three Schoenflies Conjecture is no longer the easiest of the hard cases of the Conjecture, but is now merely the hardest of the easy cases.
	www.math.ucsb.edu /~bigelow/oldseminars/S06.html (943 words)

	Atiyah–Singer index theorem - Wikipedia, the free encyclopedia
		The initial proof was based on that of the Hirzebruch-Riemann-Roch theorem (1954), and involved cobordism theory and pseudodifferential operators.
		Thom's cobordism theory gives a set of generators; for example, complex vector spaces with the trivial bundle together with certain bundles over even dimensional spheres.
		Peter B. Gilkey: Invariance Theory, the Heat Equation, and the Atiyah–Singer Theorem.
	en.wikipedia.org /wiki/Atiyah-Singer_index_theorem (3349 words)

	University of Oregon: Department of Mathematics
		The idea of using equivariant cobordism theory to study questions about the existence and structure of group actions on manifolds dates back to work of Conner and Floyd, soon after cobordism theory was invented.
		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.
	www.uoregon.edu /~dps/bordism.php (444 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 (1086 words)

	Matches for:
		Typical results are the Duistermaat-Heckman theory, the Berline-Vergne-Atiyah-Bott localization theorem in equivariant de Rham theory, and the "quantization commutes with reduction" theorem and its various corollaries.
		To formulate the idea that these theorems are all consequences of a single result involving equivariant cobordisms, the authors have developed a cobordism theory that allows the objects to be non-compact manifolds.
		This is a natural and important generalization of the notion of a moment map occurring in the theory of Hamiltonian dynamics.
	www.mathaware.org /bookstore?fn=20&arg1=survseries&item=SURV-98 (324 words)

	Projects
		In the complementary case of the rational numbers, Quillen described several distinct categories of algebraic objects, all of which are equivalent to rational homotopy theory in an appropriate sense.
		COBORDISM THEORY: Cobordism theory is a way of organising and classifying manifolds whose stable tangent bundles admit additional structure.
		It originally flowered in the hands of Rene Thom and Jack Milnor during the early 1960s, and the study of complex cobordism energised homotopy theory for the next 20 years or so; it is now finding applications in quantum field theory.
	www.ma.man.ac.uk /~nige/proje.html (1087 words)

	[No title]
		When the cohomology theory in question is real K-theory, there is a beautiful answer: there is PD for KO-theory if and only if the manifold in question is spin.
		Your comments were very helpful, as were those of Mark Hovey, who assured me that after we invert the prime 2, oriented cobordism theory and spin cobordism theory become the same, while complex cobordism becomes almost the same.
		More precisely, we've got a spectrum MU for complex cobordism theory, and we've got a spectrum MSO for oriented cobordism, but localizing away from 2 MU becomes isomorphic to MSO wedged with the double suspension of MSO.
	www.math.niu.edu /~rusin/known-math/00_incoming/ellip_coho (735 words)

	September 2 - Today in Science History
		Thom is also known for his later work developing the catastrophe theory (1972), a mathematical treatment of continuous action producing a discontinuous result.
		Soviet mathematician known for his contributions to the analytical theory of numbers, including a partial solution of the Goldbach conjecture proving that every sufficiently large odd integer can be expressed as the sum of three odd primes.
		He is credited with being the founder of the theory of abstract spaces, which generalized the traditional mathematical definition of space as a locus for the comparison of figures; in Fréchet's terms, space is defined as a set of points and the set of relations.
	www.todayinsci.com /9/9_02.htm (1568 words)

	Sergei Petrovich Novikov Summary
		His early work was in cobordism theory, in relative isolation.
		Among other advances he showed how the Adams spectral sequence, a powerful tool for proceeding from homology theory to the calculation of homotopy groups, could be adapted to the new (at that time) cohomology theory typified by cobordism and K-theory.
		This required the development of the idea of cohomology operations in the general setting, since the basis of the spectral sequence is the initial data of Ext functors taken with respect to a ring of such operations, generalising the Steenrod algebra.
	www.bookrags.com /Sergei_Petrovich_Novikov (653 words)

	Future Research Topics
		This area also involves number theory and algebraic geometry as well as finite group theory, particularly simple groups and their realisation as symmetry groups.
		Quillen's geometric interpretation of cobordism: cobordism is traditionally viewed as a relation on manifolds of finite dimension and usually applied to compact manifolds.
		However, Quillen gave a description of complex cobordism in which compactness was replaced by the use of proper maps.
	www.maths.gla.ac.uk /~ajb/andy-research.html (968 words)

	Transactions of the American Mathematical Society
		N. Baas, On bordism theory of manifolds with singularity, Math.
		P. Landweber, Coherence, flatness and cobordism of classifying spaces, Proceedings of Advanced Study Institute on Algebraic Topology, Aarhus, 1970, pp.
		D. Ravenel and W. Wilson, The Morava K theories of Eilenberg-MacLane spaces and the Conner-Floyd conjecture, Amer.
	www.ams.org /tran/2000-352-11/S0002-9947-99-02484-8/home.html (581 words)

	Graduate School of Arts and Sciences - Mathematics and Statistics
		A master's-level curriculum rich in both theory and application is available to graduate students in mathematics, the natural sciences, engineering, and mathematics education, as well as to advanced undergraduates in mathematics.
		Further topics are chosen from analytic number theory, class field theory, and the theory of Diophantine equations.
		The theory and logic in the development of nonparametric techniques including order statistics, tests based on runs, goodness of fit, rank-order (for location and scale), measures of association, analysis of variance, asymptotic relative efficiency.
	www.bu.edu /bulletins/grs/item29.html (6120 words)

	Serguei Novikov
		The qualitative theory of foliations of codimension 1 on three-dimensional manifolds.
		Periodic problems in the theory of solitons (non-linear waves) and in the spectral theory of linear operators, Riemann surfaces and theta-functions in mathematical physics.
		The theory was extended to the discrete (difference) operators on the lattices and to the very general non regular configurations see [1] and [2].
	www.ipst.umd.edu /Faculty/novikov.htm (923 words)

	Topology & Group Theory Seminar
		The cobordism theory of smooth high-dimensional sphere knots was completely solved (or at least turned into an algebraic problem) in the 1960s through the work of Kervaire, who showed that all even-dimensional knots are cobordant, and J. Levine, who showed that the cobordism type of an odd-dimensional knot is determined by its Seifert matrix.
		We study the cobordism theory of disk knots, i.e.
		We study both cobordism of disk knots and cobordism rel boundary, holding the boundary sphere knot fixed.
	www.math.vanderbilt.edu /~hughescb/TopGTSemF03.html (811 words)

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