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Topic: Representation theory of diffeomorphism groups

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	THE MEANING OF "TENSOR"
		Group theory, the area of continuous groups and (analytic) Lie groups in particular (tensor with respect to a given group); 3.
		In group theory, there is an additional important combining form called a "semidirect product", a typical example of which is the combining, in an affine Euclidean space, of a group of rotations with a group of translations.
		A representation of an abstract algebraic structure is a structure preserving homomorphism from the abstract structure into an algebra of matrices.
	graham.main.nc.us /~bhammel/MATH/tensor.html (1952 words)

	UC Davis Math: Glossary (Site not responding. Last check: 2007-09-10)
		As a space, a quantum group is defined by non-commuting operators which are analogous to coordinates on a Lie group in the same way that non-commuting operators represent measurable quantities in quantum mechanics.
		A theory in physics that postulates a counterintuitive symmetric relationship between fermions, which are particles such as electrons that obey the Pauli exclusion principle, and bosons, which are particles such as photons that enjoy being in the same state as each other.
		A linear representation of a Lie group generated by applying ladder operators to a certain starting vector, called a highest-weight vector, in the same way that one constructs spin states of a particle in quantum mechanics starting from a state of maximal spin in the z direction.
	math.ucdavis.edu /profiles/glossary.html (9932 words)

	Mathematics - Faculty Research Summaries
		I am interested in geometric methods in representation theory; in particular, the theory of constructible sheaves on varieties related to loop groups, and related questions in representation theory of quantum groups at roots of unity and p-adic groups.
		My research concerns the representation theory of Lie groups and their generalizations, most particularly the connections of this theory with algebraic and differential geometry, quantum groups, mathematical physics, and Langlands program.
		My work concerns Lie groups and their discrete subgroups, in particular the realization of these groups as diffeomorphism groups of manifolds, as well as related questions of analysis, geometry, topology, ergodic theory, and group theory.
	physical-sciences.uchicago.edu /research/mathematics/summaries.html (3063 words)

	Not Even Wrong » Blog Archive » Some History
		The close connection between the basic ideas of representation theory and of quantum mechanics was quite clear to him, so, unlike Einstein, he enthusiastically adopted the new point of view of quantum physics.
		The theory of continuous groups and their unitary representations ought to be taught in undergraduate physics courses.
		Since all unitary quantum irreps of the diffeomorphism group are anomalous, apart from the trivial one, all interesting GCQTs carry anomalous reps of the diffeomorphism group.
	www.math.columbia.edu /~woit/wordpress/?p=85 (1651 words)

	Citebase - Implementation of conformal covariance by diffeomorphism symmetry (Site not responding. Last check: 2007-09-10)
		Every locally normal representation of a local chiral conformal quantum theory is covariant with respect to global conformal transformations, if this theory is diffeomorphism covariant in its vacuum representation.
		The unitary, strongly continuous representation implementing conformal symmetry is constructed; it consists of operators which are inner in a global sense for the representation of the quantum theory.
		Conformal field theory on the half-space x>0 of Minkowski space-time ("boundary CFT") is analyzed from an algebraic point of view, clarifying in particular the algebraic structure of local algebras and the bi-localized charge structure of local fields.
	www.citebase.org /cgi-bin/citations?id=oai:arXiv.org:math-ph/0312017 (879 words)

	Physics Help and Math Help - Physics Forums - View Single Post - LQG and Category Theory
		representation is cyclic with respect to C(A-bar), of course, and is irreducible, as it should, under the action of the kinematical algebra [26].
		Thus, as far as the Gauss and diffeomorphism constraints are concerned, the H
		representation seems to qualify as an intermediate, or auxiliary, representation of kinematical variables and constraints, as required in the Dirac method.
	www.physicsforums.com /showpost.php?p=152944&postcount=2 (289 words)

	Not Even Wrong » Blog Archive » Smolin on the Anthropic Principle (Site not responding. Last check: 2007-09-10)
		One of Smolin’s concerns is to show that his theory of “cosmological natural selection” (discussed in his book “The Life of the Cosmos”), while being a theory of a “multiverse” just like the string theory Landscape, is different in that it is potentially falsifiable, unlike some recent anthropic arguments.
		In addition, for some of the groups that are experimentally known to be physically relevant (4d gauge and diffeomorphism groups), very little is known about their representations.
		The Virasoro algebra is a central extension of the diffeomorphism algebra in 1D, so generalizing it to several dimensions would be a first step towards constructing quantum reps of the diffeomorphism group in several dimensions.
	www.math.columbia.edu /~woit/wordpress/archives/000059.html (1237 words)

	CUNY Differential Geometry and Lie Theory Seminar
		Quantum representations, given by the Jones-Witten theory, provide a a generalization of this to mapping class groups of surfaces - for which Kazhdan's property is not known to hold.
		For special representations of the fundamental group we will show that it is given as the value at zero of a Ruelle zeta function.
		Weakly hyperbolic integer actions are generated by an Anosov diffeomorphism, providing motivation for our two main results: 1) ergodicity of volume preserving weakly hyperbolic actions, and 2)weak hyperbolicity is inherited by the induced action on the fundamental group for volume preserving weakly hyperbolic actions (with a fixed point) of Kazhdan property (T) groups on tori.
	comet.lehman.cuny.edu /fisher/difflie.html (1067 words)

	[No title] (Site not responding. Last check: 2007-09-10)
		In particular she will apply techniques from this area, as well as methods of representation theory and the theory of algebraic groups, to a variety of questions involving connected transformation groups and their discrete subgroups.
		Examples of such questions are invariance phenomena for geometric structures and related issues on the compactification of algebraic varieties, orbit equivalence of actions, and the structural study of the diffeomorphism group of a manifold.
		Ergodic theory in general concerns understanding the average behavior of systems whose dynamics is too complicated or chaotic to be followed in microscopic detail.
	www.cs.utexas.edu /users/yguan/NSFAbstracts/Abstracts/MPS/DMS.MPS.a9001959.txt (174 words)

	symmetries
		The representation theory of the Poincar´ group dominates relativistic physics, while the representation theory of the Galilei group dominates nonrelativistic physics.
		Moreover, the representation theory of diffeomorphism groups is hard and still poorly understood in 4 dimensions.
		Chern-Simons theory is the source of the best-understood states of quantum gravity in the loop representation.
	math.ucr.edu /home/baez/symmetries.html (2452 words)

	References > M
		Induced Representations of Groups and Quantum Mechanics Benjamin and Boringhieri 1968 [>qm].
		"Loop representations for 2+1 gravity on a torus" gq/9303019, CQG 10 (1993) 2625-2647 [>qg-can].
		"Gauge theory of diffeomorphism groups" in Bleuler and Werner 88, 345-371.
	www.phy.olemiss.edu /~luca/Refs/m.html (10621 words)

	Mathematics - Research in Progress
		My research concerns the representation theory of Lie groups and their generalizations, most particularly the connections of this theory with differential geometry.
		My research is mainly focused on interconnection between combinatorics (such as multivariate orthogonal polynomials) and asymptotic problems arising in the representation theory especially in connection with groups like the infinite symmetric and unitary groups).
		My work concerns Lie groups and their discrete subgroups, in particular the realization of these groups as diffeomorphism groups of manifolds, as well as related questions of analysis, geometry, and topology, and group theory.
	physical-sciences.uchicago.edu /research/1999/math_sum.html (2793 words)

	Two cheers for string theory \| Cosmic Variance (Site not responding. Last check: 2007-09-10)
		String theory, with all of its difficulties, is by far the most promising route to one of the most long-lasting and ambitious goals of natural science: a complete understanding of the microscopic laws of nature.
		Just as in quantum field theory, the observable spectrum of low-energy string excitations and their interactions (that is to say, particle physics) depends not only on the fundamental string physics, but on the specific vacuum state in which we find ourselves.
		If string theorists continue to develop a really outdated theory at one hand and arrogantly claim that are doing (they believe that in their infinite ignorance) the most important, the most powerful, the most fundamental theory at the other, then they would feel comfortable with the mocking of their colleagues.
	cosmicvariance.com /2005/07/21/two-cheers-for-string-theory (15193 words)

	Augustin Banyaga - Mathematician of the African Diaspora
		Torre, Carlos A.; Banyaga, Augustin A symplectic structure on coadjoint orbits of diffeomorphism subgroups.
		Banyaga, A. The Calabi invariant and the gauge groups.
		Banyaga, Augustin On the structure of the group of equivariant diffeomorphisms.
	www.math.buffalo.edu /mad/PEEPS/banyaga_augustin.html (1021 words)

	Session VB08 - Field Theory and Related Topics.
		Field and string theories and all theories based on fiber bundles inherit the Inönü-Wigner non-semisimplicity that marred Newtonian mechanics.
		Statistics is defined by a representation of the permutation group that is generated by particle swaps.
		Two-valued representations of permutations (Schur, 1911) were recognized as possible statistics for quasi-particles of the quantum Hall effect by Wilczek (1997).
	flux.aps.org /meetings/YR99/CENT99/abs/S7900.html (1789 words)

	Prof. Rudolf Schmid -- Home Page
		Lie Groups of Fourier Integral Operators on Open Manifolds (SURVEY), "Infinite Dimensional Lie Groups in Geometry and Representation Theory", Howard University, Washington DC, 17-21 Aug. 2000, A.
		The Lie Group of Fourier Integral Operators on OPEN Manifolds ; Infinite Dimensional Lie Groups in Geometry and Representation Theory.
		Diffeomorphism Groups of OPEN Manifolds; Workshop on Lie Groups and Lie Algebras in Infinite Dimensions.
	www.mathcs.emory.edu /~rudolf (1013 words)

	Mathematisches Institut der Universität Basel
		Lie groups in the diffeomorphism group of a manifold, homogeneous spaces, invariant volume forms, invariant metrics
		Representation theory of semisimple Lie groups: Weyl's theorem on the semisimplicity of finite dimensional representations and Howe--Moore's theorem on the vanishing at infinity of matrix coefficients of unitary representations
		Zimmer, Ergodic theory and semisimple groups, Birkhäuser, 1985
	www.math.unibas.ch /~iozzi/lie.html (131 words)

	Citebase - On the representation theory of Virasoro Nets (Site not responding. Last check: 2007-09-10)
		We discuss various aspects of the representation theory of the local nets of von Neumann algebras on the circle associated with positive energy representations of the Virasoro algebra (Virasoro nets).
		In particular we classify the local extensions of the c=1 Virasoro net for which the restriction of the vacuum representation to the Virasoro subnet is a direct sum of irreducible subrepresentations with finite statistical dimension (local extensions of compact type).
		We formulate conformal field theory in the setting of algebraic quantum field theory as Haag-Kastler nets of local observable algebras with diffeomorphism covariance on the two-dimensional Minkowski space.
	www.citebase.org /cgi-bin/citations?id=oai:arXiv.org:math/0306425 (1680 words)

	Links to open problems in mathematics, physics and financial econometrics
		What are the fundamental degrees of freedom of M-theory (the theory whose low-energy limit is eleven-dimensional supergravity and which subsumes the five consistent superstring theories) and does the theory describe nature?
		Problems: construct a quantum theory of gravity from some basic principles assuming noncommutative geometry (John Madore,...) or express some sector or limit of an underlying theory in terms of the language of noncommutative geometry
		Theories of extra dimensions: there are at least five dimensions and a single new particle, a gravitational boson called a graviton
	www.geocities.com /ednitou/index.html (536 words)

	[No title]
		There's already a branch of math relating such invariants to representations of groups and quantum groups, and their formula fits right in.
		Freidel and Krasnov study the volume of a single 3-simplex as an observable in the context of the Turaev-Viro model - a topological quantum field theory which is closely related to quantum gravity in spacetime dimension 3.
		We extend the theory of diffeomorphism-invariant spin network states from the real-analytic category to the smooth category.
	math.ucr.edu /home/baez/week120.html (2210 words)

	Stefan Haller Talks
		Fragmentation and deformation for modular groups of diffeomorphisms.
		On the group of diffeomorphisms preserving a locally conformal symplectic structure.
		Howard Conference on Infinite Dimensional Lie Groups in Geometry and Representation Theory, Howard University, Washington, DC, August 17-21, 2000.
	www.mat.univie.ac.at /~stefan/talks.html (424 words)

	Nitu Kitchloo's homepage (Site not responding. Last check: 2007-09-10)
		Cohomology of the classifying spaces of central quotients of rank two Kac-Moody groups.
		A homotopy construction of the Adjoint representation for Lie groups.
		Symplectomorphism groups of rational ruled surfaces (Statement of expected results).
	www.math.jhu.edu /~nitu (165 words)

	Ideas and Methods in Mathematical Analysis, Stochastics, and Applications - Cambridge University Press (Site not responding. Last check: 2007-09-10)
		The contributions vary in style, purpose, and content--some are surveys that summarize and clarify a subject area, others are new and adventurous expeditions into unknown territory.
		The topics cover most aspects of modern mathematical physics with special emphasis on methods from operator theory and stochastic analysis.
		On nonlinear equations associated with Lie algebras of diffeomorphism groups of two-dimensional manifolds; 18.
	www.cambridge.org /us/catalogue/catalogue.asp?isbn=0521419298 (458 words)

	String Reviews (Site not responding. Last check: 2007-09-10)
		hep-th/0201253: Supersymmetric Gauge Theories and the AdS/CFT Correspondence.
		hep-th/0003119: Lattice Gauge Theories and the AdS/CFT Correspondence.
		Based on lectures given in 1997 at the Isaac Newton Institute, Cambridge, the Trieste Spring School on String Theory, and at the 31rst International Symposium Ahrenshoop in Buckow).
	www.nuclecu.unam.mx /~alberto/physics/stringrev.html (5080 words)

	Amazon.ca: Quantization, Coherent States, and Complex Structures: Books (Site not responding. Last check: 2007-09-10)
		Representation of Eigenfunctions and Coherent States for the Zeeman
		Groups, Noncommutative Geometry: Quantum Coherent States and the
		Subjects > Science > Physics > Quantum Theory
	www.amazon.ca /exec/obidos/ASIN/0306452146 (300 words)

	Dispersionless Toda and Toeplitz operators, A. Bloch, F. Golse, T. Paul, A. Uribe
		[6] —, A Schur-Horn-Kostant convexity theorem for the diffeomorphism group of the annulus, Invent.
		[7] —, ``The Toda PDE and the geometry of the diffeomorphism group of the annulus'' in Mechanics Day (Waterloo, Ontario, 1992), Fields Inst.
		[21] B. Kostant, The solution to a generalized Toda lattice and representation theory, Adv.
	projecteuclid.org /Dienst/UI/1.0/Summarize/euclid.dmj/1085598341 (628 words)

	Michor, Peter, Publications
		[24] Peter W. Michor: The cohomology of the diffeomorphism group is a Gelfand-Fuks cohomology.
		The action of the diffeomorphism group on the space of immersions.
		Lie Theory 7,1 (1997), 61--99, ESI Preprint 200.
	www.mat.univie.ac.at /~michor/listpubl.html (2912 words)

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