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Topic: Pushforward

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	Diffeology - Pushforward, subduction, quotient
		The quotient diffeology is built on the following pushforward construction.
		This diffeology is called the pushforward of the diffeology of X by the map f, or simply the diffeology image.
		Proof: It is a direct consequence of associativity of pushforward (see above) and that the composition of surjective maps is surjective.
	www.umpa.ens-lyon.fr /~iglesias/articles/Diffeology/dflg_pushforward.html (1682 words)

	Pacific Journal of Mathematics - Abstract for 182-1-7 (Site not responding. Last check: 2007-09-27)
		For example, in the case of a faithful torus action, the pushforward of the Liouville measure by the moment map is a locally polynomial homogeneous function \ell(\lambda) on the polyhedral cone f(C)\subset \t
		of the G-invariant measure on O is the same that the pushforward of the Liouville measure on O associated to the symplectic form of the ambient complex vector space.
		Thus, this establishes for the case of complex reductive groups the relation, conjectured by D. Vogan, between the Fourier transform of the orbit O and multiplicities of the ring of regular functions on O.
	nyjm.albany.edu:8000 /PacJ/1998/182-1-7.html (239 words)

	[No title]
		The definition works by identifying a neighbourhood of the elliptic curve ar* ound e 2 E with its tangent space, and taking the equivariant cohomology of Xe, the fixpoints of e on X. In 2.6 this is generalised to an arbitrary compact connect *ed group.
		This data is precisely that of a complex orientation in topology, and defines in a standard manner a homology theory with pushforward maps [A].
		When this standard pushforward is applied locally on the elliptic curve, the effect of this is to twist the sheaf ES1(X), so that ß : X * *!
	www.math.purdue.edu /research/atopology/Grojnowski/deloc.txt (3474 words)

	[No title]
		Earlier on, Atiyah and Hirzebruch [2] had used pushforwards in equivariant K-* *theory to prove the rigidity of the A^-genus for spin manifolds.
		In the case when Y is a point, fE!(1) is* * the S1- equivariant elliptic genus of X. Proof.Recall (Dyer [7]) that the ordinary pushforward is defined as the composi* *tion of three maps, two of which are Thom isomorphisms, and the third is a natural one.
		The Grojnowski pushforward ßE!, if we consider it at the level of stalks at 0 2* * E, is nothing but the elliptic pushforward in HO*S1-theory, as described in Corollary 4.4.
	www.math.purdue.edu /research/atopology/Rosu/ellc.txt (11110 words)

	[No title] (Site not responding. Last check: 2007-09-27)
		A projectivization of a stable bundle on a K3 surface is hyperholomorphic, that is, holomorphic with respect to any complex structure induced by the hyperkaehler structure.
		We prove that a pushforward of a projectively hyperholomorphic bundle has projectively hyperholomorphic cohomology.
		Projectively hyperholomorphic bundles are stable; therefore, a pushforward of a projectively hyperholomorphic bundle is has stable cohomology.
	www.newton.cam.ac.uk /programmes/MTH/Verbitsky.html (157 words)

	Problem Set #5, Solutions (Site not responding. Last check: 2007-09-27)
		One can only do this if one can find a unique solution v to the pushforward equation:
		Since the pushforward of any curve will lie in the subspace, the pushforward of the vector will be tangent to the subspace.
		Thus we have to verify that a vectorfield like X is tangent to the subspace:
	www.ucolick.org /~burke/class/ps5s.html (435 words)

	Pullbacks -- Recommendations and Resources (Site not responding. Last check: 2007-09-27)
		When the smooth map is a self-diffeomorphism, then the pullback, together with the pushforward, describe the transformation properties of the manifold under a change of coordinates.
		It is the pullback of the diagram f p M--->N
		The pushforward is defined on tangent vectors and contravariant tensors while the pullback is defined on cotangent vectors and covariant tensors.
	www.becomingapediatrician.com /health/123/pullbacks.html (1018 words)

	HogBlog: Local Degree
		Suppose y is a regular value of f: at every point in the (finite) inverse image, the pushforward (or Jacobian, or derivative, or [insert notation here]) of f has rank n, making it a linear isomorphism.
		We can also consider the pushforward of f at x; it's a linear isomorphism of the tangent space at x to the tangent space at y, so we can easily check whether it preserves or reverses the orientations on these tangent spaces defined by the standard orientation of the sphere.
		I claim the local degree at x will be +1 precisely when the pushforward preserves orientations, and -1 when it reverses orientations.
	www.koschei.net /blog/archives/000675.html (776 words)

	pushforward of Lie bracket
		one elementary result that you see when you first learn differential geometry is that the pushforward of the Lie bracket of two vector fields is the Lie bracket of the pushforward of the two vector fields, i.e.
		around using the identities of the pullback and the pushforward:
		this last expression, that i obtained by naandiuml;vely applying the definition of the pushforward (presumably incorrectly), is meaningless, since it has a vector in N trying to act on a function in
	www.physicsforums.com /showthread.php?t=10505 (699 words)

	pushForward (Site not responding. Last check: 2007-09-27)
		pushForward(..., BasisElementLimit) -- stop when this number of basis elements is obtained
		pushForward(..., PairLimit) -- stop when this number of pairs is handled
		pushForward(..., StopBeforeComputation =>...) -- initialize but do not begin the computation
	www.stanford.edu /~mluciano/M2-help/0398.html (80 words)

	Generalizations of the classical Weyl and Colin de Verdiere's formulas and the orbit method -- Boyarchenko and ...
		obtained as the pushforward of the measure (2
		a general way of construction of the family Q and the pushforward
		cannot be obtained from a pushforward measure arising from
	www.pnas.org /cgi/content/full/102/16/5663 (3250 words)

	pushForward (Site not responding. Last check: 2007-09-27)
		pushForward(..., BasisElementLimit) -- stop when this number of basis elements is obtained
		pushForward(..., PairLimit) -- stop when this number of pairs is handled
		pushForward(..., StopBeforeComputation =>...) -- initialize but do not begin the computation
	www.math.rutgers.edu /Macaulay2/0398.html (80 words)

	Math 245A Topics in algebraic geometry: Introduction to intersection theory in algebraic geometry
		1): proper/projective/finite morphisms, proper pushforwards of cycles (ps, pdf).
		4): proper pushforward of rational equivalence, flat morphisms, fundamental cycle, pulling back cycles and rational equivalences, proper pushforward commutes with flat pullback (ps, pdf).
		3): the Segre class of a subscheme behaves with respect to proper pushforwards and flat pullbacks.
	math.stanford.edu /~vakil/245 (625 words)

	QFT of Charged n-Particle: Dynamics \| The n-Category Café
		In particular, if configuration spaces are literally spaces, and functions are literally complex-valued function, then you find you need a measure to do the pushforward - which is then integration.
		But there are other cases where no extra structure is needed and the Tao remains un-intervened with - basically, if
		Incidentally, I was just made aware of slides by Dan Freed where this pull-push quantization is also discussed.
	golem.ph.utexas.edu /category/2007/02/qft_of_charged_nparticle_dynam.html (7244 words)

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