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Topic: Moduli problem

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	PlanetMath: algebraic geometry
		This is a problem, as in the complex category, cohomology with constant coefficients (in fact, usually with integer coefficients) determines most of the cohomological invariants that are of interest, such as the Betti numbers.
		Their key insight was that the problem with sheaf cohomology comes from the fact that Zariski open sets are “too big”, that the induced topology is “too coarse”.
		The category of schemes has a natural notion of isomorphism, and many problems are interested in the isomorphism class of an object.
	planetmath.org /encyclopedia/AlgebraicGeometry.html (2495 words)

	Modular equation - Wikipedia, the free encyclopedia
		In mathematics, a modular equation is an algebraic equation satisfied by moduli, in the sense of moduli problem.
		That is, given a number of functions on a moduli space, a modular equation is an equation holding between them, or in other words an identity for moduli.
		Further, the 'cusps' of the moduli problem, which are the points of the modular curve not corresponding to honest elliptic curves but degenerate cases, may be difficult to read off from knowledge of P.
	en.wikipedia.org /wiki/Modular_equation (287 words)

	Matches for:
		Moduli problems in algebraic geometry date back to Riemann's famous count of the $3g-3$ parameters needed to determine a curve of genus $g$.
		Zariski's last chapter concerns the application of deformation theory to the moduli problem, including the determination of the dimension of the generic component for a particular family of curves.
		An appendix by Bernard Teissier reconsiders the moduli problem from the point of view of deformation theory.
	www.mathaware.org /bookstore?fn=20&arg1=ulectseries&item=ULECT-39 (182 words)

	Moduli space -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-02)
		In (Click link for more info and facts about algebraic geometry) algebraic geometry, the moduli problem is to describe the (A constant in the equation of a curve that can be varied to yield a family of similar curves) parameters on which (Click link for more info and facts about algebraic varieties) algebraic varieties depend.
		The moduli problem here is the prototype for moduli problems with level structure, meaning in this case some 'marking' of (Click link for more info and facts about torsion group) torsion groups of points on the curve.
		For a (The science of matter and energy and their interactions) physics-oriented description of moduli spaces, see (Click link for more info and facts about moduli) moduli.
	www.absoluteastronomy.com /encyclopedia/m/mo/moduli_space.htm (945 words)

	Encyclopedia: Moduli
		In theoretical physics, moduli are scalar fields whose different values are equally good (each one such scalar field is called a modulus).
		The reason is that the potential energy for moduli is constant, which can be guaranteed, for example, by supersymmetry (with sufficiently many supercharges).
		On the other hand, the usage of the phrase "moduli space" in mathematics is more general in the sense that the moduli describe the shape of an arbitrary algebraic variety, not necessarily a manifold relevant for compactification in string theory.
	www.nationmaster.com /encyclopedia/Moduli (582 words)

	Springer Online Reference Works
		Deformation) is concerned with the problem of classification of analytic objects of a given type (e.g.
		In the former case one speaks of the global moduli problem, while in the latter one speaks of the local moduli problem.
		An example of the global moduli problem is the problem of the classification of all complex structures on a compact Riemann surface (cf.
	eom.springer.de /a/a012430.htm (1914 words)

	Abstracts - Moduli Spaces in Geometry and Physics
		For example, we prove that the moduli space of Sasakian structures on homotopy spheres that are bounded by a parallelizable manifold has an infinite number of component.
		The moduli space of semistable vector bundles on a curve was constructed in the 60's by Narasimhan and Seshadri, and since then, its generalizations, detailed study and applications have experimented enormous growth.
		In the next decade, the moduli of semistable principal $G$-bundles on a curve was constructed by Ramanathan, but its study is not yet as developed as that of vector bundles.
	www.uam.es /personal_pdi/ciencias/vmunoz/abstracts.html (1046 words)

	MaloneyAbstract (Site not responding. Last check: 2007-11-02)
		This is a qualitatively new phenomenon with several important cosmological applications -- it ameliorates the cosmological moduli problem and may provide a dynamical solution to the vacuum selection problem.
		In particular, given suitable assumptions about the very early universe, this effect might explain why among the plethora of possible vacuum states of string theory, we appear to live in one with a large number of light particles and (spontaneously broken) symmetries.
		Moduli trapping also leads to novel effects in various time dependent scenarios of string theory, such as systems of moving D-branes.
	www.physics.ucdavis.edu /Abstracts/MaloneyAbstract.html (135 words)

	Pragmatic 2004: scientific activities (Site not responding. Last check: 2007-11-02)
		Lucia Caporaso: Aritmetic hyperbolicity and moduli of curves.
		Examples of (non-trivial) isotrivial families and non-modular rational points in the moduli space of smooth curves.
		Hyperbolicity of the functor of moduli of smooth curves (or: Non existence theorems for families of smooth curves, after Beauville).
	www.dmi.unict.it /~ragusa/docs/Pragmatic04-lectures.htm (239 words)

	[No title]
		Most of the examples considered are in fact moduli spaces, that is, their points correspond to the isomorphism classes of certain kinds of objects; these have been typically vector bundles or else finite dimensional representations of rings.
		The new technique arises from an insight that instead of looking at just one moduli space at a time it is important to look at an associated family of moduli spaces and that this family of moduli spaces is then birational to the finite dimensional representations of a (highly) noncommutative ring.
		The moduli spaces of finite dimensional representations of a ring are also parametrised by the natural numbers since the dimension of the representations parametrised by a connected variety will be constant.
	www.stats.bris.ac.uk /~maahs/res.htm (834 words)

	Student CA / AG Seminar, February 11, 1999 (Site not responding. Last check: 2007-11-02)
		Every scheme, for example, is the moduli space for a certain (even if not particularly interesting) moduli problem associated to it.
		Moduli spaces for more interesting problems include the projective space, Grassmannians, flag varieties, Hilbert schemes, Jacobians, Picard varieties, moduli spaces of curves, moduli spaces of vector bundles, etc. etc.
		I hope to give one or two followup talks later in the semester about moduli spaces of curves, which in my view constitute the richest examples in moduli theory.
	www.math.lsa.umich.edu /~fenescu/studentseminar/abstracts/pasha.html (140 words)

	Springer Online Reference Works
		In several regards, the concept of a Drinfel'd module is analogous to the concept of an elliptic curve (or more generally, of an irreducible Abelian variety), with which it shares many features.
		Among the similarities between Drinfel'd modules and elliptic curves are the respective structures of torsion points, of Tate modules and of endomorphism rings, the existence of analytic "Weierstrass uniformizations" , and the moduli theories (modular varieties, modular forms; cf.
		It is a major problem to determine the representation type of the
	eom.springer.de /D/d120270.htm (1069 words)

	Not Even Wrong » Blog Archive » Rube Goldberg’s Instruction Manual
		The moduli parameters are supposed to be dynamical elements of the theory, not something parametrizing different theories.
		The problem with this is that if you promote the moduli to dynamical fields, they naively correspond to massless fields, and thus give new long-range forces.
		Generally different methods of moduli stabilisation are quite specific, and have typical outcomes and mass scales, and I do not see any evidence at a technical level for the claim that one can get almost anything.
	www.math.columbia.edu /~woit/wordpress/?p=473 (4765 words)

	VBAC (Site not responding. Last check: 2007-11-02)
		The method will be to embed the moduli problem of sheaves into a moduli problem for representations of a quiver, which we associate to our original moduli problem.
		This construction gives perverse sheaves on the moduli space of vector bundles with parabolic structure on C. To this end we need to introduce a simple notion of coherent sheaves with parabolic structure and their moduli spaces, which in turn provide a geometric description of the Iwahori-Hecke algebra.
		Given a generically smooth stable curve over a discrete valuation ring such that its special fibre is irreducible with one double point, we construct a moduli stack over that descrete valuation ring which is a model for the moduli stack of vector bundles over the generic fibre of the curve.
	www.math.sciences.univ-nantes.fr /~sorger/vbac/luminy/program (2321 words)

	Geometry of Integrable Systems: Trieste Research Unit
		Another strictly related problem is the analytical structure of the so-called ``dispersive analogue of shock wave'', that is, of the oscillatory regimes which appear in hydrodynamics, plasma physics and nonlinear optics.
		Another class of geometrical problems related to Hitchin systems is the moduli problem for principal bundles (with a reductive group as structure group) on algebraic curves and surfaces.
		Another problem to be tackled is the construction of the moduli spaces of semi-stable principal bundles as a projective variety, when the curve and the group are defined over an algebraically closed field of positive characteristic.
	www.sissa.it /fm/cofin2001/trieste.html (3200 words)

	DevASP Moduli of Curves (Graduate Texts in Mathematics) Book - 0387984291
		This particular scheme is called the fine moduli space for the functor, as distinguished from the coarse moduli space, where the functor is not representable, i.e.
		The authors clarify the distinction between a moduli space and a parameter space, the former used for problems that involve intrinsic data, the latter for problems involving extrinsic data.
		The local properties of the moduli space are outlined, along with a discussion of to what extent the moduli space deviates from being a projective or affine variety.
	www.devasp.com /store/shop62/pd0387984291/Books_and_Software (920 words)

	[No title]
		Thus the radion presents an example of a cosmological moduli problem, which is referred to as the radion problem in Refs.
		Thus we argue that late inflation is an unlikely solution to the radion problem, which remains as one of the most severe problems for models with sub-millimeter dimensions.
		Thus, in the absence of fine-tuning, the moduli problem of sub-millimeter extra dimensions remains a difficult problem that requires a more interesting solution than a standard late inflation.
	particle.physics.ucdavis.edu /archive/lateinflation.tex (2014 words)

	Curso Gratis Quantum effects in brane world scenarios: moduli stabilization and the hierarchy problem (Site not responding. Last check: 2007-11-02)
		Hence, a complete solution to the hierarchy problem requires a stabilization mechanism that naturally fixes the radion at an appropriate value and explains why the interactions mediated by such a scalar are not observed.
		In this thesis, we consider the possibility that the Casinir energy is responsible for stabilizing the radion and generating a large hierarchy in a natural way.
		However, the generated masses for the moduli are large enough if the power of the warp factor is of order 10 or bigger.
	www.solocursosgratis.com /curso_gratis_quantum_effects_in_brane_world_scenarios_moduli_stabilization_and_the_hierarchy_problem-slccurso757899.htm (810 words)

	Moduli space - InfoSearchPoint.com (Site not responding. Last check: 2007-11-02)
		It agree with the calculation of the dimension of the space of quadratic differentials on a fixed such Riemann surface, which is suggested by deformation theory combined with Serre duality.
		Except when g=2, this is larger than the number 2g-1 of moduli of hyperelliptic curves.
		In applications to physics, the number of moduli of vector bundles and the closely related problem of the number of moduli of principal G-bundles has been found to be significant in gauge field theory.
	www.infosearchpoint.com /display/Moduli_problem (631 words)

	[No title]
		One is the usefulness of working with moduli spaces as a whole, rather than with their sets of components, if only because the moduli spaces tend to fit into fibration sequences and fibre squares.
		If X is an object of a category with weak equivalences, the moduli space M(X) is defined to be the nerve of the subcategory of C con- sisting of all objects weakly equivalent to X together with the weak equivalences between them.
		MODULI PROBLEM 13 where = Aut (G, M) and the object on the left is the union of the components of Map h(X, BG(M, n)) giving maps which induce isomor- phisms on ß1.
	hopf.math.purdue.edu /Blanc-Dwyer-Goerss/moduli.txt (11294 words)

	Lipman Bers, May 22, 1914 — October 29, 1993 \| By Irwin Kra and Hyman Bass \| Biographical Memoirs
		His subsequent work on moduli of Riemann surfaces and Kleinian groups would include some of the most important contributions to complex analysis in the second half of the twentieth century.
		One of the important open problems, the moduli problem, left to us from the nineteenth century, was to make rigorous and precise Bernhardt Riemann's claim that the complex analytic structure on a closed surface with p >= 2 handles depends on 3p–3 complex parameters.
		His studies in topology/geometry revealed new structures in what we thought were well-understood areas (for example, the structure of self maps of [two-dimensional] surfaces) and led to remarkably rapid and dramatic progress in not so well understood topics (for example, the classification of three-dimensional manifolds).
	www.nap.edu /html/biomems/lbers.html (5038 words)

	Past Algebraic Geometry Seminars (Site not responding. Last check: 2007-11-02)
		The stable compactification of moduli of parametrized J-holomorphic curves in $X$ with boundary in $L$ (with prescribed topological data) is compact and Hausdorff in Gromov's $C^\infty$ topology.
		In the special case where the expected dimension of the moduli is zero, and there is an $S^1$ action on the pair $(X,L)$ which preserves $J$ and acts on $L$ freely, we define the Euler number for this $S^1$ equivariant pair and the prescribed topological data.
		These two problems are related in that both involve the geometry of the Abel-Jacobi mapping from the space of stable maps to the intermediate jacobian.
	www.math.uiuc.edu /~ersharpe/pastag.html (1269 words)

	Frances Kirwan: research (Site not responding. Last check: 2007-11-02)
		The whole idea of a moduli problem is that you're trying to classify something in geometry, some geometrical objects.
		The moduli space is the set of equivalence classes of the objects that you want to classify, but with some natural geometrical structure on it which reflects how these objects vary.
		If I'm thinking about a problem, then I need some sort of 2-dimensional picture to get me going, even if the actual problem is in some large number of dimensions, an arbitrary number of dimensions, or an infinite number of dimensions.
	www.ma.hw.ac.uk /~nick/fom/kirwanqu.html (387 words)

	A list of open problems and questions on the moduli space of curves
		This document originated as a rough transcription of the problems and questions generated during a discussion session towards the close of the workshop.
		It was suggested at the problem session that there may exist a Morse function which would provide such a decomposition.
		For an algebraic geometer it is natural to ask whether it solves a moduli problem.
	www.aimath.org /WWN/modspacecurves/open-problems/open-problems.html (1524 words)

	Untitled (Site not responding. Last check: 2007-11-02)
		For this idealized scheme to be applicable, one should know the effective moduli of the structure.
		Given the matrix elastic moduli and the hole volume fraction, its traction-free shape remains the only factor controlling the stress distribution and hence all the relative quantities.
		The problem is posed as a full scale inverse problem of elasticity.
	www.math.technion.ac.il /pde/abstracts01/vigdergauz.html (262 words)

	A. Voronov Abstract (Site not responding. Last check: 2007-11-02)
		Any formal moduli problem is believed to be associated with a differentialgraded (DG) Lie algebra.
		Whatever object you deform, you should expect tofind a DG Lie algebra, whose first cohomology describes the space ofinfinitesimal deformations and whose moduli space of solutions of theMaurer-Cartan equation describes the moduli space in the formalneighborhood of your initial object.
		Wewill also report on some recent progress of Ginzburg and the speakertowards generalizing those results to the formal moduli problem associatedwith the tensor product of a DG commutative and a Lie algebras.
	www.math.msu.edu /~mccarthy/colloq.96-7/voronov-abstract.html (173 words)

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