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# Topic: Moduli space

###### In the News (Mon 20 May 13)

 Moduli space - Wikipedia, the free encyclopedia In algebraic geometry, a moduli space is a parameter space for families of algebraic objects (such as algebraic varieties, morphisms, vector bundles). There is also another major question, of determining moduli for vector bundles V on a fixed algebraic variety X. When X has dimension 1 and V is a line bundle, this is the theory of the Jacobian variety of a curve. 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 theory. en.wikipedia.org /wiki/Moduli_space   (796 words)

 Moduli - Wikipedia, the free encyclopedia The reason is that the potential energy for moduli is constant, which can be guaranteed, for example, by supersymmetry (with sufficiently many supercharges). The space of possible configurations (values) of all these moduli is called the moduli space (that page gives some explanation of the original, mathematical usage). 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. en.wikipedia.org /wiki/Moduli   (210 words)

 EWM Workshop on Moduli Spaces in Mathematics and Physics The second step is to fix values of the discrete invariants and to try to construct a moduli space; that is, an algebraic variety (or other appropriate space in other parts of geometry) whose points correspond to the equivalence classes of the objects to be classified, in some natural way. Moduli spaces are one of the fundamental constructions of Algebraic Geometry and they arise in connection with classification problems. Roughly speaking a moduli space of stable vector bundles on an algebraic projective variety X is a scheme whose points are in ``natural bijection" to isomorphic classes of stable vector bundles on X. www.math.helsinki.fi /EWM/meetings/moduli.html   (1037 words)

 Jonathan Weitsman Tolman and J. Weitsman: Symplectic geometry of the moduli space of flat connections on a Riemann surface: inductive decompositions and vanishing theorems. Jeffrey and J. Weitsman: Symplectic Geometry of the Moduli Space of Flat Connections on a Riemann Surface: Inductive Decompositions and Vanishing Theorems. Jeffrey and J. Weitsman: Toric structures on the moduli space of flat connections on a Riemann surface II: inductive decompostion of the moduli space, Math Ann 307 (1997), 93-108. www.math.ucsc.edu /Faculty/Weitsman.html   (357 words)

 IMPRS Working Areas Very surprisingly, these and other more complicated moduli spaces (e.g., of vector bundles, of stable maps, etc.) were discovered in recent years to play an important role in mathematical physics, especially in the theory of quantum strings, which strives to the unification of quantum field theory and the theory of gravity. The gauge-theoretic moduli spaces arising in these theories are spaces of solutions of certain partial differential equations defined in terms of geometric objects such as connections and spinors, modulo a large group of gauge symmetries. An intensive study of gauge theoretic moduli spaces was initiated in the early eighties, with Donaldson's famous result on obstructions to the existence of smooth structures on certain classes of 4-manifolds. www.imprs-modulispaces.mpg.de /working_areas.html   (1038 words)

 -Metric on the Moduli Space of (ResearchIndex)   (Site not responding. Last check: 2007-10-21) Suppose that the moduli space M(P) of irreducible self-dual connections on P is a smooth... 6 The Riemannian geometry of the YangMills moduli space (context) - Groisser, Parker - 1987 2 The geometry of the moduli space of CP 2 instantons (context) - Groisser - 1990 citeseer.ist.psu.edu /392665.html   (408 words)

 Moduli space: Facts and details from Encyclopedia Topic   (Site not responding. Last check: 2007-10-21) In mathematics, a modular equation is an algebraic equation satisfied by moduli, in the sense of moduli problem.... In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach... In abstract algebra and algebraic geometry, the spectrum of a commutative ring r, denoted by spec(r), is defined to be the set of all prime ideals... www.absoluteastronomy.com /encyclopedia/m/mo/moduli_space.htm   (2301 words)

 [No title]   (Site not responding. Last check: 2007-10-21) Note that moduli space can be thought of as the space of equivalence classes of complex structures on a fixed surface of genus g, where two complex structures are deemed "the same" if they are biholomorphically equivalent. Anyway, Teichmueller space may be defined as the space of equivalence classes of complex structures on a fixed surface of genus g, where two complex structures are counted as the same if they are biholomorphically equivalent *by a diffeomorphism connected to the identity*. By how moduli space is defined, two points in Teichmueller space define the same point in moduli space iff one is obtained from another by an element of the mapping class group. math.ucr.edu /home/baez/twf_ascii/week28   (2194 words)

 Moduli space   (Site not responding. Last check: 2007-10-21) The moduli problem here is the prototype for moduli problems withlevel structure, meaning in this case some 'marking' of torsiongroups of points on the curve. There is also another major question, of determining moduli for vectorbundles V on a fixed algebraic variety X. When X hasdimension 1 and V is a line bundle, this is the theory of the Jacobianvariety of a curve. The basic strategy isto simplify the classification problem by adding additional data in such a way that the original moduli space is the quotient of the new one by a reductive group action. www.therfcc.org /moduli-space-210504.html   (688 words)

 [No title]   (Site not responding. Last check: 2007-10-21) The first one is algebraic, and treats the case of moduli spaces of stable vector bundles on a projective curve. Using the Hecke cycles we prove that this is indeed the case, and we apply the same techniques to study the stability of the adjoint of the PoincarĂ© bundle, and of the Picard bundle. Using the analytical description of the moduli space of stable bundles over a complex variety, we also express the deformations of the PoincarĂ© bundle in terms of non-abelian cohomology. www.imar.ro /~vsavin/abstract.html   (440 words)

 [No title] For a 1-connected spectrum E, we study the mod- uli space of suspension spectra which come equipped with a weak equivalence to E. We construct a spectral sequence converging to the homotopy of the moduli space in positive degrees. MODULI OF SUSPENSION SPECTRA 3 For q 2, let Wq be the spectrum with q-action which classifies the q-th layer of the Goodwillie tower of the identity functor from based spaces to based spaces. A weak equivalence of spaces is shorthand for (a chain of) weak homotopy equivalence(s). www.math.purdue.edu /research/atopology/Klein/susp-spectra.txt   (5147 words)

 Amazon.com: Moduli of Curves (Graduate Texts in Mathematics): Books: Joe Harris,Ian Morrison   (Site not responding. Last check: 2007-10-21) 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 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. The actual construction of the moduli space is the subject of chapter 4, from the viewpoint of geometric invariant theory. www.amazon.com /exec/obidos/tg/detail/-/0387984291?v=glance   (1411 words)

 On a class of representations of the Yangian and moduli space of monopoles   (Site not responding. Last check: 2007-10-21) A new class of infinite dimensional representations of the Yangians corresponding to a complex semisimple algebra and its Borel subalgebra is constructed. In particular it is shown that the underlying symplectic leaves are isomorphic to the moduli spaces of monopoles defined as the components of the space of based maps of one-dimensional complex projective space into the generalized flag manifold. Thus the constructed representations of the Yangian may be considered as a quantization of the moduli space of the monopoles. www.maths.tcd.ie /report_series/abstracts/tcdm0415.html   (143 words)

 [No title] All these moduli spaces are important and interesting, but the moduli space of elliptic curves is a nice simple example when you're first trying to learn this stuff. Now, the quotient space H/SL(2,Z) is not a smooth manifold, because while the upper halfplane H is a manifold and the group SL(2,Z) is discrete, the action of SL(2,Z) on H is not free: i.e., certain points in H don't move when you hit them with certain elements of SL(2,Z). For any open set U in the moduli space, an object of S(U) is a family of elliptic curves over U, such that each elliptic curve in the family sits over the point in moduli space corresponding to its isomorphism class. www.math.niu.edu /~rusin/known-math/98/PSL   (2693 words)

 PUBLICATIONS (still under construction, please check back later) proved that the moduli space of coassociative deformations of a compact coassociative 4-submanifold C in a G_2-manifold M is a smooth manifold of dimension equal to b^2_+(C). In this paper, we show that the moduli space of coassociative deformations of a noncompact, asymptotically cylindrical coassociative 4-fold C in an asymptotically cylindrical G_2-manifold M is also a smooth manifold. We study the moduli space of deformations of a special kind of associative submanifolds in a G_2 manifold (which we call complex associative submanifolds); and we study the moduli space of deformations of a special kind of Cayley submanifolds (which we call complex Cayley submanifolds). www.math.northwestern.edu /~salur/preprint.html   (525 words)

 Mu Alpha Theta Math Log Imagine - Fall 2002   (Site not responding. Last check: 2007-10-21) A map of the United States is, loosely speaking, a moduli space for the set of states. A moduli space for a mathematical set might be a plane, a line, a curve, a cylinder, a hockey-puck-shaped entity, or even a wild hyperamoeba in 26-dimensional space. So we might guess that the moduli space for S is the same as the moduli space for angles: a circle. www.mualphatheta.org /Mathematical_Log/Issues/Fall_02/MAO_Imagine_Fall_02.htm   (1239 words)

 Geometry of the moduli space of Higgs bundles - Hausel (ResearchIndex)   (Site not responding. Last check: 2007-10-21) The deformation space of a holomorphic bundle E is well known to be H 1 End E ; that of a Higgs pair (E; is similar, but involves... 4 An introduction to the topology of the moduli space of stabl.. 3 the cohomology of moduli spaces of rank 2 vector bundles ove.. citeseer.ist.psu.edu /419144.html   (597 words)

 [No title]   (Site not responding. Last check: 2007-10-21) One forms a moduli space Def(M,G/H) of geometric structures on M modelled on G/H which classifies structures up to equivalence. When M is a closed surface, these moduli spaces carry invariant symplectic structures, and when G is compact the investigator conjectures that the symplectic measure is ergodic under the action of the mapping class group. When G is SL(2,R), this moduli space is related to Teichmuller space of (possibly singular) hyperbolic structures, and the investigator has used geometric structures to study the dynamics on the moduli space. www.cs.utexas.edu /users/yguan/NSFAbstracts/Abstracts/MPS/DMS.MPS.a9803518.txt   (380 words)

 [No title]   (Site not responding. Last check: 2007-10-21) A fine moduli space of sheaves on X is a family This paper is devoted to the study of such fine moduli spaces. We first give some general results, and apply them in three cases on the projective plane : the fine moduli spaces of prioritary sheaves, the fine moduli spaces consisting of simple rank 1 sheaves, and those which come from moduli spaces of morphisms. www.math.jussieu.fr /~drezet/altern.html   (113 words)

 [No title] Further examples of s* *imple "moduli spaces" of topological spaces are provided by the configuration spaces * *of n distinct points in a manifold M, which have also been studied extensively (see * *[FH ] for a comprehensive survey). MODULI SPACES OF HOMOTOPY THEORY 3 (a) For each H, we have homological classification of spaces with this cohom* *ology algebra. Nerves and moduli spaces Since ordinary (integral) homotopy types do not have any known differential g* *raded models, there is no hope of generalizing the deformation approach of sections 2* *-5 to cover them, too. www.math.purdue.edu /research/atopology/Blanc/mod03.txt   (7641 words)

 [No title]   (Site not responding. Last check: 2007-10-21) Harvard Room 507 Moduli spaces of abelian varieties in positive characteristic FRANS OORT (Universiteit Utrecht and MIT) Abstract: Abelian varieties in characteristic p have additional structures (the action of Frobenius on the p-divisible group, Newton polygons, structures of the p-kernel) which give access to geometric aspects of these moduli spaces in all characteristics. For g>2 the moduli space of abelian varieties of dimension g over the complex numbers does not contain a compact codimension g subvariety. As a corollary: a new proof of the irreducibility of the moduli space of principally polarized abelian varieties in arbitrary characteristic. www-math.mit.edu /~abuch/seminar/abstracts/020305-oort.txt   (292 words)

 Weinberger, S.: Computers, Rigidity, and Moduli: The Large-Scale Fractal Geometry of Riemannian Moduli Space. The Large-Scale Fractal Geometry of Riemannian Moduli Space He provides applications to the problem of closed geodesics, the theory of submanifolds, and the structure of the moduli space of isometry classes of Riemannian metrics with curvature bounds on a given manifold. Ultimately, geometric complexity of a moduli space forces functions defined on that space to have many critical points, and new results about the existence of extrema or equilibria follow. pup.princeton.edu /titles/7903.html   (420 words)

 The classical moduli space   (Site not responding. Last check: 2007-10-21) The vector moduli space is determined by the on the moduli, which is necessary to have a positive definite metric on the momentum space. It is now easy to determine the rest of the structure on the moduli space from the general formulae. www.phys.uu.nl /~hofman/scriptie/duality/node31.html   (1220 words)

 Brian Greene - Wikipedia, the free encyclopedia His second book, The Fabric of the Cosmos (2004), is about space, time, and the nature of the universe. He currently studies string cosmology, especially the imprints of trans-Planckian physics on the cosmic microwave background, and brane-gas cosmologies that could explain why the space around us has three large dimensions. Expanded on the suggestion of a fl hole electron, namely that the electron may be a fl hole. en.wikipedia.org /wiki/Brian_Greene   (934 words)

 DC MetaData for: Moduli Space and Structure of Noncommutative 3-Spheres   (Site not responding. Last check: 2007-10-21) Abstract: We analyse the moduli space and the structure of noncommutative 3-spheres. The moduli space of noncommutative 3-spheres is identified with equivalence classes of pairs of points in a symmetric space of unitary unimodular symmetric matrices. The scaling foliation of the moduli space is identified to the gradient flow of the character of a virtual representation of \$SO(6)\$. www.esi.ac.at /Preprint-shadows/esi1365.html   (240 words)

 Moduli space of Calabi-Yau manifolds   (Site not responding. Last check: 2007-10-21) As harmonic 2-forms are annihilated by the natural Laplacian on forms, which is the same as the Lichnerowicz Laplacian on forms, we can add B and G and identify the tangent space to the moduli space as the kernel of the Lichnerowicz Laplacian. It has to be added that although the Kähler moduli and the complex structure moduli are locally orthogonal, it may not be possible in general to define a corresponding product structure on the global moduli space. We proceed to describe the moduli space of complex structures of the Calabi-Yau manifold. www.phys.uu.nl /~hofman/scriptie/duality/node40.html   (1246 words)

 MAT 615 - Topics in Algebraic Geometry -- Spring 2004   (Site not responding. Last check: 2007-10-21) Moduli of Curves, J. Harris and I. Morrison, GTM 187, Springer A conjectural description of the tautological ring of the moduli space of curves, C. Faber, math.AG/9711218 The geometry of moduli spaces of shaves, D. Huybrechts, M. Lehn, Aspects of Math. www.math.sunysb.edu /~sorin/615   (124 words)

 The Moduli Space of Curves Resource Page   (Site not responding. Last check: 2007-10-21) Topology and geometry of the moduli space of curves The content grew out of the ARCC workshop: Topology and Geometry of the Moduli Space of Curves, March 28 to April 1, 2005, organized by Ulrike Tillmann and Ravi Vakil. This should also be an aid to students and to those who wish to learn a different camp's approach to the moduli space of curves. www.aimath.org /WWN/modspacecurves/aim/modspacecurves.html   (211 words)

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