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11
Gauge theory for embedded surfaces
 I, Topology
, 1993
"... (i) Topology of embedded surfaces. Let X be a smooth, simplyconnected 4manifold, and ξ a 2dimensional homology class in X. One of the features of topology in dimension 4 is the fact that, although one may always represent ξ as the fundamental class of some smoothly ..."
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Cited by 68 (6 self)
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(i) Topology of embedded surfaces. Let X be a smooth, simplyconnected 4manifold, and ξ a 2dimensional homology class in X. One of the features of topology in dimension 4 is the fact that, although one may always represent ξ as the fundamental class of some smoothly
K3 surfaces and string duality
"... The primary purpose of these lecture notes is to explore the moduli space of type IIA, type IIB, and heterotic string compactified on a K3 surface. The main tool which is invoked is that of string duality. K3 surfaces provide a fascinating arena for string compactification as they are not trivial sp ..."
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Cited by 62 (14 self)
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The primary purpose of these lecture notes is to explore the moduli space of type IIA, type IIB, and heterotic string compactified on a K3 surface. The main tool which is invoked is that of string duality. K3 surfaces provide a fascinating arena for string compactification as they are not trivial spaces but are sufficiently simple for one to be able to analyze most of their properties in detail. They also make an almost ubiquitous appearance in the common statements concerning string duality. We review the necessary facts concerning the classical geometry of K3 surfaces that will be needed and then we review “old string theory ” on K3 surfaces in terms of conformal field theory. The type IIA string, the type IIB string, the E8 × E8 heterotic string, and Spin(32)/Z2 heterotic string on a K3 surface are then each analyzed in turn. The discussion is biased in favour of purely geometric notions concerning the K3 surface
RiemannRoch for toric orbifolds
 J. Diff. Geom
, 1997
"... Let 1,:::, d and be elements of the integer lattice, Z n, and let N ( ) be the number of solutions, k =(k1;:::;kd), of the equation (1.1) k1 1 +:::+ kd d =; ..."
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Cited by 21 (3 self)
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Let 1,:::, d and be elements of the integer lattice, Z n, and let N ( ) be the number of solutions, k =(k1;:::;kd), of the equation (1.1) k1 1 +:::+ kd d =;
A spectral sequence for splines
 ADVANCES IN APPLIED MATHEMATICS
, 1997
"... We define a complex R=J of graded modules on a ddimensional simplicial complex #, so that the top homology module of this complex consists of piecewise polynomial functions (splines) of smoothness r on the cone of #. In a series of papers ([4], [5], [6]), Billera and Rose used a similar approach to ..."
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Cited by 6 (4 self)
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We define a complex R=J of graded modules on a ddimensional simplicial complex #, so that the top homology module of this complex consists of piecewise polynomial functions (splines) of smoothness r on the cone of #. In a series of papers ([4], [5], [6]), Billera and Rose used a similar approach to study the dimension of the spaces of splines on #, but with a complex substantially di#erent from R=J.We obtain bounds on the dimension of the homology modules H i #R=J #, for all i#d, and find a spectral sequence which relates these modules to the spline module. We use this to give simple conditions governing the projective dimension of the spline module. We also prove that if the spline module is free, then it is determined entirely by local data; that is, by the arrangements of hyperplanes incident to the various dimensional faces of #.
PU(2) monopoles. II: Toplevel SeibergWitten moduli spaces and Witten's conjecture in low degrees
 J. REINE ANGEW. MATH.
, 2001
"... In this article, a continuation of [10], we complete the proof  for a broad class of fourmanifolds  of Witten's conjecture that the Donaldson and SeibergWitten series coincide, at least through terms of degree less than or equal to c 2, where c ˆ 1 …7w ‡ 11s † and w and s are the Euler charac ..."
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Cited by 3 (0 self)
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In this article, a continuation of [10], we complete the proof  for a broad class of fourmanifolds  of Witten's conjecture that the Donaldson and SeibergWitten series coincide, at least through terms of degree less than or equal to c 2, where c ˆ 1 …7w ‡ 11s † and w and s are the Euler characteristic and signature of the four4 manifold. We use our computations of Chern classes for the virtual normal bundles for the SeibergWitten strata from the companion article [10], a comparison of all the orientations, and the PU…2 † monopole cobordism to compute pairings with the links of levelzero SeibergWitten moduli subspaces of the moduli space of PU…2 † monopoles. These calculations then allow us to compute lowdegree Donaldson invariants in terms of SeibergWitten invariants and provide a partial veri®cation of Witten's conjecture.
On hermitianholomorphic classes related to uniformization, the dilogarithm and the liouville action
 Communications in Mathematical Physics
"... Metrics of constant negative curvature on a compact Riemann surface are critical points of the Liouville action functional, which in recent constructions is rigorously defined as a class in a Čechde Rham complex with respect to a suitable covering of the surface. We show that this class is the squa ..."
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Cited by 3 (2 self)
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Metrics of constant negative curvature on a compact Riemann surface are critical points of the Liouville action functional, which in recent constructions is rigorously defined as a class in a Čechde Rham complex with respect to a suitable covering of the surface. We show that this class is the square of the metrized holomorphic tangent bundle in hermitianholomorphic Deligne cohomology. We achieve this by introducing a different version of the hermitianholomorphic Deligne complex which is nevertheless quasiisomorphic to the one introduced by Brylinski in his construction of Quillen line bundles. We reprove the relation with the determinant of cohomology construction. Furthermore, if we specialize the covering to the one provided by a Kleinian uniformization (thereby allowing possibly disconnected surfaces) the same class can be reinterpreted as the transgression of the regulator class
www.elsevier.com/locate/cam Existence of resonances in magnetic scattering
, 2001
"... The Schrodinger operator with a compactly supported magnetic eld is shown to produce in nitely many resonances, in any odd dimension ¿ 3.The proof is based on the Poisson formula for resonances and properties of the magnetic heat invariants. ..."
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Cited by 2 (0 self)
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The Schrodinger operator with a compactly supported magnetic eld is shown to produce in nitely many resonances, in any odd dimension ¿ 3.The proof is based on the Poisson formula for resonances and properties of the magnetic heat invariants.
Quotient spaces and critical points of invariant functions for C*actions
, 1993
"... Consider a linear action of the group C on X = C n+1 . We study the fundamental algebraic properties of the sheaves of invariant and basic dierential forms for such an action, and use these to dene an algebraic notion of multiplicity for critical points of functions which are invariant under ..."
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Cited by 1 (1 self)
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Consider a linear action of the group C on X = C n+1 . We study the fundamental algebraic properties of the sheaves of invariant and basic dierential forms for such an action, and use these to dene an algebraic notion of multiplicity for critical points of functions which are invariant under the C action. We also prove a theorem relating the cohomology of the Milnor bre of the critical point on the quotient space with this algebraic multiplicity. Contents 1 C actions and their quotient spaces 4 The quotient space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 The stratication of the quotient space . . . . . . . . . . . . . . . . . . . . 7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2 Invariant and basic dierential forms 10 Local cohomology calculations . . . . . . . . . . . . . . . . . . . . . . . . 13 3 Quasiacyclicity of the df^complexes 20 The invariant df^complex . . . . . . . . . . . . . . . . ....
Hyperplane Sections of Grassmannians . . .
, 2001
"... We obtain some effective lower and upper bounds for the number of (n, k)MDS linear codes over �. As a consequence, one obtains an asymptotic formula for this number. These results also apply for the number of inequivalent representations over � of the uniform matroid or, alternatively, the number o ..."
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We obtain some effective lower and upper bounds for the number of (n, k)MDS linear codes over �. As a consequence, one obtains an asymptotic formula for this number. These results also apply for the number of inequivalent representations over � of the uniform matroid or, alternatively, the number of �rational points of certain open strata of Grassmannians. The techniques used in the determination of bounds for the number of MDS codes are applied to deduce several geometric properties of certain sections of Grassmannians by coordinate hyperplanes.
GENERATORS OF H 1 ( h ; Z) FOR TRIANGULATED SURFACES: Construction And Classification
"... We consider a bounded Lipschitzpolyhedron Ω R³ of general topology equipped with a tetrahedral triangulation that induces a mesh h of the surface @ We seek a maximal set of surface edge cycles that are independent in H1( h ; Z) and bounding with respect to the exterior of We present an algori ..."
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We consider a bounded Lipschitzpolyhedron Ω R³ of general topology equipped with a tetrahedral triangulation that induces a mesh h of the surface @ We seek a maximal set of surface edge cycles that are independent in H1( h ; Z) and bounding with respect to the exterior of We present an algorithm for constructing suitable 1cycles in h : First, representatives of a basis of the homology group H1( h ; Z) are constructed, merely using the combinatorial description of the surface mesh h . Then, a duality pairing based on linking numbers is used to determine those combinations that are bounding w.r.t. R n This is the key to circumventing a triangulation of the exterior region R in the computations. For shaperegular, quasiuniform families of meshes, the asymptotic complexity of the algorithm is shown to be O(N ), where N is the number of edges of h . The scheme