Results 1  10
of
60
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 114 (7 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
Dbranes in LandauGinzburg models and algebraic geometry
 J. High Energy Phys
"... geometry ..."
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How many zeros of a random polynomial are real
 Bull. Amer. Math. Soc. (N.S
, 1995
"... Abstract. We provide an elementary geometric derivation of the Kac integral formula for the expected number of real zeros of a random polynomial with independent standard normally distributed coefficients. We show that the expected number of real zeros is simply the length of the moment curve (1, t, ..."
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Cited by 81 (0 self)
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Abstract. We provide an elementary geometric derivation of the Kac integral formula for the expected number of real zeros of a random polynomial with independent standard normally distributed coefficients. We show that the expected number of real zeros is simply the length of the moment curve (1, t,..., t n) projected onto the surface of the unit sphere, divided by π. The probability density of the real zeros is proportional to how fast this curve is traced out. We then relax Kac’s assumptions by considering a variety of random sums, series, and distributions, and we also illustrate such ideas as integral geometry and the FubiniStudy metric. Contents 1.
Universality and scaling of correlations between zeros on complex manifolds
 Invent. Math
"... Abstract. We study the limit as N → ∞ of the correlations between simultaneous zeros of random sections of the powers L N of a positive holomorphic line bundle L over a compact complex manifold M, when distances are rescaled so that the average density of zeros is independent of N. We show that the ..."
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Cited by 80 (26 self)
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Abstract. We study the limit as N → ∞ of the correlations between simultaneous zeros of random sections of the powers L N of a positive holomorphic line bundle L over a compact complex manifold M, when distances are rescaled so that the average density of zeros is independent of N. We show that the limit correlation is independent of the line bundle and depends only on the dimension of M and the codimension of the zero sets. We also provide some explicit formulas for pair correlations. In particular, we provide an alternate derivation of Hannay’s limit pair correlation function for SU(2) polynomials, and we show that this correlation function holds for all compact Riemann surfaces.
The Complex Hyperbolic Geometry of the Moduli Space of Cubic Surfaces
 In preparation
, 1998
"... Contents 1. Introduction 2. Moduli of smooth cubic surfaces 3. Moduli of stable cubic surfaces 4. Proofs of lemmas 5. Topology of nodal degenerations 6. Fractional dierentials and extension of the period map 7. The monodromy group and hyperplane conguration 8. Cuspidal degenerations 9. Proo ..."
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Cited by 70 (17 self)
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Contents 1. Introduction 2. Moduli of smooth cubic surfaces 3. Moduli of stable cubic surfaces 4. Proofs of lemmas 5. Topology of nodal degenerations 6. Fractional dierentials and extension of the period map 7. The monodromy group and hyperplane conguration 8. Cuspidal degenerations 9. Proof of the main theorem 10. The universal cubic surface 11. Automorphisms of cubic surfaces 12. Index of notation 1: Introduction A classical theorem of great beauty describes the connection between cubic curves and hyperbolic geometry: the moduli space of the former is a quotient of the complex hyperbolic line (or real hyperbolic plane). The purpose of this paper is to exhibit a similar connection for cubic surfaces: their space of moduli is a quotient of complex hyperbolic fourspace. We will make a precise statement below. The theorem for cubic curves can be established using periods of integrals, or, in modern language, Hodge structures. Indeed
An analytic proof of the geometric quantization conjecture
 of GuilleminSternberg, Invent. Math 132
, 1998
"... Abstract. We present a direct analytic approach to the GuilleminSternberg conjecture [GS] that ‘geometric quantization commutes with symplectic reduction’, which was proved recently by Meinrenken [M1], [M2] and Vergne [V1], [V2] et al. Besides providing a new proof of this conjecture, our methods a ..."
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Cited by 57 (8 self)
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Abstract. We present a direct analytic approach to the GuilleminSternberg conjecture [GS] that ‘geometric quantization commutes with symplectic reduction’, which was proved recently by Meinrenken [M1], [M2] and Vergne [V1], [V2] et al. Besides providing a new proof of this conjecture, our methods also lead immediately to further extensions in various contexts. Let M;x be a closed symplectic manifold such that there is a Hermitian line bundle L over M admitting a Hermitian connection rL with the property that ÿ1p
Moduli spaces of parabolic Higgs bundles and parabolic K(D) pairs over smooth curves: I
, 1998
"... This paper concerns the moduli spaces of rank two parabolic Higgs bundles and parabolic K(D) pairs over a smooth curve. Precisely which parabolic bundles occur in stable K(D) pairs and stable Higgs bundles is determined. Using Morse theory, the moduli space of parabolic Higgs bundles is shown to b ..."
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Cited by 39 (2 self)
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This paper concerns the moduli spaces of rank two parabolic Higgs bundles and parabolic K(D) pairs over a smooth curve. Precisely which parabolic bundles occur in stable K(D) pairs and stable Higgs bundles is determined. Using Morse theory, the moduli space of parabolic Higgs bundles is shown to be a noncompact, connected, simply connected manifold, and a computation of its Poincaré polynomial is given.
Degrees of real Wronski maps
 Discrete Comput. Geom
, 1975
"... We study the map which sends vectors of polynomials into their Wronski determinants. This defines a projection map of a Grassmann variety which we call a Wronski map. Our main result is computation of degrees of the real Wronski maps. Connections with real algebraic geometry and control theory are d ..."
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Cited by 28 (5 self)
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We study the map which sends vectors of polynomials into their Wronski determinants. This defines a projection map of a Grassmann variety which we call a Wronski map. Our main result is computation of degrees of the real Wronski maps. Connections with real algebraic geometry and control theory are described.
A survey of moving frames
 Computer Algebra and Geometric Algebra with Applications. Volume 3519 of Lecture Notes in Computer Science, 105–138
, 2005
"... Abstract. This article presents the equivariant method of moving frames for finitedimensional Lie group actions, surveying a variety of applications, including geometry, differential equations, computer vision, numerical analysis, the calculus of variations, and invariant flows. 1. Introduction. Acc ..."
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Cited by 24 (3 self)
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Abstract. This article presents the equivariant method of moving frames for finitedimensional Lie group actions, surveying a variety of applications, including geometry, differential equations, computer vision, numerical analysis, the calculus of variations, and invariant flows. 1. Introduction. According to Akivis, [1], the method of moving frames originates in work of the Estonian mathematician Martin Bartels (1769–1836), a teacher of both Gauss and Lobachevsky. The field is most closely associated with Élie Cartan, [21], who forged earlier contributions by Darboux, Frenet, Serret, and Cotton into a powerful tool for analyzing the geometric
Constant terms of powers of a Laurent polynomial
"... this paper we prove the conjecture for commutative K. Therefore, from now on ..."
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Cited by 24 (0 self)
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this paper we prove the conjecture for commutative K. Therefore, from now on