Showing papers on "Geometry and topology published in 1999"

Generalized View‐Dependent Simplification

Abstract: We propose a technique for performing view-dependent geometry and topology simplifications for level-of-detailbased renderings of large models. The algorithm proceeds by preprocessing the input dataset into a binary tree, the view-dependence tree of general vertex-pair collapses. A subset of the Delaunay edges is used to limit the number of vertex pairs considered for topology simplification. Dependencies to avoid mesh foldovers in manifold regions of the input object are stored in the view-dependence tree in an implicit fashion. We have observed that this not only reduces the space requirements by a factor of two, it also highly localizes the memory accesses at run time. The view-dependence tree is used at run time to generate the triangles for display. We also propose a cubic-spline-based distance metric that can be used to unify the geometry and topology simplifications by considering the vertex pos itions and normals in an integrated manner.

Supersymmetric quantum theory and non-commutative geometry

Abstract: Classical differential geometry can be encoded in spectral data, such as Connes' spectral triples, involving supersymmetry algebras. In this paper, we formulate non-commutative geometry in terms of supersymmetric spectral data. This leads to generalizations of Connes' non-commutative spin geometry encompassing non-commutative Riemannian, symplectic, complex-Hermitian and (Hyper-) Kahler geometry. A general framework for non-commutative geometry is developed from the point of view of supersymmetry and illustrated in terms of examples. In particular, the non-commutative torus and the non-commutative 3-sphere are studied in some detail.

New Trends in Algebraic Geometry: Mordell–Weil lattices for higher genus fibration over a curve

Abstract: Introduction, notation Let K = k(C) be the function field of an algebraic curve C over an algebraically closed ground field k. Let Γ/K be a smooth projective curve of genus g > 0 with a k -rational point O ∈ Γ(K), and let J/K denote the Jacobian variety of Γ/K . Further let (τ, B) be the K/k :-trace of J (see §2 below and). Then the Mordell-Weil theorem (in the function field case) states that the group of K -rational points J(K) modulo the subgroup τB(k) is a finitely generated Abelian group. Now, given Γ/K, there is a smooth projective algebraic surface S with genus g fibration f : S → C which has Γ as its generic fibre and which is relatively minimal in the sense that no fibres contain an exceptional curve of the first kind (−1-curve). It is known that the correspondence Γ/K ↔ ( S, f ) is bijective up to isomorphisms (cf.). The main purpose of this paper is to give the Mordell-Weil group M = J(K)/τB(k) (modulo torsion) the structure of Euclidean lattice via intersection theory on the algebraic surface S. The resulting lattice is the Mordell- Weil lattice (MWL) of the Jacobian variety J/K, which we sometimes call MWL of the curve Γ/K or of the fibration f : S → C . For this, we first establish the relationship between the Mordell-Weil group and the Neron-Severi group NS( S ) of S (Theorem 1, stated in §2 and proved in §3). Then (in §4) we introduce the structure of lattice on the Mordell-Weil group by defining a natural pairing in terms of the intersection pairing on NS( S ).

Complex symplectic geometry with applications to ordinary differential operators

Abstract: Complex symplectic spaces, and their Lagrangian subspaces, are defined in accord with motivations from Lagrangian classical dynamics and from linear ordinary differential operators; and then their basic algebraic properties are established. After these purely algebraic developments, an Appendix presents a related new result on the theory of self-adjoint operators in Hilbert spaces, and this provides an important application of the principal theorems. 1. Fundamental definitions for complex symplectic spaces, and three motivating illustrations Complex symplectic spaces, as defined below, are non-trivial generalizations of the real symplectic spaces of Lagrangian classical dynamics [AM], [MA]. Further, these complex spaces provide important algebraic structures clarifying the theory of boundary value problems of linear ordinary differential equations, and the theory of the associated self-adjoint linear operators on Hilbert spaces [AG], [DS], [NA]. These fundamental concepts are introduced in connection with three examples or motivating discussions in this first introductory section, with further technical details and applications presented in the Appendix at the end of this paper. The new algebraic results are given in the second and main section of this paper, which developes the principal theorems of the algebra of finite dimensional complex symplectic spaces and their Lagrangian subspaces. A preliminary treatment of these subjects, with full attention to the theory of self-adjoint operators, can be found in the earlier monograph of these authors [EM]. Definition 1. A complex symplectic space S is a complex linear space, with a prescribed symplectic form [:], namely a sesquilinear form (i) u, v → [u : v], S × S → C, so [c1u + c2v : w] = c1[u : w] + c2[v : w], (1.1) which is skew-Hermitian, (ii) [u : v] = −[v : u], so [u : c1v + c2w] = c̄1[u : v] + c̄2[u : w] Received by the editors August 19, 1997. 1991 Mathematics Subject Classification. Primary 34B05, 34L05; Secondary 47B25, 58F05.

The Design and Implementation of Planar Maps in CGAL

Abstract: Planar maps are fundamental structures in computational geometry. They are used to represent the subdivision of the plane into regions and have numerous applications. We describe the planar map package of CGAL1 -- the Computational Geometry Algorithms Library. We discuss problems that arose in the design and implementation of the package and report the solutions we have found for them. In particular we introduce the two main classes of the design--planar maps and topological maps that enable the convenient separation between geometry and topology. We also describe the geometric traits which make our package flexible by enabling to use it with any family of curves as long as the user supplies a small set of operations for the family. Finally, we present the algorithms we implemented for point location in the map, together with experimental results that compare their performance.

Existence of close pseudoholomorphic disks for almost complex manifolds and an applications to Kobayashi-Royden pseudonorm

Abstract: Dette er forfatternes aksepterte versjon. This is the author’s final accepted manuscript.

Heat kernels, symplectic geometry, moduli spaces, and finite groups

Abstract: In this note we apply heat kernels to derive some localization formula in sympletcic geometry, to study moduli spaces of flat connections on a Riemann surface, to obtain the push-forward measures for certain maps between Lie groups and to solve equations in finite groups.

Grothendieck's reconstruction principle and 2-dimensional topology and geometry

Abstract: This paper attempts to relate some ideas of Grothendieck in his Esquisse d'un programme and some of the recent results on 2-dimensional topology and geometry. Especially, we shall discuss the Teichmuller theory, the mapping class groups, $SL(2, \bold C)$ representation variety of surface groups, and Thurston's theory of measured laminations.

Singularity Theory: Multiplicities of Zero-Schemes in Quasihomogeneous Corank-1 Singularities C n → C n

Abstract: How many cusps does a swallowtail have, After it becomes a stable map, And how many swallowtails does a butterfly have, After it . . . (with apologies to B. Dylan)