Spectral partitioning methods use the Fiedler vector---the eigenvector of the secondsmallest eigenvalue of the Laplacian matrix---to find a small separator of a graph. These methods are important components of many scientific numerical algorithms and have been demonstrated by experiment to work extremely well. In this paper, we show that spectral partitioning methods work well on bounded-degree planar graphs and finite element meshes--- the classes of graphs to which they are usually applied. While naive spectral bisection does not necessarily work, we prove that spectral partitioning techniques can be used to produce separators whose ratio of vertices removed to edges cut is O( p n) for bounded-degree planar graphs and two-dimensional meshes and O i n 1=d j for well-shaped d-dimensional meshes. The heart of our analysis is an upper bound on the second-smallest eigenvalues of the Laplacian matrices of these graphs. 1. Introduction Spectral partitioning has become one of the mos...

We present NC approximation schemes for a number of graph problems when restricted to geometric graphs including unit disk graphs and graphs drawn in a civilized manner. Our approximation schemes exhibit the same time versus performance trade-off as the best known approximation schemes for planar graphs. We also define the concept of -precision unit disk graphs and show that for such graphs the approximation schemes have a better time versus performance trade-off than the approximation schemes for arbitrary unit disk graphs. Moreover, compared to unit disk graphs, we show that for -precision unit disk graphs, many more graph problems have efficient approximation schemes. Our NC approximation schemes can also be extended to obtain efficient NC approximation schemes for several PSPACE-hard problems on unit disk graphs specified using a restricted version of the hierarchical specification language of Bentley, Ottmann and Widmayer. The approximation schemes for hierarchically specified un...

Thurston and many others developed the theory of geometrically finite ends of hyperbolic 3–manifolds. It remained to understand those ends which are not geometrically finite; such ends are called geometrically infinite.

Abstract. A collection of n balls in d dimensions forms a k-ply system if no point in the space is covered by more than k balls. We show that for every k-ply system �, there is a sphere S that intersects at most O(k 1/d n 1�1/d) balls of � and divides the remainder of � into two parts: those in the interior and those in the exterior of the sphere S, respectively, so that the larger part contains at most (1 � 1/(d � 2))n balls. This bound of O(k 1/d n 1�1/d) is the best possible in both n and k. We also present a simple randomized algorithm to find such a sphere in O(n) time. Our result implies that every k-nearest neighbor graphs of n points in d dimensions has a separator of size O(k 1/d n 1�1/d). In conjunction with a result of Koebe that every triangulated planar graph is isomorphic to the intersection graph of a disk-packing, our result not only gives a new geometric proof of the planar separator theorem of Lipton and Tarjan, but also generalizes it to higher dimensions. The separator algorithm can be used for point location and geometric divide and conquer in a fixed dimensional space.

Abstract. In an earlier paper, we used the absolute grading on Heegaard Floer homology HF + to give restrictions on knots in S 3 which admit lens space surgeries. The aim of the present article is to exhibit stronger restrictions on such knots, arising from knot Floer homology. One consequence is that all the non-zero coefficients of the Alexander polynomial of such a knot are ±1. This information in turn can be used to prove that certain lens spaces are not obtained as integral surgeries on knots. In fact, combining our results with constructions of Berge, we classify lens spaces L(p, q) which arise as integral surgeries on knots in S 3 with |p | ≤ 1500. Other applications include bounds on the four-ball genera of knots admitting lens space surgeries (which are sharp for Berge’s knots), and a constraint on three-manifolds obtained as integer surgeries on alternating knots, which is closely to related to a theorem of Delman and Roberts. 1.

In 1960 Sasaki [Sas] introduced a type of metric-contact structure which can be thought of as the odd-dimensional version of Kähler geometry. This geometry became known as Sasakian geometry, and although it has been studied fairly extensively ever

Abstract. Let (X,d) be a tree (T) of hyperbolic metric spaces satisfying the quasi-isometrically embedded condition. Let v be a vertex of T. Let (Xv, dv) denote the hyperbolic metric space corresponding to v. Then i: Xv → X extends continuously to a map î: ̂Xv → ̂X. This generalizes a Theorem of Cannon and Thurston. The techniques are used to give a new proof of a result of Minsky: Thurston’s ending lamination conjecture for certain Kleinian groups. Applications to graphs of hyperbolic groups and local connectivity of limit sets of Kleinian groups are also given. 1.

. Explicit families of entire circle patterns with the combinatorics of the square grid are constructed, and it is shown that the collection of entire, locally univalent circle patterns on the sphere is infinite dimensional. In Particular, Doyle's conjecture is false in this setting. Mobius invariants of circle patterns are introduced, and turn out to be discrete analogs of the Schwarzian derivative. The invariants satisfy a nonlinear discrete version of the Cauchy-Riemann equations. A global analysis of the solutions of these equations yields a rigidity theorem characterizing the Doyle spirals. It is also shown that by prescribing boundary values for the Mobius invariants, and solving the appropriate Dirichlet problem, a locally univalent meromorphic function can be approximated by circle patterns. 1991 Mathematics Subject Classification. 30C99, 05B40, 30D30, 31A05, 31C20, 30G25. Key words and phrases. Meromorphic functions, Schwarzian derivative, rigidity, error function, Dirichlet ...

Delaunay triangulations and Voronoi diagrams are one of the most thoroughly studies objects in computational geometry, with numerous applications including nearest-neighbor searching, clustering, finite-element mesh generation, deformable surface modeling, and surface reconstruction. Many algorithms in these application domains begin by constructing the Delaunay triangulation or Voronoi diagram of a set of points in R³. Since three-dimensional Delaunay triangulations can have complexity Ω(n²) in the worst case, these algorithms have worst-case running time \Omega (n2). However, this behavior is almost never observed in practice except for highly-contrived inputs. For all practical purposes, three-dimensional Delaunay triangulations appear to have linear complexity. This frustrating

We discuss gauge theory with a topological N = 2 symmetry. This theory captures the de Rham complex and Riemannian geometry of some underlying moduli space M and the partition function equals the Euler number χ(M) of M. We explicitly deal with moduli spaces of instantons and of flat connections in two and three dimensions. To motivate our constructions we explain the relation between the Mathai-Quillen formalism and supersymmetric quantum mechanics and introduce a new kind of supersymmetric quantum mechanics based on the Gauss-Codazzi equations. We interpret the gauge theory actions from the Atiyah-Jeffrey point of view and relate them to supersymmetric quantum mechanics on spaces of connections. As a consequence 1