Abstract. We provide a translation between Chekanov’s combinatorial theory for invariants of Legendrian knots in the standard contact R 3 and a relative version of Eliashberg and Hofer’s Contact Homology. We use this translation to transport the idea of “coherent orientations ” from the Contact Homology world to Chekanov’s combinatorial setting. As a result, we obtain a lifting of Chekanov’s differential graded algebra invariant to an algebra over Z[t, t −1] with a full Z grading. 1.

products; for example the zeroset of a smooth function on a manifold is not necessarily a manifold, and the non-transverse intersection of submanifolds is

Abstract. The algebraic definition of charges for symmetry-preserving D-branes in Wess-Zumino-Witten models is shown to coincide with the geometric definition, for all simple Lie groups. The charge group for such branes is computed from the ambiguities inherent in the geometric definition.

Little is known about the global structure of the basins of attraction of Newton’s method in two or more complex variables. We make the first steps by focusing on the specific Newton mapping to solve for the common roots of P(x, y) = x(1 − x) and Q(x,y) = y 2 + Bxy − y. There are invariant circles S0 and S1 within the lines x = 0 and x = 1 which are superattracting in the x-direction and hyperbolically repelling within the vertical line. We show that S0 and S1 have local super-stable manifolds, which when pulled back under iterates of N form global super-stable spaces W0 and W1. By blowing-up the points of indeterminacy p and q of N and all of their inverse images under N we prove that W0 and W1 are real-analytic varieties. We define linking between closed 1-cycles in Wi (i = 0, 1) and an appropriate positive closed (1,1) current providing a homomorphism lk: H1(Wi, Z) → Q. If Wi intersects the critical value locus of N, this homomorphism has dense image, proving that H1(Wi, Z) is infinitely generated. Using the Mayer-Vietoris exact sequence and an algebraic trick, we show that the same is true for the closures of the basins of the roots W(ri). Key Words. Complex dynamics, Newton’s Method, homology, linking numbers, invariant currents.

arXiv version: fonts, pagination and layout may vary from GT published version

Sophia-Antipolis, France, during the summer of 2007. We are grateful to Nicholas Ayache and the members of the ASCLEPIOS group for their hospitality. Some people have read incomplete versions of the manuscript, pointed out several mistakes, and given useful suggestions. In particular, we thank Dimas Martínez, Luis Gustavo Nonato, and Luiz Velho for that. All of them are currently working with us on some extensions of the results described in this manuscript. Almost all triangle meshes used in the experiments described in this manuscript belong to the repository of 3D digital models of the AIM@SHAPE project. We also relied upon several open source and freely available softwares, such as MeshLab, Geomview, SurfRemesh and Blender, for preprocessing the triangle meshes we used. Luiz Velho also provided us with a software for mesh simplification based on his Four-Face Clusters algorithm. Marcelo Siqueira would like to thank CNPq for partially supporting his research (Grant 475703/2006-

Abstract not found

Abstract. We compare the K-theories of symplectic quotients with respect to a compact connected Lie group and with respect to its maximal torus, and in particular we give a method for computing the former in terms of the latter. More specifically, let G be a compact connected Lie group with no torsion in its fundamental group, let T be a maximal torus of G, and let M be a compact Hamiltonian G-space. Let M//G and M//T denote the symplectic quotient of M by G and by T, respectively. Using Hodgkin’s Künneth spectral sequence for equivariant K-theory, we express the K-theory of M//G in terms of the elements in the K-theory of M//T which are invariant under the action of the Weyl group, in addition to the Euler class e of a natural Spin c vector bundle over M//T. This Euler class e is induced by the denominator in the Weyl character formula, viewed as a virtual representation of T; this is relevant for our proof. Our results are K-theoretic analogues of similar (unpublished) results by Martin for rational cohomology. However, our results and approach differ from his in three significant ways. First, Martin’s method involves integral formulæ, but the corresponding index formulæ in K-theory are too coarse a tool, as they cannot detect torsion. Instead, we carefully

We give the homotopy classification and compute the index of boundary value problems for elliptic equations. The classical case of operators that satisfy the Atiyah–Bott condition is studied first. We also consider the general case of boundary value problems for operators that do not necessarily satisfy the Atiyah–Bott condition. Keywords: elliptic boundary value problems, Atiyah–Bott condition, index theory,

We show that a locally Ahlfors 2-regular and locally linearly locally contractible metric surface is locally quasisymmetrically equivalent to the disk. We also discuss an application of this result to the problem of characterizing surfaces embedded in some Euclidean space that are locally bi-Lipschitz equivalent to a ball in the plane.