No abstract

IntroductionOne of the underlying principles in the study of Kleinian groups is that aspects of the complex projective geometry of quotients ofĈ by these groups reflect properties of the three-dimensional hyperbolic geometry of the quotients of H 3 by these groups. Yet, even though it has been over thirty-five years since Lipman Bers exhibited a holomorphic embedding of the Teichmüller space of Riemann surfaces in terms of the projective geometry of a Teichmüller space of quasi-Fuchsian manifolds, no corresponding embedding in terms of the three-dimensional hyperbolic geometry has been presented. One of the goals of this paper is to give such an embedding. This embedding is straightforward and has been expected for some time ([Ta97], [Mc98]): to each member of a Bers slice of the space QF of quasi-Fuchsian 3-manifolds, we associate the bending measured lamination of the convex hull facing the fixed "conformal" end.The geometric relationship between a boundary component of a convex hull and the projective surface at infinity for its end is given by a process known as grafting, an operation on projective structures on surfaces that traces its roots back at least to Klein [Kl33, p. 230 ). The main technical tool in our proof that bending measures give coordinates for Bers slices, and the second major goal of this paper, is the completion of the proof of the "Grafting Conjecture". This conjecture states that for a fixed measured lamination λ, the self-map of Teichmüller space induced by grafting a surface along λ is a homeomorphism of Teichmüller space; our contribution to this argument is a proof of the injectivity of the grafting map. While the principal application of this result that we give is to geometric coordinates on the Bers slice of QF , one expects that the grafting homeomorphism might lead to other systems of geometric coordinates for other families of Kleinian groups (see §5.2); thus we feel that this result is of interest in its own right.We now state our results and methods more precisely. Throughout, S will denote a fixed differentiable surface which is closed, orientable, and of genus g ≥ 2. Let T g be the corresponding Teichmüller space of marked conformal structures on S, and let P g denote the deformation space of (complex) projective structures on S (see §2 for definitions).

We initiate the study of classical knots through the homotopy class of the nth evaluation map of the knot, which is the induced map on the compactified n-point configuration space. Sending a knot to its nth evaluation map realizes the space of knots as a subspace of what we call the nth mapping space model for knots. We compute the homotopy types of the first three mapping space models, showing that the third model gives rise to an integer-valued invariant. We realize this invariant in two ways, in terms of collinearities of three or four points on the knot, and give some explicit computations. We show this invariant coincides with the second coefficient of the Conway polynomial, thus giving a new geometric definition of the simplest finite-type invariant. Finally, using this geometric definition, we give some new applications of this invariant relating to quadrisecants in the knot and to complexity of polygonal and polynomial realizations of a knot. r

Many languages in Africa are written using Latin-based scripts, but often with extra diacritics (e.g. dots below in Igbo: i _ ; o _ ; u _) or modifications to the letters themselves (e.g. open vowels "e" and "o" in Lingala: ɛ, ɔ). While it is possible to render these characters accurately in Unicode, oftentimes keyboard input methods are not easily accessible or are cumbersome to use, and so the vast majority of electronic texts in many African languages are written in plain ASCII. We call the process of converting an ASCII text to its proper Unicode form unicodification. This paper describes an open-source package which performs automatic unicodification, implementing a variant of an algorithm described in previous work of De Pauw, Wagacha, and de Schryver. We have trained models for more than 100 languages using web data, and have evaluated each language using a range of feature sets.

We give the first explicit computations of rational homotopy groups of spaces of "long knots" in Euclidean spaces. We define a spectral sequence which converges to these rational homotopy groups whose E 1 term is defined in terms of braid Lie algebras. For odd k we establish a vanishing line for this spectral sequence, show the Euler characteristic of the rows of this E 1 term is zero, and make calculations of E 2 in a finite range.