Catalan numbers and branched coverings by the Riemann sphere
We prove that the number of PGL(2, C) equivalence classes of degree d rational maps with a fixed branch set is generically equal to the “Catalan number” ϱ(d) = ( 1 d )( 2d−2 d−1 ) . Several (previously known) applications of Catalan numbers to combinatorics and algebraic geometry are discussed throughout the course of the article.

title: Fixed points of polynomial maps. I. Rotation subsets of the circles
abstract: — We give a combinatorial analysis of rational rotation subsets of the circle. These are invariant subsets that have well-defined rational rotation numbers under the standard self-covering maps of S. This analysis has applications to the classification of dynamical systems generated by polynomials in one complex variable.

title: $k$-flat structures and exotic characteristic classes
abstract: We generalize the concept of "foliation" and define k-flat structures; these are smooth vector bundles with afflne connections whose characteristic forms vanish above a certain dimension. Using semisimplicial techniques we construct a classifying space for k-flat structures, and prove a classification theorem for these structures on smooth manifolds. Techniques from rational homotopy theory are used to relate the exotic characteristic classes of foliations to the rational homotopy groups and cohomology of the classifying space.  A familiar technique in algebraic topology is to reduce the probD lem of studying structures on a large collection of objects to the study of a universal example which "contains" all the others. Consider the problem of classifying vector bundles over CW complexes, in this case we need only look at spaces B0k and BUk, and their canonical k-plane bundles . A nice feature of this example is that the algebraic invariants of k-plane bundles (i.e. Stiefel-Whitney classes, Chern classes, Euler class) are contained in the cohomology of the classifying spaces. In this paper, we look at a set of invariants which can be attached to foliated smooth manifolds. These are the so-called "exotic characteristic classes" of foliations which were constructed in the late 1960's independently by 13ott, Haefliger, Gelfand-Fuks, Kamber-Tondeur, and Malgrange; they can be defined using purely geometric techniques (i.e. see ). At about the same time, Haefliger  developed a topological classification of foliations using Br spaces, and has applied the theory to smooth manifolds via theorems of Thurston and Gromov-Phillips. One would like to demonstrate a relationship between the exotic characteristic classes and the cohomology of the classifying spaces as in the example above, although this has proven to be difficult. Nevertheless, a lot is known about the homotopy type of Brk, largely through the efforts of Hutsch, Hurder, Kamber-Tondeur, Rasmussen, Thurston, et al. In §1, we introduce k-flat structures, which generalize the concept of foliations. These are smooth vector bundles with affine connections whose characteristic forms vanish above dimension 2k; they constitute the largest classes of objects for which exotic characteristic classes can be defined. k-flat structures and exotic characteristic classes were studied by Lehmann in . Using semisimplicial techniques, we will construct classifying spaces for k-flat structures and prove a classification theorem for these structures on smooth manifolds. §2 contains a brief review of semisimplicial methods, while §3 contains the constructions and proofs. Received by the editors March 29, 1984 and, in revised form, November 18, 1985.

title: On the shape of the unit sphere in Q(Δ)
abstract: We show that the unit sphere in the Banach space of L 1 holomorphic quadratic differentials on the disk is weakly uniformly convex with exponent 1/2 at certain points

title: Fixed points of polynomial maps. Part II. Fixed point portraits
abstract: — Douady, Hubbard and Branner have introduced the concept of a "limb" in the Mandelbrot set. A quadratic map f(z) = z + c belongs to the p/q-limb if and only if there exist q external rays of its Julia set which land at a common fixed point of/, and which are permuted by/with combinatorial rotation number ^^Q/Z, p / q ^ O . ( This note will make a similar analysis of higher degree polynomials by introducing the concept of the "fixed point portrait" of a monic polynomial map.

title: The mapping class group of a generic quadratic rational map and automorphisms of the 2-shift
abstract: Mixtures of polyisocyanates are prepared in one step by reacting phosgene with a slurry of selected polyamines in an inert solvent.

title: Volatility of the short rate in the rational lognormal model
abstract: A recent article of Flesaker and Hughston introduces a one factor interest rate model called the rational lognormal model. This model has a lot to recommend it including guaranteed finite positive interest rates and analytic tractability. Consequently, it has received a lot of attention among practioners and academics alike. However, it turns out to have the undesirable feature of predicting that the asymptotic value of the short rate volatility is zero. This theoretical result is proved rigorously in this article. The outcome of an empirical study complementing the theoretical result is discussed at the end of the article. European call options are valued with the rational lognormal model and a comparably calibrated mean reverting Gaussian model. unsurprisingly, rational lognormal option values are considerably lower than the analogous mean reverting Gaussian option values. In other words, the volatility in the rational lognormal model declines so quickly that options are severely undervalued.