R. C. Penner

The decorated Teichmüller space of punctured surfaces

A principal ℝ+5-bundle over the usual Teichmüller space of ans times punctured surface is introduced. The bundle is mapping class group equivariant and admits an invariant foliation. Several coordinatizations of the total space of the bundle are developed. There is furthermore a natural cell-decomposition of the bundle. Finally, we compute the coordinate action of the mapping class group on the total space; the total space is found to have a rich (equivariant) geometric structure. We sketch some connections with arithmetic groups, diophantine approximations, and certain problems in plane euclidean geometry. Furthermore, these investigations lead to an explicit scheme of integration over the moduli spaces, and to the construction of a “universal Teichmüller space,” which we hope will provide a formalism for understanding some connections between the Teichmüller theory, the KP hierarchy and the Virasoro algebra. These latter applications are pursued elsewhere.

Bounds on least dilatations

We consider the collection of all pseudo-Anosov homeomorphisms on a surface of fixed topological type. To each such homeomorphism is associated a real-valued invariant, called its dilatation (which is greater than one), and we define the spectrum of the surface to be the collection of logarithms of dilatations of pseudo-Anosov maps supported on the surface. The spectrum is a natural object of study from the topological, geometric, and dynamical points of view. We are concerned in this paper with the least element of the spectrum, and explicit upper and lower bounds on this least element are derived in terms of the topological type of the surface; train tracks are the main tool used in establishing our estimates.

Topology and prediction of RNA pseudoknots

MOTIVATION Several dynamic programming algorithms for predicting RNA structures with pseudoknots have been proposed that differ dramatically from one another in the classes of structures considered. RESULTS Here, we use the natural topological classification of RNA structures in terms of irreducible components that are embeddable in the surfaces of fixed genus. We add to the conventional secondary structures four building blocks of genus one in order to construct certain structures of arbitrarily high genus. A corresponding unambiguous multiple context-free grammar provides an efficient dynamic programming approach for energy minimization, partition function and stochastic sampling. It admits a topology-dependent parametrization of pseudoknot penalties that increases the sensitivity and positive predictive value of predicted base pairs by 10-20% compared with earlier approaches. More general models based on building blocks of higher genus are also discussed. AVAILABILITY The source code of gfold is freely available at http://www.combinatorics.cn/cbpc/gfold.tar.gz. CONTACT duck@santafe.edu SUPPLEMENTARY INFORMATION Supplementary data are available at Bioinformatics online.

Arc operads and arc algebras

Several topological and homological operads based on families of projectively weighted arcs in bounded surfaces are introduced and studied. The spaces underlying the basic operad are identied with open subsets of a combinatorial compactication due to Penner of a space closely related to Riemann’s moduli space. Algebras over these operads are shown to be Batalin{Vilkovisky algebras, where the entire BV structure is realized simplicially. Furthermore, our basic operad contains the cacti operad up to homotopy. New operad structures on the circle are classied and combined with the basic operad to produce geometrically natural extensions of the algebraic structure of BV algebras, which are also computed.

Closed/open string diagrammatics

Abstract We introduce a combinatorial model based on measured foliations in surfaces which captures the phenomenology of open/closed string interactions. The predicted equations are derived in this model, and new equations can be discovered as well. In particular, several new equations together with known transformations generate the combinatorial version of open/closed duality. On the topological and chain levels, the algebraic structure discovered is new, but it specializes to a modular bi-operad on the level of homology.

The moduli space of a punctured surface and perturbative series

0. Introduction. Let F denote the oriented genus g surface with s punctures, 2g — 2 + s > 0 , s > l , and choose a distinguished puncture P of F*. Let Tg be the Teichmüller space of conformai classes of complete finitearea metrics on F* (see [A]), and let MCg denote the mapping class group of orientation-preserving difFeomorphisms of F (fixing P) modulo isotopy (see [B]). When 0, s are understood, we omit their mention. In §1 and §2, we report on joint work with D. B. A. Epstein [EP] where new and useful coordinates on Tg s are given (Theorem 2) and a MC^-equivariant cell decomposition of Tg is described (Theorem 3). There is thus an induced cell decomposition of the quotient M g = Tg s /MC*, which is the usual moduli space of F* in case 8 = 1. In §3, we describe a remarkable connection (see [P]) between this cell-decomposition for s = 1 and a technique from quantum field theory, which allows the computation of certain numerical invariants of M, (Corollary 6). Analogues of Theorem 3 have been obtained independently by [BE and H] using different techniques. Furthermore, Corollary 7 is in agreement with some recent work in [HZ]. Let M denote Minkowskii 3-space with bilinear pairing (•,•) of type (+ ,+ , ) , and let L+ C M denote the (open) positive light-cone. The uniformization theorem (see [A]) allows us to identify Tg s with the space of (conjugacy classes of faithful and discrete) representations of iri(F£) in SO(2,1) (as a Fuchsian group of the first kind in the component of the identity).

Topological classification and enumeration of RNA structures by genus

To an RNA pseudoknot structure is naturally associated a topological surface, which has its associated genus, and structures can thus be classified by the genus. Based on earlier work of Harer–Zagier, we compute the generating function

Torelli groups, extended Johnson homomorphisms, and new cycles on the moduli space of curves

\mathbf{D}_{g,\sigma }(z)=\sum _{n}\mathbf{d}_{g,\sigma }(n)z^n