Asymptotic enumeration and limit laws for graphs of fixed genus

It is shown that the number of labelled graphs with n vertices that can be embedded in the orientable surface S_g of genus g grows asymptotically like $c^{(g)}n^{5(g-1)/2-1}\gamma^n n!$ where $c^{(g)}>0$, and $\gamma \approx 27.23$ is the exponential growth rate of planar graphs. This generalizes the result for the planar case g=0, obtained by Gimenez and Noy. An analogous result for non-orientable surfaces is obtained. In addition, it is proved that several parameters of interest behave asymptotically as in the planar case. It follows, in particular, that a random graph embeddable in S_g has a unique 2-connected component of linear size with high probability

TFT construction of RCFT correlators II: Unoriented world sheets

A full rational CFT, consistent on all orientable world sheets, can be constructed from the underlying chiral CFT, i.e. a vertex algebra, its representation category C, and the system of chiral blocks, once we select a symmetric special Frobenius algebra A in the category C [I]. Here we show that the construction of [I] can be extended to unoriented world sheets by specifying one additional datum: a reversion on A - an isomorphism from the opposed algebra of A to A that squares to the twist. A given full CFT on oriented surfaces can admit inequivalent reversions, which give rise to different amplitudes on unoriented surfaces, in particular to different Klein bottle amplitudes. We study the classification of reversions, work out the construction of the annulus, Moebius strip and Klein bottle partition functions, and discuss properties of defect lines on non-orientable world sheets. As an illustration, the Ising model is treated in detail.Comment: 112 pages, table of contents, several figures. v2: typos corrected, version to be published in Nucl.Phys.

Periodic minimal surfaces of cubic symmetry

A survey of cubic minimal surfaces is presented, based on the concept of fundamental surface patches and their relation to the asymmetric units of the space groups. The software Surface Evolver has been used to test for stability and to produce graphic displays. Particular emphasis is given to those surfaces that can be generated by a finite piece bounded by straight lines. Some new varieties have been found and a systematic nomenclature is introduced, which provides a symbol (a ‘gene’) for each triply-periodic minimal surface that specifies the surface unambiguously

TFT construction of RCFT correlators IV: Structure constants and correlation functions

We compute the fundamental correlation functions in two-dimensional rational conformal field theory, from which all other correlators can be obtained by sewing: the correlators of three bulk fields on the sphere, one bulk and one boundary field on the disk, three boundary fields on the disk, and one bulk field on the cross cap. We also consider conformal defects and calculate the correlators of three defect fields on the sphere and of one defect field on the cross cap. Each of these correlators is presented as the product of a structure constant and the appropriate conformal two- or three-point block. The structure constants are expressed as invariants of ribbon graphs in three-manifolds.Comment: 98 pages, some figures; v2 (version published in NPB): typos correcte

On the Property F Conjecture

This thesis solves the following question posed by Etingof, Rowell, and Witherspoon: Are the images of mapping class group representations associated to the modular category Mod-D^w (G) always finite? We answer this question in the affirmative, generalizing their work in the braid group case. Our approach is to translate the problem into manipulation of colored graphs embedded in the given surface as defined by Kirillov. To do this translation, we use the fact that any such representation associated to a finite group G and 3-cocycle ɯ is isomorphic to a Turaev-Viro-Barrett-Westbury (TVBW) representation associated to the spherical fusion category Vecw/G of twisted G-graded vector spaces. As shown by Kirillov, the representation space for this TVBW representation is canonically isomorphic to a vector space spanned by Vecw/G-colored graphs embedded in the surface. By analyzing the action of the Birman generators on a finite spanning set of colored graphs, we find that the mapping class group acts by permutations on a slightly larger finite spanning set. This implies that the representation has finite image

Real algebraic surfaces with isolated singularities

Given a real algebraic surface S in RP3, we propose a constructive procedure to determine the topology of S and to compute non-trivial topological invariants for the pair (RP3, S) under the hypothesis that the real singularities of S are isolated. In particular, starting from an implicit equation of the surface, we compute the number of connected components of S, their Euler characteristics and the weighted 2-adjacency graph of the surface

Free subgroups of free products and combinatorial hypermaps

We derive a generating series for the number of free subgroups of finite index in $\Delta^+ = \mathbb{Z}_p*\mathbb{Z}_q$ by using a connection between free subgroups of $\Delta^+$ and certain hypermaps (also known as ribbon graphs or "fat" graphs), and show that this generating series is transcendental. We provide non-linear recurrence relations for the above numbers based on differential equations that are part of the Riccati hierarchy. We also study the generating series for conjugacy classes of free subgroups of finite index in $\Delta^+$, which correspond to isomorphism classes of hypermaps. Asymptotic formulas are provided for the numbers of free subgroups of given finite index, conjugacy classes of such subgroups, or, equivalently, various types of hypermaps and their isomorphism classes.Comment: 27 pages, 3 figures; supplementary SAGE worksheets available at http://sashakolpakov.wordpress.com/list-of-papers