Moduli Spaces

We prove that chromatic graph homology for commutative dg algebras, due to Helme-Guizon and Rong, can be extended to brace algebras, at least when the graph is a planar tree. Examples of brace algebras include the cochain algebra of a simplicial set and the Hochschild cochain complex of an associative algebra.

DEFORMATIONS OF G2-STRUCTURES, STRING DUALITIES AND FLAT HIGGS BUNDLES Rodrigo de Menezes Barbosa Tony Pantev, Advisor We study M-theory compactifications on G2-orbifolds and their resolutions given by total spaces of coassociative ALE-fibrations over a compact flat Riemannian 3manifold Q. The flatness condition allows an explicit description of the deformation space of closed G2-structures, and hence also the moduli space of supersymmetric vacua: it is modeled by flat sections of a bundle of Brieskorn-Grothendieck resolutions over Q. Moreover, when instanton corrections are neglected, we also have an explicit description of the moduli space for the dual type IIA string compactification. The two moduli spaces are shown to be isomorphic for an important example involving A1-singularities, and the result is conjectured to hold in generality. We also discuss an interpretation of the IIA moduli space in terms of “flat Higgs bundles” on Q and explain how it suggests a new approach to SYZ...

The goal of this article is to consider the role played by finite‐order elements in the mapping class groups and special loci on moduli spaces, within the framework of Grothendieck–Teichmüller theory, and in particularly in the genus zero case. Quotienting topological surfaces by finite‐order automorphisms induces certain morphisms between moduli spaces; we consider the corresponding special homomorphisms between mapping class groups. In genus zero, these morphisms are always defined over ℚ, so that the canonical outer Galois action on profinite genus zero mapping class groups respects the induced homomorphisms. For simplicity, we consider only the subgroup $ \widehat {GT} ^1 _{0,0} $ of elements F = (λ , f ) ∈ $ \widehat {GT} $ with λ = 1 and conditions on the Kummer characters ρ 2(F ) = ρ 3(F ) = 0. We define a subgroup $ \widehat {GS} ^1 _{0,0} $ ⊂ $ \widehat {GT} ^1 _{0,0} $ by considering only elements of $ \widehat {GT} ^1 _{0,0} $ respecting these homomorphisms on the first t...

The connection between the partial quotients of the regular continued fraction and the number of left and right cuts of the cutting sequence of a geodesic across the triangles of the Farey tessellation has been established by Series. To find a similar connection for the nearest integer continued fraction, we require a completely new approach that combines aspects of well-known work on continued fractions. Firstly, we discuss the impact of actually truncating the continued fraction and explore alternate methods that use Möbius maps. We then animate the roles of various vertices, geodesics, and Farey triangles, in the geometrical figures that are commonly used, by developing a geometrical animation of the process that underlies the continued fraction of an irrational real number ξ. Finally, we reveal the interplay between the orbits of three vertices, namely zero, one, and infinity, as they move toward ξ, through rational values, under the action of successive partial products of Möbius maps derived from the nearest integer continued fraction.

We study, from the point of view of abelian and Kummer surfaces and their moduli, the special quintic threefold known as Nieto's quintic. It is, in some ways, analogous to the Segre cubic and the Burkhardt quartic and can be interpreted as a moduli space of certain Kummer surfaces. It contains 30 planes and has 10 singular points: we describe how some of these arise from bielliptic and product abelian surfaces and their Kummer surfaces.

Multi-environmental yield trial is very vital in assessing newly developed rice lines for its adaptability and stability across environments especially prior to release of the newly developed variety for commercial cultivation. The growth performance and phenotypic variability of these genotypes are the combination of environment, genotype and genotype by environment (G×E) interaction factors. Thus, evaluation creates an opportunity for effective selection of superior genotypes. The objectives of this study were to evaluate the newly developed blast resistant rice lines in varied environmental conditions, precisely measure the response of the advanced lines in multiple environments and classify the genotypes into groups that could serve as varieties for commercial cultivation. Genetic materials included 18 improved blast resistant rice lines and the recipient parent MR219. The total of 19 newly developed genotypes was evaluated under four varied environments in Peninsular Malaysia. ...

We show how the quantum trace map of Bonahon and Wong can be constructed in a natural way using the skein algebra of Muller, which is an extension of the Kauffman bracket skein algebra of surfaces. We also show that the quantum Teichmüller space of a marked surface, defined by Chekhov–Fock (and Kashaev) in an abstract way, can be realized as a concrete subalgebra of the skew field of the skein algebra.

We are interested in the first prolongational limit set of the boundary of parallelizable regions of a given flow of the plane which has no fixed points. We prove that for every point from the boundary of a maximal parallelizable region, there exists exactly one orbit contained in this region which is a subset of the first prolongational limit set of the point. Using these uniquely determined orbits, we study the structure of maximal parallelizable regions.

This paper investigates the kinematic motions of space-like and time-like curves specified by acceleration fields in Minkowski space ℝ2,1. Through the motion, the relationship between the acceleration fields and velocity fields is determined. In this study, we focus on studying the flows of inextensible space-like curves with a space-like principal normal vector specified by a normal acceleration that equals the curvature of the curve. Through the motion of the inextensible space-like curve with normal acceleration, we prove that the position vector of the curve satisfies a one-dimensional wave equation. We present some novel applications and visualize the flows of curves and their curvatures.

We make calculations in graph homology which further understanding of the topology of spaces of string links, in particular calculating the Euler characteristics of finite-dimensional summands in their homology and homotopy. In doing so, we also determine the supercharacter of the symmetric group action on the positive arity components of the modular envelope of $L_\infty$.

We construct stable sheaves over K3 fibrations using a relative Fourier-Mukai transform which describes the sheaves in terms of spectral data similar to the construction for elliptic fibrations. On K3 fibered Calabi–Yau threefolds we show that the Fourier-Mukai transform induces an embedding of the relative Jacobian of spectral line bundles on spectral covers into the moduli space of sheaves of given invariants. This makes the moduli space of spectral sheaves a generic torus fibration over the moduli space of curves of the given arithmetic genus on the Calabi–Yau manifold.