An infinite surface with the lattice property I: Veech groups and coding geodesics

We study infinite translation surfaces which are ℤ-covers of compact translation surfaces. We obtain conditions ensuring that such surfaces have Veech groups which are Fuchsian of the first kind and give a necessary and sufficient condition for recurrence of their straight-line flows. Extending results of Hubert and Schmithüsen, we provide examples of infinite non-arithmetic lattice surfaces, as well as surfaces with infinitely generated Veech groups.

This thesis uses dynamical systems to study two types of objects: number expansions (Chapters 1 and 2) and translation surfaces (Chapter 3). Our first chapter builds a broad, unifying theory for a large class of continued fraction algorithms producing what we call contracted Farey expansions. These algorithms are based on three ideas: (i) contraction of generalised continued fractions, (ii) induced transformations, and (iii) the natural extension of the Farey tent map. Within this theory, we find several well-studied algorithms; a new subfamily of superoptimal continued fractions with arbitrarily good convergence and approximation properties; and a unifying framework to prove several old and new results in Diophantine approximation. Chapter 2 introduces a new, one-parameter family of functions called skewed symmetric golden maps. Using tools from ergodic theory, we study the relative frequencies of digits typically occurring in number expansions produced by these maps. The central tool for our analysis is a mysterious phenomenon of our functions called matching, which has been recently observed and exploited to understand several other families of functions generating number expansions. Our final chapter deals with translation surfaces, i.e., surfaces obtained by gluing pairs of parallel, equal-length, and oppositely oriented edges of planar polygons. The Veech group of a translation surface is the group of Jacobians of its orientation-preserving affine automorphisms. We develop a novel algorithm to construct translation surfaces with prescribed lattice Veech groups in given strata. Our ideas are also used to obtain obstructions for the realisability of certain Veech groups in certain strata. In particular, we show that the square torus is the only translation surface in any minimal stratum whose Veech group is the modular group.

The Veech group of a translation surface is the group of Jacobians of orientation-preserving affine automorphisms of the surface. We present an algorithm which constructs all translation surfaces with a given lattice Veech group in any given stratum. In developing this algorithm, we give a new proof of a finiteness result of Smillie and Weiss, namely that there are only finitely many unit-area translation surfaces in any stratum with the same lattice Veech group. Our methods can be applied to obtain obstructions of lattices being realized as Veech groups in certain strata; in particular, we show that the square torus is the only translation surface in any minimal stratum whose Veech group is all of textrm{SL}_2mathbb {Z}.

The local constraint of double twist cannot lead to long range order in our physical space ℝ. Cubic blue phases can be interpreted as an array of disclination lines. The symmetry 1432, P4232 and 14,32 of cubic blue phases can be deduced from the symmetry groups of the P, F and G infinite periodic minimal surfaces by removing all the mirror operations incompatible with chirality. Axes of double twist cylinders are shown to be the same as the core of the labyrinths of infinite periodic minimal surface. The blue phase configuration is then driven by boundary conditions on the axes of double twist cylinders and on the minimal surface. We show that the configuration which minimizes the elastic free energy on the minimal surfaces is defined by the asymptotic lines.

Motivated by questions in property testing, we search for linear error-correcting codes that have the “single local orbit” property, i.e., they are specified by a single local constraint and its translations under the symmetry group of the code. We show that the dual of every “sparse” binary code whose coordinates are indexed by elements of $\mathbb{F}_{2^n}$ for prime $n$ and whose symmetry group includes the group of nonsingular affine transformations of $\mathbb{F}_{2^n}$ has the single local orbit property. (A code is said to be sparse if it contains polynomially many codewords in its block length.) In particular this class includes the dual-BCH codes for whose duals (i.e., for BCH codes) simple bases were not known. Our result gives the first short ($O(n)$-bit, as opposed to the natural $\exp(n)$-bit) description of a low-weight basis for BCH codes. The interest in the single local orbit property comes from the recent result of Kaufman and Sudan (STOC 2008) that shows that the duals of codes that have the single local orbit property under the affine symmetry group are locally testable. When combined with our main result, this shows that all sparse affine-invariant codes over the coordinates $\mathbb{F}_{2^n}$ for prime $n$ are locally testable. If, in addition to $n$ being prime, $2^n-1$ does not have large divisors, then we get that every sparse cyclic-invariant code also has the single local orbit. In particular this implies that BCH codes of such length are generated by a single low-weight codeword and its cyclic shifts.

A tiling of the Euclidean plane, by regular polygons, is called 2-uniform tiling if it has two orbits of vertices under the action of its symmetry group. There are 20 distinct 2-uniform tilings of the plane. Plane being the universal cover of torus and Klein bottle, it is natural to ask about the exploration of maps on these two surfaces corresponding to the 2-uniform tilings. We call such maps as doubly semiequivelar maps. In the present study, we compute and classify (up to isomorphism) doubly semiequivelar maps on torus and Klein bottle. This classification of semiequivelar maps is useful in classifying a category of symmetrical maps which have two orbits of vertices, named as 2-uniform maps.

Some remarks on algebraic and scaling properties of 3-dimensional bravais lattices

APPLICATION OF ELLIPTIC RIEMANNIAN GEOMETRY TO PROBLEMS OF CRYSTALLOGRAPHY

[Cr(NH3)6] [ZnCI4]CI, trigonal, R-3c; a= 7.5415 (5), c=44.959 (3) A (on hexagonal axes); Z = 6, Do = 1.78, De= 1.785 g cm -3. The compound was prepared in an attempt to produce the trigonal-bipyramidal [ZnCIs] 3anion. The structure consists of [Cr(NH3)6] 3 +, [ZnC14] zand CIions. The [ZnCI4] 2ion is severely distorted from a regular tetrahedral geometry by the close approach of the CIion in the incipient formation of a fifth Zn-CI bond. [Cr(NH3)6] [CoCI4]CI is isostructural. Introduction. Initial cell dimensions for hexagonal axes were determined from precession photographs (Mo Ka radiation). Systematic absences (hkl absent unless h + k + l= 3n, hhl absent unless l= 2n) indicated space group R3c (No. 167) or R3c (No. 161). A yellow crystal displaying point-group symmetry 3m, with faces of the forms {001} and {104}, was sealed in a Lindemann glass tube and mounted on a Hilger & Watts Y290 four-circle diffractometer in a general orientation. Unit-cell parameters and the orientation matrix were refined by least squares from 12 reflexions (20>38 °, MoK~ radiation, 2=0.71069 A, Zr filter) (Busing & Levy, 1967). Intensities were collected for two equivalent sets of reflexions with 20<55 ° (0-20 scan mode, 4s count at each of 80 steps of 0.01 ° in 0, background counts of 80s at each end of the scan). Three standard reflexions showed < 2 % intensity decay during the collection; linear decay functions were fitted and applied. The crystal was measured with a travelling microscope and absorption corrections applied by a Gaussian integration method (/z= 33 cm -1, 312 grid points, transmission factors range from 0.411 to 0.716). 987 reflexions with positive net recorded intensity were merged to give 521 unique reflexions, with an internal consistency index ~(w~+w2)(F~F2)2/2~(wtF2+ w2F22)=30% [ w = 1/a2(F), based on counting statistics]. The statistical distribution of E values indicated the centrosymmetric space group R3e. This would require the Cr and Zn atoms to lie on special positions with site symmetries 3($6) and 32(D3), which implies a regular trigonal-bipyramidal [ZnCIs] 3anion. A Fourier synthesis phased by the metal atoms, however, showed the Zn atom to be disordered over two equal-weight symmetry-related sites close to the 32 position. All attempts to find a solution in lower-symmetry space groups were unsuccessful. With a twofold disorder model for Zn in R'3c, the CI and N atoms were located, and the structure refined by full-matrix least squares to a minimum value of Y.wd 2 (A = IFolIFcl). Complex scattering factors for Cr a+, Zn 2+, CI-, N and H were those of Cromer & Waber (1974) and Cromer & Ibers (1974). With anisotropic thermal parameters, R was reduced to 9.12 % and Rw[=(~wA2/~wlFol2) 1/2] to 8.27%. It was then possible to resolve the 'equatorial' CI atom [C1(2)] into two half-weight symmetry-related components. Further refinement (R=7.15, Rw=5.90%) followed by a difference map revealed the positions of the H atoms. These were also refined, with individual isotropic temperature factors, and the final values of R and Rw were 4.95 and 3.77%. The number of parameters refined was 41. All shift/e.s.d, ratios in the final cycle were < 0.005. The weighting scheme was shown to be consistent by an analysis of variance (Hamilton, 1974). Atomic parameters are shown in Table 1. It was not possible to resolve CI(1) into disorder components, and the reasonable U values indicate that any disorder for CI(1) is insignificant. A final difference synthesis showed no significant features.* Discussion. Although [CuC15] 3(Raymond, Meek & [bers, 1968), [CdCIs] 3(Long, Herlinger, Epstein & * A list of structure factors has been deposited with the British Library Lending Division as Supplementary Publication No. SUP 31844 (5 pp.). Copies may be obtained through The Executive Secretary, International Union of Crystallography, 13 White Friars, Chester CH1 1NZ, England. Table 1. Atomic coordinates

Efficacious symmetry-adapted atomic displacement method for lattice dynamical studies

In this article, a framework is presented that allows the systematic derivation of planar edge-to-edge k-isocoronal tilings from tile-s-transitive tilings, s ≤ k. A tiling {\cal T} is k-isocoronal if its vertex coronae form k orbits or k transitivity classes under the action of its symmetry group. The vertex corona of a vertex x of {\cal T} is used to refer to the tiles that are incident to x. The k-isocoronal tilings include the vertex-k-transitive tilings (k-isogonal) and k-uniform tilings. In a vertex-k-transitive tiling, the vertices form k transitivity classes under its symmetry group. If this tiling consists of regular polygons then it is k-uniform. This article also presents the classification of isocoronal tilings in the Euclidean plane.

In this paper we compute spaces of chiral operators in general two-dimensional (0,2) nonlinear sigma models, both in theories twistable to the A/2 or B/2 model, as well as in non-twistable theories, and apply them to check recent duality conjectures. The fact that in a nonlinear sigma model, the Fock vacuum can act as a section of a line bundle on the target space plays a crucial role in our (0,2) computations, so we begin with a review of this property. We also take this opportunity to show how even in (2,2) theories, the Fock vacuum encodes in this way choices of target space spin structures, and discuss how such choices enter the A and B model topological field theories. We then compute chiral operators in general (0,2) nonlinear sigma models, and apply them to the recent Gadde-Gukov-Putrov triality proposal, which says that certain triples of (0,2) GLSMs should RG flow to nontrivial IR fixed points. We find that different UV theories in the same proposed universality class do not necessarily have the same space of chiral operators — but, the mismatched operators do not contribute to elliptic genera and are in non-integrable representations of the proposed IR affine symmetry groups, suggesting that the mismatched states become massive along RG flow. We find this state matching in examples not only of different geometric phases of the same GLSMs, but also in phases of different GLSMs, indirectly verifying the triality proposal, and giving a clean demonstration that (0,2) chiral rings are not topologically protected. We also check proposals for enhanced IR affine E 6 symmetries in one such model, verifying that (matching) chiral states in phases of corresponding GLSMs transform as 27’s, $$ \overline{\mathbf{27}}'\mathrm{s} $$ .

Hecke algebras of semidirect products and the finite part of the Connes–Marcolli [formula omitted]-algebra

On discrete symmetries and fundamental domains of target space

By noticing the fact that the charged leptons and quarks in the standard model are chirality-based Dirac spinors since their weak interaction violates maximally parity symmetry though they behave as Dirac fermions in electromagnetic interaction, we show that such a chirality-based Dirac spinor possesses not only electric charge gauge symmetry U(1) but also inhomogeneous spin gauge symmetry WS(1,3) = SP(1,3)$\rtimes$W$^{1,3}$, which reveals the nature of gravity and spacetime. The gravitational force and spin gauge force are governed by the gauge symmetries $W^{1,3}$ and SP(1,3), respectively, and a biframe spacetime with globally flat Minkowski spacetime as base spacetime and locally flat gravigauge spacetime as a fiber is described by the gravigauge field through emergent non-commutative geometry. The gauge-geometry duality and renormalizability in gravitational quantum field theory (GQFT) are carefully discussed. A detailed analysis and systematic investigation on gravidynamics and spinodynamics as well as electrodynamics are carried out within the framework of GQFT. A full discussion on the generalized Dirac equation and Maxwell equation as well as Einstein equation and spin gauge equation is made in biframe spacetime. New effects of gravidynamics as extension of general relativity are particularly analyzed. All dynamic equations of basic fields are demonstrated to preserve the spin gauge covariance and general coordinate covariance due to the spin gauge symmetry and emergent general linear group symmetry GL(1,3,R), so they hold naturally in any spinning reference frame and motional reference frame.