Combinatorial Properties of Triangle-Free Rectangle Arrangements and the Squarability Problem

We consider arrangements of axis-aligned rectangles in the plane. A geometric arrangement specifies the coordinates of all rectangles, while a combinatorial arrangement specifies only the respective intersection type in which each pair of rectangles intersects. First, we investigate combinatorial contact arrangements, i.e., arrangements of interior-disjoint rectangles, with a triangle-free intersection graph. We show that such rectangle arrangements are in bijection with the 4-orientations of an underlying planar multigraph and prove that there is a corresponding geometric rectangle contact arrangement. Moreover, we prove that every triangle-free planar graph is the contact graph of such an arrangement. Secondly, we introduce the question whether a given rectangle arrangement has a combinatorially equivalent square arrangement. In addition to some necessary conditions and counterexamples, we show that rectangle arrangements pierced by a horizontal line are squarable under certain sufficient conditions.Comment: 15 pages, 13 figures, extended version of a paper to appear at the International Symposium on Graph Drawing and Network Visualization (GD) 201

Applications of Graph Theory to Separability

Let S be a surface with a triangular tiling T. Let R be a reflection a side of one of the triangles; so that R is an orientation reversing isometry of the surface. Define M = {s in S |S : Rs = s}. We then say that the surface S separates along the reflection R if S-R has two components. This paper considers the applications of graph theoretic methods to determining whether a reflection is separating or not and compares the algorithmic efficiency of these methods to the current known methods

Enlarging properties of graphs

The subjects of this thesis are the enlarging and magnifying properties of graphs. Upper bounds for the isoperimetric number i(G) of a graph G are determined with respect to such elementary graph properties as order, valency, and the number of three and four-cycles. The relationship between i(G) and the genus of G is studied in detail, and a class of graphs called finite element graphs is shown never to supply enlarging families. The magnifying properties of Hamiltonian cubic graphs are investigated, and a class of graphs known as shift graphs is defined. These are shown never to form enlarging families, using a technical lemma derived from Klawe's Theorem on non-expanding families of graphs. The same lemma is used, in conjunction with some elementary character theory, to prove that several important classes of Cayley graphs do not form enlarging families, and to derive a lower bound on the subdominant eigenvalue of a vertex-transitive graph. The problem of finding Ramanujan graphs is discussed. Some necessary conditions for a graph to be Ramanujan, depending on the automorphism group of the graph, and the number of certain reduced walks in the graph, are derived. Finally, the techniques of Buck are used to construct an infinite number of families of linear expanders, deploying free subgroups of the group SL(2, Z).<p

Free nilpotent and HH-type Lie algebras. Combinatorial and orthogonal designs

The aim of our paper is to construct pseudo $H$-type algebras from the covering free nilpotent two-step Lie algebra as the quotient algebra by an ideal. We propose an explicit algorithm of construction of such an ideal by making use of a non-degenerate scalar product. Moreover, as a bypass result, we recover the existence of a rational structure on pseudo $H$-type algebras, which implies the existence of lattices on the corresponding pseudo $H$-type Lie groups. Our approach substantially uses combinatorics and reveals the interplay of pseudo $H$-type algebras with combinatorial and orthogonal designs. One of the key tools is the family of Hurwitz-Radon orthogonal matrices

Branes, graphs and singularities

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Physics, 2009.Cataloged from PDF version of thesis.Includes bibliographical references (p. 341-354).In this thesis, we study various aspects of string theory on geometric and nongeometric backgrounds in the presence of branes. In the first part of the thesis, we study non-compact geometries. We introduce "brane tilings" which efficiently encode the gauge group, matter content and superpotential of various quiver gauge theories that arise as low-energy effective theories for D-branes probing singular non-compact Calabi-Yau spaces with toric symmetries. Brane tilings also offer a generalization of the AdS/CFT correspondence. A technique is developed which enables one to quickly compute the toric vacuum moduli space of the quiver gauge theory. The equivalence of this procedure and the earlier approach that used gauged linear sigma models is explicitly shown. As an application of brane tilings, four dimensional quiver gauge theories are constructed that are AdS/CFT dual to infinite families of Sasaki-Einstein spaces. Various checks of the correspondence are performed. We then develop a procedure that constructs the brane tiling for an arbitrary toric Calabi-Yau threefold. This solves a longstanding problem by computing superpotentials for these theories directly from the toric diagram of the singularity. A different approach to the low-energy theory of D-branes uses exceptional collections of sheaves associated to the base of the threefold. We provide a dictionary that translates between the language of brane tilings and that of exceptional collections. Geometric compactifications represent only a very small subclass of the landscape: the generic vacua are non-geometric. In the second part of the thesis, we study perturbative compactifications of string theory that rely on a fibration structure of the extra dimensions. Non-geometric spaces preserving .A = 1 supersymmetry in four dimensions are obtained by using T-dualities as monodromies. Several examples are discussed, some of which admit an asymmetric orbifold description. We explore the possibility of twisted reductions where left-moving spacetime fermion number Wilson lines are turned on in the fiber.by David Vegh.Ph.D

Machine-Learning and Data Science Techniques in String and Gauge Theories

Techniques from supervised and unsupervised machine learning, along with those from data and network science, are applied to generated datasets of mathematical objects relevant to string and gauge theories. Investigations show success in identifying and learning new structure associated to these objects. Datasets considered in the research work completed for this thesis include: dessins d’enfants, quivers, Hilbert series, amoebae, polytopes, Calabi-Yau manifolds, brane webs, and cluster algebras

Duality and dynamics of supersymmetric field theories from D-branes on singularities

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Physics, 2005.Includes bibliographical references (p. 359-373).We carry out various investigations regarding gauge theories on the worldvolume of D-branes probing toric singularities. We first study the connection that arises in Toric Duality between different dual gauge theory phases and the multiplicity of fields in the gauged linear sigma models associated with the probed geometries. We introduce a straightforward procedure for the determination of toric dual theories and partial resolutions based on the (p, q) web description of toric singularities. We study the non-conformal theories that arise in the presence of fractional branes. We introduce a systematic procedure to study the resulting cascading RG flows, including the effect of anomalous dimensions on beta functions. Supergravity solutions dual to logarithmic RG flows are constructed, validating the field theory analysis of the cascades. We systematically study the IR dynamics of cascading gauge theories. We show how the deformation in the dual geometries is encoded in a quantum modification of the moduli space. We construct an infinite family of superconformal quiver gauge theories which are AdS/CFT dual to Sasaki-Einstein horizons with explicit metrics. The gauge theory and geometric computations of R-charges and central charges are shown to agree. We introduce new Type IIB brane constructions denoted brane tilings which are dual to D3-branes probing arbitrary toric singularities. Brane tilings encode both the quiver and superpotential of the gauge theories on the D-brane probes. They give a connection with the statistical model of dimers.(cont.) They provide the simplest known method for computing toric moduli spaces of gauge theories, which reduces to finding the determinant of the Kasteleyn matrix of a bipartite graph.by Sebastián Federico Franco.Ph.D