Showing papers on "Geometry and topology published in 2008"

Comparison theorems in Riemannian geometry

Abstract: Basic concepts and results Toponogov's theorem Homogeneous spaces Morse theory Closed geodesics and the cut locus The sphere theorem and its generalizations The differentiable sphere theorem Complete manifolds of nonnegative curvature Compact manifolds of nonpositive curvature Bibliography Additional bibliography Index.

An introduction to contact topology

Abstract: This text on contact topology is a comprehensive introduction to the subject, including recent striking applications in geometric and differential topology: Eliashberg's proof of Cerf's theorem via the classification of tight contact structures on the 3-sphere, and the Kronheimer-Mrowka proof of property P for knots via symplectic fillings of contact 3-manifolds. Starting with the basic differential topology of contact manifolds, all aspects of 3-dimensional contact manifolds are treated in this book. One notable feature is a detailed exposition of Eliashberg's classification of overtwisted contact structures. Later chapters also deal with higher-dimensional contact topology. Here the focus is on contact surgery, but other constructions of contact manifolds are described, such as open books or fibre connected sums. This book serves both as a self-contained introduction to the subject for advanced graduate students and as a reference for researchers.

Quantum foam and topological strings

Abstract: We find an interpretation of the recent connection found between topological strings on Calabi-Yau threefolds and crystal melting: Summing over statistical mechanical configuration of melting crystal is equivalent to a quantum gravitational path integral involving fluctuations of Kahler geometry and topology. We show how the limit shape of the melting crystal emerges as the average geometry and topology of the quantum foam at the string scale. The geometry is classical at large length scales, modified to a smooth limit shape dictated by mirror geometry at string scale and is a quantum foam at area scales ~ gsα'.

Algebraic models in geometry

Abstract: 1. Lie Groups and Homogeneous Spaces 2. Minimal Models 3. Manifolds 4. Complex and Symplectic Manifolds 5. Geodesics 6. Curvature 7. G-Spaces 8. Blow-ups and Intersection Products 9. A Florilege of Geometric Applications APPENDICES A. De Rham Forms B. Spectral Sequences C. Basic Homotopy Recollections

Complex Nonlinearity: Chaos, Phase Transitions, Topology Change and Path Integrals

Abstract: Basics of Nonlinear and Chaotic Dynamics.- Phase Transitions and Synergetics.- Geometry and Topology Change in Complex Systems.- Nonlinear Dynamics of Path Integrals.- Complex Nonlinearity: Combining It All Together.

Notes on toric Sasaki-Einstein seven-manifolds and AdS_4/CFT_3

Abstract: We study the geometry and topology of two infinite families Y^{p,k} of Sasaki-Einstein seven-manifolds, that are expected to be AdS_4/CFT_3 dual to families of N=2 superconformal field theories in three dimensions. These manifolds, labelled by two positive integers p and k, are Lens space bundles S^3/Z_p over CP^2 and CP^1 x CP^1, respectively. The corresponding Calabi-Yau cones are toric. We present their toric diagrams and gauged linear sigma model charges in terms of p and k, and find that the Y^{p,k} manifolds interpolate between certain orbifolds of the homogeneous spaces S^7, M^{3,2} and Q^{1,1,1}.

Notes on toric Sasaki-Einstein seven-manifolds and AdS4/CFT3

Abstract: We study the geometry and topology of two infinite families Yp,k of Sasaki-Einstein seven-manifolds, that are expected to be AdS4/CFT3 dual to families of = 2 superconformal field theories in three dimensions. These manifolds, labelled by two positive integers p and k, are Lens space bundles S3/p over P2 and P1 × P1, respectively. The corresponding Calabi-Yau cones are toric. We present their toric diagrams and gauged linear sigma model charges in terms of p and k, and find that the Yp,k manifolds interpolate between certain orbifolds of the homogeneous spaces S7,M3,2 and Q1,1,1.

The horofunction boundary of the Hilbert geometry

Abstract: We investigate the horofunction boundary of the Hilbert geometry defined on an arbitrary finite-dimensional bounded convex domain D. We determine its set of Busemann points, which are those points that are the limits of `almost-geodesics'. In addition, we show that any sequence of points converging to a point in the horofunction boundary also converges in the usual sense to a point in the Euclidean boundary of D. We prove that all horofunctions are Busemann points if and only if the set of extreme sets of the polar of D is closed in the Painleve-Kuratowski topology.

On existence and uniqueness of the carrying simplex for competitive dynamical systems.

Abstract: Certain dynamical models of competition are shown to have a unique invariant hypersurface Σ, having simple geometry and topology, such that every non-zero tractory is asymptotic to a trajectory in Σ.

The "M"-ellipsoid, symplectic capacities and volume

Abstract: In this work we bring together tools and ideology from two different fields, symplectic geometry and asymptotic geometric analysis, to arrive at some new results. Our main result is a dimension-independent bound for the symplectic capacity of a convex body ,

On existence and uniqueness of the carrying simplex for competitive dynamical systems

Abstract: Certain dynamical models of competition have a unique invariant hypersurface to whichevery nonzero tractory is asymptotic, having simple geometry and topology.

Ring of simple polytopes and differential equations

Abstract: Simple polytopes are a classical object of convex geometry. They play a key role in many modern fields of research, such as algebraic and symplectic geometry, toric topology, enumerative combinatorics, and mathematical physics. In this paper, the results of a new approach based on a differential ring of simple polytopes are described. This approach allows one to apply the theory of differential equations to the study of combinatorial invariants of simple polytopes.

Geometry and Topology of Coadjoint Orbits of Semisimple Lie Groups

Abstract: Orbits of coadjoint representations of classical compact Lie groups have a lot of applications. They appear in representation th eory, geometrical quantization, theory of magnetism, quantum optics etc. As geometric objects the orbits were the subject of much study. However, they remain hard for calculation and application. We propose simple solutions for the following problems: an explicit parameterization of the orbit by means of a general- ized stereographic projection, obtaining a Kahlerian stru cture on the orbit, introducing basis two-forms for the cohomology group of the orbit.

Curved Spaces: From Classical Geometries to Elementary Differential Geometry

Abstract: This self-contained 2007 textbook presents an exposition of the well-known classical two-dimensional geometries, such as Euclidean, spherical, hyperbolic, and the locally Euclidean torus, and introduces the basic concepts of Euler numbers for topological triangulations, and Riemannian metrics. The careful discussion of these classical examples provides students with an introduction to the more general theory of curved spaces developed later in the book, as represented by embedded surfaces in Euclidean 3-space, and their generalization to abstract surfaces equipped with Riemannian metrics. Themes running throughout include those of geodesic curves, polygonal approximations to triangulations, Gaussian curvature, and the link to topology provided by the Gauss-Bonnet theorem. Numerous diagrams help bring the key points to life and helpful examples and exercises are included to aid understanding. Throughout the emphasis is placed on explicit proofs, making this text ideal for any student with a basic background in analysis and algebra.

Discrete distortion in triangulated 3-manifolds

Abstract: We introduce a novel notion, that we call discrete distortion, for a triangulated 3-manifold. Discrete distortion naturally generalizes the notion of concentrated curvature defined for triangulated surfaces and provides a powerful tool to understand the local geometry and topology of 3-manifolds. Discrete distortion can be viewed as a discrete approach to Ricci curvature for singular flat manifolds. We distinguish between two kinds of distortion, namely, vertex distortion, which is associated with the vertices of the tetrahedral mesh decomposing the 3-manifold, and bond distortion, which is associated with the edges of the tetrahedral mesh. We investigate properties of vertex and bond distortions. As an example, we visualize vertex distortion on manifold hypersurfaces in R4 defined by a scalar field on a 3D mesh. distance fields.

Quantitative characterisation of the geometry and topology of pore space in 3D rock images

Abstract: In this thesis, a suite of techniques and algorithms is presented to tackle three main tasks. Firstly, many existing image-related approaches (processing or analysis) need to be extended from low-dimensional space (e.g. 2D) to a higher-dimensional space (3D). In addition, they often also need to be improved to achieve better accuracy and more efficiency to enable processing of massive volumetric images. Frequently new techniques or algorithms also need to be developed to cover the gap in these previous requirements. Based on these approaches, the second task is to extract the geometric and topological properties of the pore space directly from 3D images of rock samples. The third task is then to study and to establish the relationship between the microstructure and the macroscopic properties by constructing realistic network structures for network models or by conducting some numerical experiments such as mercury injection etc. In the framework of the methodology presented in this thesis, many commonly used image processing and analysis approaches form the basis of the pore space quantification procedure. These primarily include 3D Euclidean distance transformations, 3D geodesic distance transformations, component labelling (clustering), and morphological operations. Among these techniques, some are either unavailable in 3D discrete space or are of too low-efficiency for handling the huge size of rock samples, and others simply did not exist prior to my work. The next level of the methodology is to quantify the pore space. In order to process 3D images efficiently thus, firstly, the medial axis (skeleton) of the object (e.g. the pore space) is generated so that simple and compact basic information of the object remains while irrelevant redundant information is neglected in the resultant skeleton image. Having obtained the skeleton of an object, most of the geometric and topological quantities of this object can then be easily derived. After reviewing many existing algorithms, a more accurate and efficient thinning algorithm is presented to meet the specific requirements for the study of pore microstructure. Furthermore, general geometric and topological properties of the pore space are calculated and analysed, including pore size distribution, bond (or node) radiillengthlvolume, shape factor and coordination number etc. As an important contribution, a novel algorithm to compute the Euler-Poincare characteristic (Euler number) is presented and a new topological descriptor is introduced to overcome the limitations of the Euler number and the coordination number. To validate the methodology and to carry out some basic analysis of the microstructure of porous media, I investigate the geometric and topologic features directly from 3D binary images of rock samples. The volumetric pore size distribution is obtained, and the frequency of pore inscribed radii (or diameter) is calculated, the shape of cross sections along pore channels is quantified as the shape factor and the corresponding algorithm is created. In this study, many quantities for describing the morphological properties of porous media have been successfully introduced. To carry this novel methodology into the use of network models for the prediction of flow processes, three rock samples are selected and analysed. A new approach is developed for partitioning the pore space into the network of nodes and bonds. This partitioning differs from existing methods and it aims to solve some specific problems which often occur in unconsolidated (high porosity) porous media. Following this some single/multi-phase properties are calculated for these three rock samples, such as absolute permeabilities and relative permeabilities. A number of relations between pore size and the absolute permeability, or between pore connectivity and absolute permeability, are explored. The comprehensive relation between pore size, connectivity and absolute permeabilities is also studied and preliminary results are given. This research has created new tools that will play important roles in the analysis of porous media.

The Finite Non-periodic Toda Lattice: A Geometric and Topological Viewpoint

Abstract: In 1967, Japanese physicist Morikazu Toda published the seminal papers exhibiting soliton solutions to a chain of particles with nonlinear interactions between nearest neighbors. In the decades that followed, Toda's system of particles has been generalized in different directions, each with its own analytic, geometric, and topological characteristics that sets it apart from the others. These are known collectively as the Toda lattice. This survey describes and compares several versions of the finite non-periodic Toda lattice from the perspective of their geometry and topology.

The Three Gap Theorem and Riemannian geometry

Abstract: The classical Three Gap Theorem asserts that for a natural number n and a real number p, there are at most three distinct distances between consecutive elements in the subset of [0,1) consisting of the reductions modulo 1 of the first n multiples of p. Regarding it as a statement about rotations of the circle, we find results in a similar spirit pertaining to isometries of compact Riemannian manifolds and the distribution of points along their geodesics.

Core varieties, extensivity, and rig geometry

Abstract: The role of the Frobenius operations in analyzing finite spaces, as well as the extended algebraic geometry over rigs, depend partly on varieties (Birkhoan inclusions of algebraic categories) that have coreflections as well as reflections and whose dual category of ane spaces is extensive. Even within the category of those rigs where 1+1 = 1, not only distributive lattices but also the function algebras of tropical geometry (where x + 1 = 1) and the dimension rigs of separable prextensive categories (where x + x 2 = x 2 ) enjoy those features. (Talk given at CT08, Calais.) Algebraic geometry, analytic geometry, smooth geometry, and also simplicial topology, all enjoy the axiomatic cohesion described in my recent article (4). The cohesion theory aims to assist the development of those subjects by revealing characteristic ways in which their categories dier from others. (Such considerations will be important in order to carry out Grothendieck's 1973 program (2) for simplifying the foundations of algebraic geometry.) An axiomatic theory often captures more examples than originally intended. In the present case not only the smooth generalization (SDG) but also some semi-combinatorial ones are of interest. Some of these can be approached via sites of definition that can be handled in ways very closely analogous to Grothendieck's algebraic geometry construc- tions. Even ultra-basic properties of cohesive categories, such as extensivity, may not be true in the sites themselves; a modest step toward the treatment of that problem is the recognition of some algebraic categories as core varieties within others. But first I recall another remarkable feature of many cohesive toposes that, although discovered through its relevance to dierential equations, nevertheless points toward the power of map spaces in building combinatorial objects from simple ingredients.

An invitation to toric degenerations

Abstract: This is an expository paper which explores the ideas of the authors' paper "From Affine Geometry to Complex Geometry", arXiv:0709.2290. We explain the basic ideas of the latter paper by going through a large number of concrete, increasingly complicated examples.

On A-tensors in Riemannian geometry

Abstract: We present examples, both compact and non-compact complete, of lo- cally non-homogeneous proper A-manifolds.

Lectures on Riemannian Geometry, Part II: Complex Manifolds

Abstract: This is a set of introductory lecture notes on the geometry of complex manifolds. It is the second part of the course on Riemannian Geometry given at the MRI Masterclass in Mathematics, Utrecht, 2008. The first part was given by Prof. E. van den Ban, and his lectures notes can be found on the web-site of this course, http://www.math.uu.nl/people/ban/riemgeom2008/riemgeom2008.html. Topics that we discuss in these lecture notes are : almost complex structures and complex

Natural differential operations on manifolds: an algebraic approach

Abstract: Natural algebraic differential operations on geometric quantities on smooth manifolds are considered. A method for the investigation and classification of such operations is described, the method of IT-reduction. With it the investigation of natural operations reduces to the analysis of rational maps between -jet spaces, which are equivariant with respect to certain algebraic groups. On the basis of the method of IT-reduction a finite generation theorem is proved: for tensor bundles all the natural differential operations of degree at most can be algebraically constructed from some finite set of such operations. Conceptual proofs of known results on the classification of natural linear operations on arbitrary and symplectic manifolds are presented. A non-existence theorem is proved for natural deformation quantizations on Poisson manifolds and symplectic manifolds.Bibliography: 21 titles.