Showing papers on "Geometry and topology published in 2006"

Geometry of Quantum States: Frontmatter

Abstract: a ) p. 131 The discussion between eqs. (5.14) and (5.15) is incorrect (dA should be made as large as possible!). b ) p. 256 In the figure, the numbers 6) and 7) occur twice. c ) p. 292 At the end of section 12.5, it should be the space of isospectral 0Hermitian matrices. d ) p. 306 A ”Tr” is missing in eq. (13.43). e ) p. 327, Eq. (14.64b) is 〈Trρ〉B = N(14N+10) (5N+1)(N+3) should be 〈Trρ〉B = 8N+7 (N+2)(N+4)

Gauge theories from toric geometry and brane tilings

Abstract: We provide a general set of rules for extracting the data defining a quiver gauge theory from a given toric Calabi-Yau singularity. Our method combines information from the geometry and topology of Sasaki-Einstein manifolds, AdS/CFT, dimers, and brane tilings. We explain how the field content, quantum numbers, and superpotential of a superconformal gauge theory on D3-branes probing a toric Calabi-Yau singularity can be deduced. The infinite family of toric singularities with known horizon Sasaki-Einstein manifolds La,b,c is used to illustrate these ideas. We construct the corresponding quiver gauge theories, which may be fully specified by giving a tiling of the plane by hexagons with certain gluing rules. As checks of this construction, we perform a-maximisation as well as Z-minimisation to compute the exact R-charges of an arbitrary such quiver. We also examine a number of examples in detail, including the infinite subfamily La,b,a, whose smallest member is the Suspended Pinch Point.

Elementary prerequisites for E-infinity . (Recommended background readings in nonlinear dynamics, geometry and topology)

Abstract: The present work constitutes a guided tour through the mathematics needed for a proper understanding of E -infinity theory as applied to high energy physics and quantum gravity.

Geometry of, and via, symmetries

Abstract: It is well known that Lie groups and homogeneous spaces provide a rich source of interesting examples for a variety of geometric aspects. Likewise it is often the case that topological and geometric restrictions yield the existence of isometries in a more or less direct way. The most obvious example of this is the group of deck transformations of the universal cover of a nonsimply connected manifold. More subtle situations arise in the contexts of rigidity problems and of collapsing with bounded curvature. Our main purpose here is to present the view point that the geometry of isometry groups provide a natural and useful link between theory and examples in Riemannian geometry. This fairly unexplored territory is fascinating and interesting in its own right. At the same time it enters naturally when such groups arise in settings as above. More importantly, perhaps, this study also provides a systematic search for geometrically interesting examples, where the group of isometries is short of acting transitively in contrast to the case of homogeneous spaces mentioned above. Although the general philosophy presented here applies to many different situations, we will illustrate our point of view primarily within the context of manifolds with nonnegative or positive curvature. We have divided our presentation into five sections. The first section is concerned with basic equivariant Riemannian geometry of smooth compact transformation groups, including a treatment of Alexandrov geometry of orbit spaces. Section two is the heart of the subject. It deals with the geometry and topology in the presence of symmetries. It is here we explain our guiding principle which provides a systematic search for new constructions and examples of manifolds of positive or nonnegative curvature. In the third section we exhibit all the known constructions and examples of such manifolds. The topic of section four is geometry via symmetries. We display three different types of problems in which symmetries are not immediately present from the outset, but where their emergence is crucial to their solutions. In the last section we discuss a number of open problems and conjectures related either directly, potentially or at least in spirit to the subject presented here. Our exposition assumes basic knowledge of Riemannian geometry, and a rudimentary familiarity with Lie groups. Although we use Alexandrov geometry of spaces with a lower curvature bound our treatment does not require prior knowledge of this subject. Our intentions have been that anyone with these prerequisites will be able to get an impression of the subject, and guided by the references provided here will be able to go as far as their desires will take them.

Analysis, Geometry and topology of elliptic operators

Abstract: On the Mathematical Work of Krzysztof P Wojciechowski: Selected Aspects of the Mathematical Work of Krzysztof P Wojciechowski (M Lesch) Gluing Formulae of Spectral Invariants and Cauchy Data Spaces (J Park) Topological Theories: The Behavior of the Analytic Index under Nontrivial Embedding (D Bleecker) Critical Points of Polynomials in Three Complex Variables (L I Nicolaescu) Chern-Weil Forms Associated with Superconnections (S Paycha & S Scott) Heat Kernel Calculations and Surgery: Non-Laplace Type Operators on Manifolds with Boundary (I G Avramidi) Eta Invariants for Manifold with Boundary (X Dai) Heat Kernels of the Sub-Laplacian and the Laplacian on Nilpotent Lie Groups (K Furutani) Remarks on Nonlocal Trace Expansion Coefficients (G Grubb) An Anomaly Formula for L2-Analytic Torsions on Manifolds with Boundary (X Ma & W Zhang) Conformal Anomalies via Canonical Traces (S Paycha & S Rosenberg) Noncommutative Geometry: An Analytic Approach to Spectral Flow in von Neumann Algebras (M T Benameur et al) Elliptic Operators on Infinite Graphs (J Dodziuk) A New Kind of Index Theorem (R G Douglas) Noncommutative Holomorphic and Harmonic Functions on the Unit Disk (S Klimek) Star Products and Central Extensions (J Mickelsson) An Elementary Proof of the Homotopy Equivalence between the Restricted General Linear Group and the Space of Fredholm Operators (T Wurzbacher) Theoretical Particle, String and Membrane Physics, and Hamiltonian Dynamics: T-Duality for Non-Free Circle Actions (U Bunke & T Schick) A New Spectral Cancellation in Quantum Gravity (G Esposito et al) A Generalized Morse Index Theorem (C Zhu)

The geometry of a bi-Lagrangian manifold

Abstract: This is a survey on bi-Lagrangian manifolds, which are symplectic manifolds endowed with two transversal Lagrangian foliations We also study the non-integrable case (ie, a symplectic manifold endowed with two transversal Lagrangian distributions) We show that many different geometric structures can be attached to these manifolds and we carefully analyze the associated connections Moreover, we introduce the problem of the intersection of the two leaves, one of each foliation, through a point and show a lot of significative examples

Pursuit and evasion in non-convex domains of arbitrary dimensions

Abstract: Most results in pursuit-evasion games apply only to planar domains or perhaps to higher-dimensional domains which must be convex. We introduce a very general set of techniques to generalize and extend certain results on simple pursuit to non-convex domains of arbitrary dimension which satisfy a coarse curvature condition (the CAT(0) condition). I. PURSUIT / EVASION There is a significant literature on pursuit-evasion games, with natural motivations coming from robotics [10], [14], [24]. Such games involve one or more evaders in a fixed domain being hunted by one or more pursuers who win the game if the appropriate capture criteria are satisfied. Such criteria may be physical capture (the pursuers move to where the evaders are located) [12], [13], [20] or visual capture (there is a line-of-sight between a pursuer and an evader) [10], [23]. The types of pursuit games are many and varied: continuous or discrete time, bounded or unbounded speed, and constraints on admissible acceleration, energy expenditure, strategy, and sensing. For a quick introduction to the literature on pursuit games, see, e.g., [15], [10]. This paper focuses on one particular variable in pursuit games: the geometry and topology of the domain on which the game is played. The vast majority of the known results on pursuit-evasion are dependent on having domains which are two-dimensional or, if higher-dimensional, then convex. There has of late been a limited number of results for pursuit games on surfaces of revolution [11], cones [18], and round spheres [16]. Our results are complementary to these, in the sense that we work with domains having dimension higher than two, without constraints on being smooth or a manifold. The principal contribution of this work is a significant extension of known results on convex or planar domains to domains of arbitrary dimension which satisfy a type of curvature constraint known as the CAT(0) condition. Roughly speaking, the CAT(0) condition is a measure of what triangles in a metric space (X, d) look like, and, in particular, how a triangle compares to a Euclidean triangle with the same three side lengths. A simple mnemonic for a CAT(0) space is that it is a metric space, all of whose geodesic triangles have an angle sum no greater than π. Examples of CAT(0) domains are numerous and include the following: 1) convex Euclidean domains; 2) simply-connected subsets of E; 3) simply-connected Riemannian manifolds with nonpositive sectional curvature; 4) smooth Euclidean domains with boundaries having no more than one non-convex direction at each point; 5) simply-connected piecewise-Euclidean cubical complexes with no positive discrete curvature at the vertices; 6) Euclidean rectangular prisms with certain cylindrical sets removed; 7) simply-connected unions of convex sets which have no triple intersections. Our goal in this paper is to motivate the adoption of CAT(0) techniques in this and other areas of robotics in which the generalization of results from 2-d to higher dimensions is problematic. Decades of work by geometers in CAT(0) and more general Alexandrov geometry (geometry of spaces of bounded curvature) forms a powerful set of tools which are not very visible outside of mathematics departments (see [4], [5] and §III below.). The proofs of pursuit/evasion results in this paper are all very simple and very short, given the appropriate standard results from comparison geometry. We extend results to CAT(0) spaces in a dimension-independent manner, and, often, examples which are of high dimension are no more difficult than those with dimension two: the same proofs cover all cases. After giving a motivational example of a simple pursuit problem in the plane (§II-B), we motivate the notion of comparison triangles, total curvature bounds, and their utility in simple pursuit problems. We then present a brief primer on CAT(0) geometry in §III, followed by a more technical result on growth rates of total curvature in §IV. These tools are used in §V to solve problems involving simple pursuit curves. We conclude this note with results on escape criteria (§VI), contrasts with the positive curvature case (§VII), and a series of remarks and open directions (§VIII). II. A QUICK SUMMARY OF PURSUIT

The universal Kobayashi-Hitchin correspondence on Hermitian manifolds

Abstract: We prove a very general Kobayashi-Hitchin correspondence on arbitrary compact Hermitian manifolds. This correspondence refers to moduli spaces of "universal holomorphic oriented pairs". Most of the classical moduli problems in complex geometry (e. g. holomorphic bundles with reductive structure groups, holomorphic pairs, holomorphic Higgs pairs, Witten triples, arbitrary quiver moduli problems) are special cases of this universal classification problem. Our Kobayashi-Hitchin correspondence relates the complex geometric concept "polystable oriented holomorphic pair" to the existence of a reduction solving a generalized Hermitian-Einstein equation. The proof is based on the Uhlenbeck-Yau continuity method. We also investigate metric properties of the moduli spaces in this general non-Kaehlerian framework. We discuss in more detail moduli spaces of oriented connections, Douady Quot spaces and moduli spaces of non-abelian monopoles.

A Sampler of Riemann-Finsler Geometry

Abstract: Finsler geometry generalizes Riemannian geometry in the same sense that Banach spaces generalize Hilbert spaces. This book presents an expository account of seven important topics in Riemann�Finsler geometry, which have recently undergone significant development but have not had a detailed pedagogical treatment elsewhere. Each article will open the door to an active area of research and is suitable for a special topics course in graduate-level differential geometry.

Geometry and topology of symplectic resolutions

Abstract: This is an overview of math.AG/0310186, math.AG/0309290, math.AG/0501247, math.AG/0401002 and math.AG/0504584 written for the Proceedings of the AMS Meeting on Algebraic Geometry, Seattle, 2005.

Cantorian space–time and Hilbert space: Part II—Relevant consequences

Abstract: In this paper, we will show the consequences of the link between e (∞) and H (∞) . Starting from El Naschie’s e (∞) nature shows itself as an arena where the laws of physics appear at each scale in a self–similar way, linked to the resolution of the observations; while Hilbert’s space H (∞) is the mathematical support to describe the interaction between the observer and dynamical systems. The present formulation of space–time, based on the non-classical, Cantorian geometry and topology of the space–time, automatically solves the paradoxical outcome of the two-slit experiment and duality. The experimental fact that a wave-particle duality exists is an indirect confirmation of the existence of e (∞) . Another direct consequence of the fact that real space–time is the infinite dimensional hierarchical e (∞) is the existence of the scaling law R ( N ). The present author proposed it as a generalization of the Compton wavelength. This rule gives an answer to segregation of matter at different scales; it shows the role of fundamental constants like the speed of light and Plank’s constant h in the fundamental lengths scale without invoking the methodology of quantum mechanics. In addition, we consider the genesis of E -Infinity. A Cantorian potential theory can be formulated to take into account the geometry and topology of e (∞) in the context of gravitational theories. Consequently, we arrive at the result of the existence of gravitational channels.

A1-algebraic topology

Abstract: We present some recent results in A1-algebraic topology, which means both in A1-homotopy theory of schemes and its relationship with algebraic geometry. This refers to the classical relationship between homotopy theory and (differential) topology. We explain several examples of �motivic� versions of classical results: the theory of the Brouwer degree, the classification ofA1-coverings through theA1-fundamental group, the Hurewicz Theorem and the A1-homotopy of algebraic spheres, and the A1-homotopy classification of vector bundles. We also give some applications and perspectives.

Higher index theory of elliptic operators and geometry of groups

Abstract: The Atiyah�Singer index theorem has been vastly generalized to higher index theory for elliptic operators in the context of noncommutative geometry. Higher index theory has important applications to problems in differential topology and differential geometry such as the Novikov Conjecture on homotopy invariance of higher signatures and the existence problem of Riemannian metrics with positive scalar curvature. In this article, I will give a survey on recent development of higher index theory, its applications, and its fascinating connection to the geometry of groups and metric spaces.

Symplectic field theory and its applications

Abstract: Symplectic field theory (SFT) attempts to approach the theory of holomorphic curves in symplectic manifolds (also called Gromov-Witten theory) in the spirit of a topological field theory. This naturally leads to new algebraic structures which seems to have interesting applications and connections not only in symplectic geometry but also in other areas of mathematics, e.g. topology and integrable PDE. In this talk we sketch out the formal algebraic structure of SFT and discuss some current work towards its applications.

Riemannian Holonomy and Algebraic Geometry

Abstract: This survey paper is devoted to Riemannian manifolds with special holonomy. To any Riemannian manifold of dimension n is associated a closed subgroup of SO(n), the holonomy group; this is one of the most basic invariants of the metric. A famous theorem of Berger gives a complete (and rather small) list of the groups which can appear. Surprisingly, the compact manifolds with holonomy smaller than SO(n) are all related in some way to Algebraic Geometry. This leads to the study of special algebraic varieties (Calabi-Yau, complex symplectic or complex contact manifolds) for which Riemannian geometry rises interesting questions.

Complexity of 3-manifolds

Abstract: We give a summary of known results on Matveev's complexity of compact 3-manifolds. The only relevant new result is the classification of all closed orientable irreducible 3-manifolds of complexity 10.

Buildings and their Applications in Geometry and Topology

Abstract: Buildings were first introduced by J. Tits in 1950s to give systematic geometric inter- pretations of exceptional Lie groups and have been generalized in various ways: Euclidean buildings (Bruhat-Tits buildings), topological buildings, R-buildings, in particular R-trees. They are useful for many different applications in various subjects: algebraic groups, finite groups, finite geometry, representation theory over local fields, algebraic geometry, Arakelov intersection for arithmetic va- rieties, algebraic K-theories, combinatorial group theory, global geometry and algebraic topology, in particular cohomology groups, of arithmetic groups and S-arithmetic groups, rigidity of cofinite subgroups of semisimple Lie groups and nonpositively curved manifolds, classification of isoparamet- ric submanifolds in R n of high codimension, existence of hyperbolic structures on three dimensional manifolds in Thurston's geometrization program. In this paper, we survey several applications of buildings in differential geometry and geometric topology. There are four underlying themes in these applications: 1. Buildings often describe the geometry at infinity of symmetric spaces and locally symmetric spaces and also appear as limiting objects under degeneration or scaling of metrics. 2. Euclidean buildings are analogues of symmetric spaces for semisimple groups defined over local fields and their discrete subgroups. 3. Buildings of higher rank are rigid and hence objects which contain or induce higher rank buildings tend to be rigid. 4. Additional structures on buildings, for example, topological buildings, are important in applications for infinite groups.

Sub-Finsler geometry in dimension three

Abstract: We define the notion of sub-Finsler geometry as a natural generalization of sub-Riemannian geometry with applications to optimal control theory. We compute a complete set of local invariants, geodesic equations, and the Jacobi operator for the three-dimensional case and investigate homogeneous examples.

Homotopical dynamics in symplectic topology

Abstract: This is mainly a survey of recent work on algebraic ways to "measure" moduli spaces of connecting trajectories in Morse and Floer theories as well as related applica- tions to symplectic topology The paper also contains some new results In particular, we show that the methods of Barraud and Cornea (2003) continue to work in gen- eral symplectic manifolds (without any connectivity conditions) but under the bubbling threshold

Non-commutative localization in algebra and topology

Abstract: Dedication Preface Historical perspective Conference participants Conference photo Conference timetable 1. On flatness and the Ore condition J. A. Beachy 2. Localization in general rings, a historical survey P. M. Cohn 3. Noncommutative localization in homotopy theory W. G. Dwyer 4. Noncommutative localization in group rings P. A. Linnell 5. A non-commutative generalisation of Thomason's localisation theorem A. Neeman 6. Noncommutative localization in topology A. A. Ranicki 7. L2-Betti numbers, isomorphism conjectures and noncommutative localization H. Reich 8. Invariants of boundary link cobordism II. The Blanchfield-Duval form D. Sheiham 9. Noncommutative localization in noncommutative geometry Z. Skoda.