Showing papers on "Geometry and topology published in 2003"

Compactness results in Symplectic Field Theory

Abstract: This is one in a series of papers devoted to the foundations of Symplectic Field Theory sketched in (4). We prove compactness results for moduli spaces of holomorphic curves arising in Symplectic Field Theory. The theorems generalize Gromov's compactness theorem in (8) as well as compactness theorems in Floer homology theory, (6, 7), and in contact geometry, (9, 19).

Hodge Theory and Complex Algebraic Geometry II

Abstract: The 2003 second volume of this account of Kaehlerian geometry and Hodge theory starts with the topology of families of algebraic varieties. Proofs of the Lefschetz theorem on hyperplane sections, the Picard–Lefschetz study of Lefschetz pencils, and Deligne theorems on the degeneration of the Leray spectral sequence and the global invariant cycles follow. The main results of the second part are the generalized Noether–Lefschetz theorems, the generic triviality of the Abel–Jacobi maps, and most importantly Nori's connectivity theorem, which generalizes the above. The last part of the book is devoted to the relationships between Hodge theory and algebraic cycles. The book concludes with the example of cycles on abelian varieties, where some results of Bloch and Beauville, for example, are expounded. The text is complemented by exercises giving useful results in complex algebraic geometry. It will be welcomed by researchers in both algebraic and differential geometry.

Cohomology of Vector Bundles and Syzygies

Abstract: The central theme of this book is an exposition of the geometric technique of calculating syzygies. It is written from a point of view of commutative algebra, and without assuming any knowledge of representation theory the calculation of syzygies of determinantal varieties is explained. The starting point is a definition of Schur functors, and these are discussed from both an algebraic and geometric point of view. Then a chapter on various versions of Bott's Theorem leads on to a careful explanation of the technique itself, based on a description of the direct image of a Koszul complex. Applications to determinantal varieties follow, plus there are also chapters on applications of the technique to rank varieties for symmetric and skew symmetric tensors of arbitrary degree, closures of conjugacy classes of nilpotent matrices, discriminants and resultants. Numerous exercises are included to give the reader insight into how to apply this important method.

Ricci-flat Metrics, Harmonic Forms and Brane Resolutions

Abstract: We discuss the geometry and topology of the complete, non-compact, Ricci-flat Stenzel metric, on the tangent bundle of S n+1 . We obtain explicit results for all the metrics, and show how they can be obtained from first-order equations derivable from a superpotential. We then provide an explicit construction for the harmonic self-dual (p, q)-forms in the middle dimension p+q=(n+1) for the Stenzel metrics in 2(n+1) dimensions. Only the (p, p)-forms are L 2 -normalisable, while for (p, q)-forms the degree of divergence grows with . We also construct a set of Ricci-flat metrics whose level surfaces are U(1) bundles over a product of N Einstein-Kahler manifolds, and we construct examples of harmonic forms there. As an application, we construct new examples of deformed supersymmetric non-singular M2-branes with such 8-dimensional transverse Ricci-flat spaces. We show explicitly that the fractional D3-branes on the 6-dimensional Stenzel metric found by Klebanov and Strassler is supported by a pure (2,1)-form, and thus it is supersymmetric, while the example of Pando Zayas-Tseytlin is supported by a mixture of (1,2) and (2,1) forms. We comment on the implications for the corresponding dual field theories of our resolved brane solutions.

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 g_s \alpha'.

Introduction to Möbius differential geometry

Abstract: This book introduces the reader to the geometry of surfaces and submanifolds in the conformal n-sphere. Various models for Mobius geometry are presented: the classical projective model, the quaternionic approach, and an approach that uses the Clifford algebra of the space of homogeneous coordinates of the classical model; the use of 2-by-2 matrices in this context is elaborated. For each model in turn applications are discussed. Topics comprise conformally flat hypersurfaces, isothermic surfaces and their transformation theory, Willmore surfaces, orthogonal systems and the Ribaucour transformation, as well as analogous discrete theories for isothermic surfaces and orthogonal systems. Certain relations with curved flats, a particular type of integrable system, are revealed. Thus this book will serve both as an introduction to newcomers (with background in Riemannian geometry and elementary differential geometry) and as a reference work for researchers.

Modular groups in Cantorian E(∞) high-energy physics

Abstract: This paper proposes that the geometry and topology of quantum spacetime is shadowed closely by the Mobius geometry of quasi-Fuschian and Kleinian groups and that is the cause behind the phenomena of high-energy particle physics. In addition, on the large scale measurement of, for instance, the microwave background temperature, the universality of the Merger sponge provides an excellent limit set model for the Charlier–Zeldovich proposal of the fracticality of the universe today and the rather accurate estimate Tc=(ln20/ln3)=2.726k. In particular the paper shows the link between the fix points of the modular groups of the vacuum and the golden mean φ=(1/(1+φ))=( 5 −1)/2 of E(∞) spacetime by analytical continuation of a Mobius transformation.

Stable Modules and the D(2)-Problem

Abstract: This 2003 book is concerned with two fundamental problems in low-dimensional topology. Firstly, the D(2)-problem, which asks whether cohomology detects dimension, and secondly the realization problem, which asks whether every algebraic 2-complex is geometrically realizable. The author shows that for a large class of fundamental groups these problems are equivalent. Moreover, in the case of finite groups, Professor Johnson develops general methods and gives complete solutions in a number of cases. In particular, he presents a complete treatment of Yoneda extension theory from the viewpoint of derived objects and proves that for groups of period four, two-dimensional homotopy types are parametrized by isomorphism classes of projective modules. This book is carefully written with an eye on the wider context and as such is suitable for graduate students wanting to learn low-dimensional homotopy theory as well as established researchers in the field.

Some problems of integral geometry on Anosov manifolds

Abstract: In this paper we prove that on an Anosov manifold the space of symmetric m -tensor fields of vanishing energy is finite dimensional modulo the space of potential tensor fields for an arbitrary m and coincides with the latter for m=0 and m=1 . For m=2 this question relates to the spectral rigidity problem.

Symplectic surgeries from singularities

Abstract: We describe a variety of symplectic surgeries (not a priori compatible with Kaehler structures) which are obtained by combining local Kaehler degenerations and resolutions of singularities. The effect of the surgeries is to replace configurations of Lagrangian spheres with symplectic submanifolds. We discuss several examples in detail, relating them to existence questions for symplectic manifolds with c1 > 0, c1 = 0, c1 < 0 in four and six dimensions.

Small contractions and symplectic 4-folds

Abstract: We classify small contractions of (holomorphically) symplectic 4-folds. AMS MSC: 14E30, 14J35.

Global Riemannian Geometry: Curvature and Topology

Abstract: Distance Geometric Analysis on Manifolds (Steen Markvorsen).- The Dirac Operator in Geometry and Physics (Maung Min-Oo).

Noncommutative localization in topology

Abstract: A survey of the applications of the noncommutative Cohn localization of rings to the topology of manifolds with infinite fundamental group, with particular emphasis on the algebraic K- and L-theory of generalized free products.

Geometry of Reduced Sextics of Torus Type

Abstract: In [7], we gave a classification of the configurations of singularities of irreducible sextic of torus type. In this paper, we give a classification of the configurations of singularities on reducible sextics of torus type. We determine the component types and the geometry of the components for each configuration.

Quaternionic Geometry of Matroids

Abstract: Building on a recent joint paper with Sturmfels, here we argue that the combinatorics of matroids is intimately related to the geometry and topology of toric hyperkaehler varieties. We show that just like toric varieties occupy a central role in Stanley's proof for the necessity of McMullen's conjecture (or g-inequalities) about the classification of face vectors of simplicial polytopes, the topology of toric hyperkaehler varieties leads to new restrictions on face vectors of matroid complexes. Namely in this paper we give two proofs that the injectivity part of the Hard Lefschetz theorem survives for toric hyperkaehler varieties. We explain how this implies the g-inequalities for rationally representable matroids. We show how the geometrical intuition in the first proof, coupled with results of Chari, leads to a proof of the g-inequalities for general matroid complexes, which is a recent result of Swartz. The geometrical idea in the second proof will show that a pure O-sequence should satisfy the g-inequalities, thus showing that our result is in fact a consequence of a long-standing conjecture of Stanley.

The changing shape of geometry : celebrating a century of geometry and geometry teaching

Abstract: Introduction 1. The nature of geometry 2. Desert island theorems - Greek geometry 3. The history of geometry 4. Desert island theorems - elementary Euclidean geometry 5. Pythagoras' theorem 6. Desert island theorems - advanced Euclidean geometry 7. The golden ratio 8. Desert island theorems - non-Euclidean geometry and topology 9. Recreational geometry 10. Desert island theorems - geometrical physics 11. The teaching of geometry Appendices.

Orbit configuration spaces associated to discrete subgroups of PSL(2,R)

Abstract: The purpose of this article is to analyze several Lie algebras associated to “orbit configuration spaces” obtained from a group G acting freely, and properly discontinuously on the upper half-plane H 2 . The Lie algebra obtained from the descending central series for the associated fundamental group is shown to be isomorphic, up to a regrading, to 1. the Lie algebra obtained from the higher homotopy groups of analogous constructions associated to H 2 × C q modulo torsion, as well as 2. the Lie algebra obtained from horizontal chord diagrams for surfaces. The resulting Lie algebras are similar to those studied in [T. Kohno, Linear representations of braid groups and classical Yang-Baxter equations, Contemp. Math. 78 (1988) 339–363; T. Kohno, Vassiliev invariants and de Rham complex on the space of knots, Contemp. Math. 179 (1994) 123–138; T. Kohno, Elliptic KZ system, braid groups of the torus and Vassiliev invariants, Topology and its Applications 78 (1997) 79–94; D.C. Cohen, Monodromy of fiber-type arrangements and orbit configuration spaces, Forum Math. 13 (2001) 505–530; F.R. Cohen, M. Xicotencatl, On orbit configuration spaces associated to the Gaussian integers: homology and homotopy groups, Topology Appl. 118 (2002) 17–29; E. Fadell and S. Husseini, The space of loops on configuration spaces and the Majer-Terracini index, Topol. Methods Nonlinear Anal. J. Julius Schauder Center 11 (1998), 249–271; E. Fadell and S. Husseini, Geometry and Topology of Configuration Spaces, in: Springer Monographs in Mathematics, Springer-Verlag, 2001; F.R. Cohen and T. Sato, On groups of morphisms of coalgebras, (submitted for publication)]. The structure of a related graded Poisson algebra defined below and obtained from an analogue of the infinitesimal braid relations parametrized by G is also addressed.

Discretization of functionally based heterogeneous objects

Abstract: The presented approach to discretization of functionally defined heterogeneous objects is oriented towards applications associated with numerical simulation procedures, for example, finite element analysis (FEA). Such applications impose specific constraints upon the resulting surface and volume meshes in terms of their topology and metric characteristics, exactness of the geometry approximation, and conformity with initial attributes. The function representation of the initial object is converted into the resulting cellular representation described by a simplicial complex. We consider in detail all phases of the discretization algorithm from initial surface polygonization to final tetrahedral mesh generation and its adaptation to special FEA needs. The initial object attributes are used at all steps both for controlling geometry and topology of the resulting object and for calculating new attributes for the resulting cellular representation.

On the geometry and topology of singular optimal control problems and their solutions

Abstract: The existence of singular arcs for optimal control problems is studied by using a geometric recursive algorithm inspired in Dirac’s theory of constraints. It is shown that singular arcs must lie in the singular locus of a projection map into the coestate space. After applying the geometrical recursive constraints algorithm, we arrive to a reduced set of hamiltonian equations that replace Pontriaguine’s maximum principle. Finally, a global singular perturbation theory is used to obtain nearly optimal solutions.