Showing papers on "Geometry and topology published in 1996"

The Seiberg-Witten equations and applications to the topology of smooth four-manifolds

Abstract: 1Introduction12Clifford Algebras and Spin Groups53Spin Bundles and the Dirac Operator234The Seiberg-Witten Moduli Space555Curvature Identities and Bounds696The Seiberg-Witten Invariant877Invariants of Kahler Surfaces109Bibliography127

Renormalization and 3-Manifolds Which Fiber over the Circle (AM-142), Volume 142

Abstract: Many parallels between complex dynamics and hyperbolic geometry have emerged in the past decade. Building on work of Sullivan and Thurston, this book gives a unified treatment of the construction of fixed-points for renormalization and the construction of hyperbolic 3- manifolds fibering over the circle.Both subjects are studied via geometric limits and rigidity. This approach shows open hyperbolic manifolds are inflexible, and yields quantitative counterparts to Mostow rigidity. In complex dynamics, it motivates the construction of towers of quadratic-like maps, and leads to a quantitative proof of convergence of renormalization.

Index theory, coarse geometry, and topology of manifolds

Abstract: Index theory (Chapter 1) Coarse geometry (Chapter 2) $C*$-algebras (Chapter 3) An example of a higher index theorem (Chapter 4) Assembly (Chapter 5) Surgery (Chapter 6) Mapping surgery to analysis (Chapter 7) The coarse Baum-Connes conjecture (Chapter 8) Methods of computation (Chapter 9) Coarse structures and boundaries (Chapter 10) References Index.

Positive Curvature, Macroscopic Dimension, Spectral Gaps and Higher Signatures

Abstract: Our journey starts with a macroscopic view of Riemannian manifolds with positive scalar curvature and terminates with a glimpse of the proof of the homotopy invariance of some Novikov higher signatures of non-simply connected manifolds. Our approach focuses on the spectra of geometric differential operators on compact and non-compact manifolds V where the link with the macroscopic geometry and topology is established with suitable index theorems for our operators twisted with almost flat bundles over V. Our perspective mainly comes from the asymptotic geometry of infinite groups and foliations.

Riemannian Manifolds of Conullity Two

Abstract: Definition and early development local structure of semi-symmetric spaces explicit treatment of foliated semi-symmetric spaces curvature homogeneous semi-symmetric spaces asymptotic distributions and algebraic rank three-dimensional Riemannian manifolds of conullity two asymptotically foliated semi-symmetric spaces elliptic semi-symmetric spaces complete foliated semi-symmetric spaces application - local rigidity problems for hypersurfaces with type number two in IR4 three-dimensional Riemannian manifolds of relative conullity two appendix - more about curvature homogeneous spaces.

A convergence theorem in the geometry of alexandrov spaces

Abstract: The fibration theorems in Riemannian geometry play an important role in the theory of convergence of Riemannian manifolds. In the present paper, we extend them to the Lipschitz submersion theorem for Alexandrov spaces, and discuss some applications.

Low degree polynomial equations: arithmetic, geometry and topology

Abstract: These are the notes of my lectures at the 1996 European Congress of Mathematicians. {} Polynomials appear in mathematics frequently, and we all know from experience that low degree polynomials are easier to deal with than high degree ones. It is, however, not clear that there is a well defined class of "low degree" polynomials. For many questions, polynomials behave well if their degree is low enough, but the precise bound on the degree depends on the concrete problem. {} It turns out that there is a collection of basic questions in arithmetic, algebraic geometry and topology all of which give the same class of "low degree" polynomials. The aim of this lecture is to explain these properties and to provide a survey of the known results.

Formal deformations of symplectic manifolds with boundary.

Abstract: Suppose that M is a manifold with boundary, supplied with a 6-symplectic structure. For the purpose of this introduction we will mean by that a Poisson structure {,} on C°°(M), non degenerate on vector fields on M whose restriction to dM is in (3 M, TM). An algebra /\\(M) is a formal deformation of (M, {,}) if it is isomorphic äs linear space to C°°(Af )[ft] and has an associative product * which is local, e.g. the jet of /* g at each point depends only on the jets of / and g at that point, and such that the Lie algebra ( ^ , ^]) is a formal deformation of the Lie algebra (C°°(M),{,}). For the precise definition see Section 4. One can think of A* (M) äs the algebra of asymptotic Symbols (in the sense of [9]) of *PDO's constructed out of Hamiltonian vector fields of {,}. The main properties of A(M) are äs follows.

Geometry, Topology and Quantization

Abstract: Preface. 1. Manifold and Differential Forms. 2. Spinor Structure and Twistor Geometry. 3. Quantization. 4. Quantization and Gauge Field. 5. Fermions and Topology. 6. Topological Field Theory. References. Index.

Non-Linear Similarity Revisited

Abstract: Let G be a finite group and V, V ′ finite dimensional real orthogonal representations of G. Then V is said to be topologically similar to V ′ (V ∼t V ) if there exists a homeomorphism h : V → V ′ which is G-equivariant. If V, V ′ are topologically similar, but not linearly isomorphic, then such a homeomorphism is called a nonlinear similarity. The topological classification of G-representations was first studied by de Rham [18]. He proved that if a topological similarity h : V → V ′ of orthogonal representations preserves the unit spheres and restricts to a diffeomorphism between S(V ) and S(V ), then V and V ′ are linearly isomorphic. In 1973, Kuiper and Robbin [11] obtained positive results on the general problem and conjectured that topological equivalence implies linear equivalence for all finite groups G. However, in 1981 Cappell and Shaneson [1] constructed the first examples of non-linear similarities. The simplest occurs for G = Z/8, but they also constructed a large class of examples for cyclic groups of the form G = Z/4q. Further results can be found in [4], [2], [3], and [13]. On the other hand, Hsiang and Pardon [10] and Madsen and Rothenberg [12] independently proved the conjecture for all odd–order groups. In addition, the main theorem of [10] ruled out some non-linear similarities for even-order groups G. The purpose of this paper is to give new restrictions on the existence of non-linear similarities using techniques from bounded topology.

Quantization of systems with a general phase space equipped with a Riemannian metric

Abstract: Quantization on phase spaces of general geometry devoid of any special symmetry properties is discussed on the basis of phase spaces endowed with a symplectic structure, a Riemannian geometry, and a structure. Using techniques from differential geometry, and especially exploiting the Dirac operator, we are able to offer a fully geometric quantization procedure for a wide class of symmetry free phase spaces. Our procedure leads to the conventional results in cases where the phase space is a symmetric space for which alternative quantization techniques suffice.