Showing papers on "Geometry and topology published in 1991"

The topology of torus actions on symplectic manifolds

Abstract: This is an extended second edition of "The Topology of Torus Actions on Symplectic Manifolds" published in this series in 1991. The material and references have been updated. Symplectic manifolds and torus actions are investigated, with numerous examples of torus actions, for instance on some moduli spaces. Although the book is still centered on convexity theorems, it contains much more results, proofs and examples. Chapter I deals with Lie group actions on manifolds. In Chapters II and III, symplectic geometry and Hamiltonian group actions are introduced, especially torus actions and action-angle variables. The core of the book is Chapter IV which is devoted to applications of Morse theory to Hamiltonian group actions, including convexity theorems. As a family of examples of symplectic manifolds, moduli spaces of flat connections are discussed in Chapter V. Then, Chapter VI centers on the Duistermaat-Heckman theorem. In Chapter VII, a topological construction of complex toric varieties is presented, and the last chapter illustrates the introduced methods for Hamiltonian circle actions on 4-manifolds.

Geometry of Low-dimensional Manifolds: Filling by holomorphic discs and its applications

Abstract: The survey is devoted to application of the technique of filling by holomorphic discs to different symplectic and complex analytic problems. COMPLEX AND SYMPLECTIC RECOLLECTIONS J -Convexity Let X, J be an almost complex manifold of the real dimension 4 and Σ be an oriented hypersurface in X of the real codimension 1. Each tangent plane T x (Σ), x ∈ Σ, contains a unique complex line ξ x ⊂ T x (Σ) which we will call a complex tangency to Σ at x . The complex tangency is canonically oriented and, therefore, cooriented. Hence the tangent plane distribution ξ on Σ can be defined by an equation α = 0 where the 1-form α is unique up to multiplication by a positive function. The 2-form d α ∣ ξ is defined up to the multiplication by the same positive factor. We say that Σ is J-convex (or pseudo-convex) if d α( T, JT ) > 0 for any non-zero vector T ∈ ξ x , x ∈ Σ. We use the word “pseudo-convex” when the almost complex structure J is not specified. An important property of a J -convex hypersurface Σ is that it cannot be touched inside (according to the canonical coorientation of Σ) by a J -holomorphic curve. In particular, if Ω is a domain in X bounded by a smooth J -convex boundary ∂Ω then all interior points of a J -holomorphic curve C ⊂ X with ∂ C ⊂ ∂Ω belong to IntΩ. Moreover, C is transversal to ∂Ω in all regular points of its boundary ∂ C .

Collapsing and pinching under a lower curvature bound

Abstract: In this paper we are concerned with collapsing phenomena and pinching problems of Riemannian manifolds whose sectional curvatures are uniformly bounded from below. For a positive integer n and for D > 0, let X/ be the set of compact Riemannian n-manifolds M with sectional curvatures KM ? -1 and diameters diam(M) < D. By the weak compactness theorem of Gromov [GLP], X' is relatively compact in the set of compact metric spaces with respect to the Hausdorff distance. Furthermore Gromov proved in [G2] that the sum of Betti numbers of any element in X is uniformly bounded in terms of the given constants. These results suggest that X/ is an object of study on which we could develop some geometry and topology. In fact, when volumes are uniformly bounded away from zero, the following results are known. Let #(v) be the subset of X/ with volume vol(M) ? v. Grove and Petersen [GP1] proved that the set of homotopy types of elements in 4(v) is finite. (Related results are in [Y2].) For pinching problems Otsu, Shiohama and the author ([OSY], [Y3]) obtained some differential sphere theorems in the class X(v). According to a recent paper [GPW], if n # 3,4, then (v) contains at most finitely many diffeomorphism types. Let Mi (i = 1, 2,...) be a convergent sequence in A, and X be the limit of them. We say that Mi collapses to X if the Hausdorff dimension of X is less than n. By the volume condition, no collapsing occurs in 4(v). From this point of view it is quite natural to ask what collapsing phenomena occur and what types of pinching theorems hold in the class X/. In this situation it seems difficult to determine the singularities of X (some examples are given in Section 1). In this paper we study collapsing phenomena in the case when X is a Riemannian manifold and establish a pinching theorem. For a Riemannian manifold M we denote by inj(M) the injectivity radius of M. When N is written in place of X and the metric of N is normalized, our

Algebraic Curves over Finite Fields

Abstract: 1 Algebraic curves and function fields 2 The Riemann-Roch theorem 3 Zeta functions 4 Applications to exponential sums and zeta functions 5 Applications to coding theory Bibliography

Geometry of phase spaces

Abstract: Philosophical preliminaries geometry of bilinear forms and affine spaces symplectic spaces and symplectic Pfaff problem symplectic manifolds Newton mechanics, Galilean symmetry and the origin of the Hamilton theory basic concepts of the Hamilton dynamics constraints on dynamics contact spaces statistical concepts and the quantum-classical correspondence affinely-rigid body and ellipsoidal figures of equilibrium.

A proof of Thurston's uniformization theorem of geometric orbifolds.

Abstract: The authors prove that every geometric orbifold is good. More precisely, let X be a smooth connected manifold, and let G be a group of diffeomorphisms of X with the property that if any two elements of G agree on a nonempty open subset of X, then they coincide on X. If Q is an orbifold which is locally modelled on quotients of open subsets of X by finite subgroups of G, then the authors prove that the universal orbifold covering of Q is a (G,X)-manifold. A similar theorem was stated, and the proof sketched, in W. Thurston's lecture notes on the geometry and topology of 3-manifolds.

Geometry of Low-dimensional Manifolds: An introduction to polyhedral metrics of non-positive curvature on 3-manifolds

Abstract: The photographic printer of the present invention stores measured gamma values for the photosensitive medium upon which prints are to be made. The gamma values modify exposure times so that the operator button corrections always correspond to a known increment in density. When the photosensitive medium is changed, new gamma values are entered and the printer again modifies exposure times by the appropriate amount so that button correction increments remain constant despite the change in paper gamma.

A geometric approach to total envisioning

Abstract: Conventional envisioners proposed in qualitative physics have two difficulties in common: ambiguities in prediction and inability of reasoning about global behaviors. We take a geometric approach to overcome these difficulties and have implemented a program PSX2NL which can reason about global behaviors by analyzing geometry and topology of solution curves of ordinary differential equations in the phase space. In this paper, we highlight a flow grammar which specifies possible patterns of solution curves one may see in the phase space. The role of a flow grammar in PSX2NL is twofold: firstly, it allows PSX2NL to reason about complex patterns in a uniform manner; secondly, it allows PSX2NL to switch to an approximate, top-down algorithm when complete geometric clues are not available due to the difficulty of mathematical problems encountered.

The geometry and topology of quotient varieties

Abstract: Let X be a nonsingular projective variety with an algebraic action of a complex torus (c*)n. We study in this thesis the symplectic quotients (reduced phase spaces) and the quotients in a more general sense. As a part of our program, we have developed a general procedure for computing the intersection homology groups of the quotient varieties. In particular, we obtained an explicit inductive formula for the intersection Poincare polynomial of an arbitrary quotient. Also, explicit resultswere obtained in the case of the maximal torus actions on the flag varieties G/ B. Thesis Supervisor: Robert MacPherson Title: Professor of Mathematics

Geometry of homogeneous hadamard manifolds

Abstract: Using result of R. Azencott and E. Wilson on the algebraic structure of homogeneous manifolds of nonpositive sectional curvature, we discuss the geometry of these manifolds and give several conditions how to distinguish the symmetric spaces among them.

Geometry of Low-dimensional Manifolds: Extremal immersions and the extended frame bundle

Abstract: We present a computationally powerful formulation of variational problems that depend on the extrinsic and intrinsic geometry of immersions into a manifold. The approach is based on a lift of the action integral to a larger space and proceeds by systematically constraining the variations to preserve the foliation of a Pfaffian system on an extended frame bundle. Explicit Euler-Lagrange equations are computed for a very general class of Lagrangians and the method illustrated with examples relevant to recent developments in theoretical physics. The method provides a means of determining spatial boundary conditions for immersions with boundary and enables a construction to be made of constants of the motion in terms of Euler- Lagrange solutions and admissible symmetry vectors.

Dynamics of surface homeomorphisms : braid types and coexistence of periodic orbits

Abstract: In this Thesis, we discuss the following general problem in dynamical systems: given a surface homeomorphism, and some information about its periodic orbits, what else can we deduce about its periodic orbit structure? Using the concept of the ‘braid type’ of a periodic orbit, its relation to Artin’s braid group, and the Nielsen-Thurston classification of surface homeomorphisms, we examine problems pertaining to the coexistence of periodic orbits, in particular for homeomorphisms of the disc, annulus and 2-torus. We aim to elucidate the underlying geometry and topology in such systems. The main original results are the following: • classification of braid types for periodic orbits of diffeomorphisms of genus one surfaces with topological entropy zero (Theorems 2.5 and 2.6). • lower bounds on the size of the rotation sets of annulus homeomorphisms which possess certain periodic orbits or finite invariant sets (Theorems 3.17 and 3.19, Theorem 3.20). • bounds on the size and shape of rotation sets of torus homeomorphisms possessing certain periodic orbits. (Theorems 3.24 and 3.25). • the coexistence of periodic orbits in the disc, for periodic orbits of prime period (Theorem 4.2), of period 4 (Theorem 4.10), and for 3-point invariant sets (Theorem 4.11). • the coexistence of periodic orbits in the annulus (Theorem 4.4), and of the sphere with a 4-point invariant set (Theorem 4.12). • given a torus homeomorphism isotopic to the identity which possesses a fixed point, it is isotopic to the identity relative to that fixed point (Theorem 5.6). • given a periodic orbit of a disc homeomorphism of period 3, the coexistence of a strongly linked fixed point (Theorem 5.10). • given a periodic orbit of the annulus homeomorphism of pseudo-Anosov braid type, its rotation number lies in the interior of the rotation set (Theorem 6.1). • amongst certain sets of braid types of the annulus and disc, the existence of minimal elements, which any other element dominates (Theorems 7.4 and 7.15).