Topological Interpretation Research Articles

A note on the theta characteristics of a compact Riemann surface

AbstractLet X be a compact connected Riemann surface and ξ a square root of the holomorphic contangent bundle of X. Sending any line bundle L over X of order two to the image of dim H0(X, ξ ⊗ L) − dim H0(X, ξ) in Z/2Z defines a quadratic form on the space of all order two line bundles. We give a topological interpretation of this quadratic form in terms of index of vector fields on X.

Quantization of 2D dilaton supergravity with matter

General N = (1,1) dilaton supergravity in two dimensions allows a background independent exact quantization of the geometric part, if these theories are formulated as specific graded Poisson-sigma models. The strategy developed for the bosonic case can be carried over, although considerable computational complications arise when the hamiltonian constraints are evaluated in the presence of matter. Nevertheless, the constraint structure is the same as in the bosonic theory. In the matterless case gauge independent nonlocal correlators are calculated non-perturbatively. They respect local quantum triviality and allow a topological interpretation. In the presence of matter the ensuing nonlocal effective theory is expanded in matter loops. The lowest order tree vertices are derived and discussed, entailing the phenomenon of virtual black holes which essentially determine the corresponding S-matrix. Not all vertices are conformally invariant, but the S-matrix is invariant, as expected. Finally, the proper measure for the 1-loop corrections is addressed. It is argued how to exploit the results from fixed background quantization for our purposes.

Sur la L 2 -cohomologie des variétés à courbure négative

We give a topological interpretation of the space of L 2 -harmonic forms of finite-volume manifolds with sufficiently pinched negative curvature. We give examples showing that this interpretation fails if the curvature is not sufficiently pinched and that our result is sharp with respect to the pinching constants. The method consists first in comparing L 2 -cohomology with weighted L 2 -cohomology thanks to previous works done by T. Ohsawa, and then in identifying these weighted spaces.

Essential coordinate components of characteristic varieties

In this note we give an algebraic and topological interpretation of essential coordinate components of characteristic varieties and illustrate their importance with an example.

Intuitionistic logic and modality via topology

In the pioneering article and two papers, written jointly with McKinsey, Tarski developed the so-called algebraic and topological frameworks for the Intuitionistic Logic and the Lewis modal system. In this paper, we present an outline of modern (non-Lewis) systems with a topological tinge. We consider topological interpretation of basic systems GL and G RZ of the provability logic in terms of the Cantor derivative and the Hausdorff residue.

Reasoning About Space: The Modal Way

We investigate the topological interpretation of modal logic in modern terms, using a new notion of bisimulation. We look at modal logics with interesting topological content, presenting, among others, a new proof of McKinsey and Tarski's theorem on completeness of S4 with respect to the real line, and a completeness proof for the logic of finite unions of convex sets of reals. We conclude with a broader picture of extended modal languages of space, for which the main logical questions are still wide open.

N=2super Yang-Mills action as a Becchi-Rouet-Stora-Tyutin term, topological Yang-Mills action, and instantons

By constructing a nilpotent extended Becchi-Rouet-Stora-Tyutin (BRST) operator $\overline{s}$ that involves the $N=2$ global supersymmetry transformations of one chirality, we show that the standard $N=2$ off-shell super Yang-Mills action can be represented as an exact BRST term $\overline{s}\ensuremath{\Psi},$ if the gauge fermion $\ensuremath{\Psi}$ is allowed to depend on the inverse powers of supersymmetry (SUSY) ghosts. By using this nonanalytical structure of the gauge fermion (via inverse powers of supersymmetry ghosts), we give field redefinitions in terms of composite fields of SUSY ghosts and $N=2$ fields and we show that Witten's topological Yang-Mills (TYM) theory can be obtained from the ordinary Euclidean $N=2$ super Yang-Mills (SYM) theory directly by using such field redefinitions. In other words, TYM theory is obtained as a change of variables (without twisting). As a consequence it is found that physical and topological interpretations of $N=2$ SYM theory are intertwined together due to the requirement of analyticity of global SUSY ghosts. Moreover, after an instanton-inspired truncation of the model is used, we show that the given field redefinitions yield the Baulieu-Singer formulation of topological Yang-Mills theory.

Topography of molecular scalar fields. I. Algorithm and Poincaré–Hopf relation

A new algorithm for locating the critical points (CP’s) of a three-dimensional molecular scalar field is discussed. The algorithm is based on a ray search from the surface extrema of appropriately defined atom-centered spheres. The algorithm is tested for molecular electrostatic potentials and electron densities of a few test molecules such as tetrahedrane, cubane, anthracene, diborane, etc. Furthermore, the Poincaré–Hopf relationship is examined for the set of CP’s thus obtained. A topological interpretation of the Euler characteristic of a given isosurface is employed for a stronger regional check on the number of CP’s enclosed in the isosurface.

A Topological Look at the Quantum Hall Effect

The amazingly precise quantization of Hall conductance in a two-dimensional electron gas can be understood in terms of a topological invariant known as the Chern number.

Luttinger surgery along Lagrangian tori and non-isotopy for singular symplectic plane curves

We discuss the properties of a certain type of Dehn surgery along a Lagrangian torus in a symplectic 4-manifold, known as Luttinger's surgery, and use this construction to provide a purely topological interpretation of a non-isotopy result for symplectic plane curves with cusp and node singularities due to Moishezon [9].

The smooth Whitehead spectrum of a point at odd regular primes

Let p be an odd regular prime, and assume that the Lichtenbaum-Quillen conjecture holds for K(Z[1/p]) at p. Then the p-primary homotopy type of the smooth Whitehead spectrum Wh(*) is described. A suspended copy of the cokernel-of-J spectrum splits off, and the torsion homotopy of the remainder equals the torsion homotopy of the fiber of the restricted S^1-transfer map t: SigmaCP^infty--> S. The homotopy groups of Wh(*) are determined in a range of degrees, and the cohomology of Wh(*) is expressed as an A-module in all degrees, up to an extension. These results have geometric topological interpretations, in terms of spaces of concordances or diffeomorphisms of highly connected, high dimensional compact smooth manifolds.

Large-scale spiral structures in turbulent thermal convection between two vertical plates.

By means of a three-dimensional numerical simulation of the Boussinesq equations, it is observed that the turbulent flow induced by thermal convection between two differentially heated vertical plates can generate strongly spiral structures on large scales. In this paper, the flow patterns and the temporal evolution of such large-scale spiral structures are manifested. It is shown that the length scale of the spiral structures is comparable to the width of the convection layer and their lifetime is about two or three orders of magnitude longer than that of small-scale structures of the turbulence. To understand the physical mechanism of the emergence of such structures, a topological interpretation of generating the structures and the role of helicity in preserving the structures are investigated and discussed.

The Forest Metrics for Graph Vertices

We propose a new graph metric and study its properties. In contrast to the standard distance in connected graphs, it takes into account all paths between vertices. Formally, it is defined as d(i,j)=q_{ii}+q_{jj}-q_{ij}-q_{ji}, where q_{ij} is the (i,j)-entry of the {\em relative forest accessibility matrix} Q(\epsilon)=(I+\epsilon L)^{-1}, L is the Laplacian matrix of the (weighted) (multi)graph, and \epsilon is a positive parameter. By the matrix-forest theorem, the (i,j)-entry of the relative forest accessibility matrix of a graph provides the specific number of spanning rooted forests such that i and j belong to the same tree rooted at i. Extremely simple formulas express the modification of the proposed distance under the basic graph transformations. We give a topological interpretation of d(i,j) in terms of the probability of unsuccessful linking i and j in a model of random links. The properties of this metric are compared with those of some other graph metrics. An application of this metric is related to clustering procedures such as "centered partition." In another procedure, the relative forest accessibility and the corresponding distance serve to choose the centers of the clusters and to assign a cluster to each non-central vertex. The notion of cumulative weight of connections between two vertices is proposed. The reasoning involves a reciprocity principle for weighted multigraphs. Connections between the resistance distance and the forest distance are established.

Topological interpretation of fuzzy sets and intervals

Fuzzy set theory and interval mathematics were independently invented and developed in the mid 1960s as tools for a quantitative analysis of approximations to mathematically exact values, which may not be observable, representable or computable. We show that both approaches are covered by a general topological theory developed almost two decades earlier. We start with the concepts of naive fuzzy set and interval theory and discuss the underlying common features. Then we represent the theory of topological filter bases, their homomorphisms and the rounding of filter bases. Fuzzy set theory and interval mathematics can be described in terms of this theory. In addition, some of the concepts of classical fuzzy and interval theory are extended.

The Topological Interpretation of the Core Group of a Surface inS4

AbstractWe give a topological interpretation of the core group invariant of a surface embedded inS4[F-R], [Ro]. We show that the group is isomorphic to the free product of the fundamental group of the double branch cover ofS4with the surface as a branched set, and the infinite cyclic group. We present a generalization for unoriented surfaces, for other cyclic branched covers, and other codimension two embeddings of manifolds in spheres.

On the reductive Borel-Serre compactification: L p -cohomology of arithmetic groups (for large p )

The L 2 -cohomology of an arithmetic quotient of an Hermitian symmetric space (i.e., of a locally symmetric variety) is known to have the topological interpretation as the intersection homology of its Baily-Borel Satake compactification. In this article, we observe that even without the Hermitian hypothesis, the L p -cohomology of an arithmetic quotient, for p finite and sufficiently large, is isomorphic to the ordinary cohomology of its reductive Borel-Serre compactification. We use this to generalize a theorem of Mumford concerning homogeneous vector bundles, their invariant Chern forms and the canonical extensions of the bundles; here, though, we are referring to canonical extensions to the reductive Borel-Serre compactification of any arithmetic quotient. To achieve that, we give a systematic discussion of vector bundles and Chern classes on stratified spaces.

Local Cohomology of Stanley–Reisner Rings with Supports in General Monomial Ideals

We study the local cohomology modules HiIΣ(k[Δ]) of the Stanley–Reisner ring k[Δ] of a simplicial complex Δ with support in the ideal IΣ⊂k[Δ] corresponding to a subcomplex Σ⊂Δ. We give a combinatorial topological formula for the multigraded Hilbert series, and in the case where the ambient complex is Gorenstein, compare this with a second combinatorial formula that generalizes results of Mustata and Terai. The agreement between these two formulae is seen to be a disguised form of Alexander duality. Other results include a comparison of the local cohomology with certain Ext modules, results about when it is concentrated in a single homological degree, and combinatorial topological interpretations of some vanishing theorems.

Topological interpretation of subharmonic mode locking in coupled oscillators with inertia

A topological argument is constructed and applied to explain subharmonic mode locking in a system of coupled oscillators with inertia. Via a series of transformations, the system is shown to be described by a classical $\mathrm{XY}$ model with periodic bond angles, which is in turn mapped onto a tight-binding particle in a periodic gauge field. It is then revealed that subharmonic quantization of the average phase velocity follows as a manifestation of topological invariance. Ubiquity of multistability and associated hysteresis are also pointed out.

On the topological interpretation of gravitational anomalies

We consider the mixed gravitational Yang–Mills anomaly as the coupling between the K-theory and K-homology of a C ∗ -algebra crossed product. The index theorem of Connes–Moscovici allows to compute the Chern character of the K-cycle by local formulae involving connections and curvatures. It gives a topological interpretation to the anomaly, in the sense of noncommutative algebras.

Robert Legendre and Henri Werlé: Toward the Elucidation of Three-Dimensional Separation

▪ Abstract The description and the physical understanding of three-dimensional separated flows are challenging problems mainly because of the use of inappropriate terms linked to the consideration of two-dimensional flows. This fact was realized in the early 1950s by Robert Legendre, who introduced the basic concepts of the Critical Point Theory to provide a rational definition of separation in three-dimensional flows. In parallel, demonstrative experiments were executed by Henri Werlé in the Onera water tunnel laboratory. From the close cooperation between these two scientists resulted the construction of a powerful theoretical tool allowing the elucidation of the structure of largely separated three-dimensional fields. The importance of their contribution to fluid mechanics is illustrated here by the consideration of basic configurations: flow past wings or elongated bodies, in front of obstacles, and behind a base. For each case, the flow organization is discussed by considering representative water tunnel visualizations and corresponding topological interpretations.