Curvature and distance function from a manifold

This paper is concerned with the relations between the differential invariants of a smooth manifold embedded in the Euclidean space and the square of the distance function from the manifold. In particular, we are interested in curvature invariants like the mean curvature vector and the second fundamental form. We find that these invariants can be computed in a very simple way using the third order derivatives of the squared distance function. Moreover, we study a general class of functionals depending on the derivatives up to a given order γ of the squared distance function and we find an algorithm for the computation of the Euler equation. Our class of functionals includes as particular cases the well-known area functional (γ = 2), the integral of the square of the quadratic norm of the second fundamental form (γ = 3), and the Willmore functional.     

Quasiconformal groups, Patterson-Sullivan theory, and local analysis of limit sets

We extend the part of Patterson-Sullivan theory to discrete quasiconformal groups that relates the exponent of convergence of the Poincaré series to the Hausdorff dimension of the limit set. In doing so we define new bi-Lipschitz invariants that localize both the exponent of convergence and the Hausdorff dimension. We find these invariants help to expose and explain the discrepancy between the conformal and quasiconformal setting of Patterson-Sullivan theory.

     

Contact-equivalence problem for linear hyperbolic equations

We consider the local equivalence problem for the class of linear second-order hyperbolic equations in two independent variables under an action of the pseudo-group of contact transformations. é. Cartan’s method is used for finding the Maurer-Cartan forms for symmetry groups of equations from the class and computing structure equations and complete sets of differential invariants for these groups. The solution of the equivalence problem is formulated in terms of these differential invariants. __________ Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 25, pp. 119–142, 2005.     

<Emphasis Type="Italic">r</Emphasis>-Tuple almost product structures

A generalization of an almost product structure and an almost complex structure on smooth manifolds is constructed. The set of tensor differential invariants of type (2, 1) and the set of differential 2-forms for such structures are constructed. We show how these tensor invariants can be used to solve the classification problem for Monge–Ampère equations and Jacobi equations.     

Monodromy Invariants – From Symplectic to Smooth Manifolds

Recently, together with Auroux and Donaldson, we have introduced some new invariants of four-dimensional symplectic manifolds. Building on the Moishezon–Teicher braid factorization techniques, we show how to compute fundamental groups of compliments to a ramification curve of generic projection. We also show that these fundamental groups are only homology invariants and outline the computations in some examples.Demonstrating the ubiquity of algebra, we go further and, using Braid factorization, we compute invariants of a derived category of representations of the quiver associated with the Fukaya–Seidel category of the vanishing cycles of a Lefschetz pencil and a structure of a symplectic four-dimensional manifold. This idea is suggested by the homological mirror symmetry conjecture of Kontsevich. We do not use it in our computations, although everything is explicit. We outline a procedure for finding homeomorphic, nonsymplectomorphic, four-dimensional symplectic manifolds with the same Saiberg–Witten invariants. This procedure defines invariants in the smooth category as well.     

Configurations of flags and representations of surface groups in complex hyperbolic geometry

In this work, we describe a set of coordinates on the PU(2,1)-representation variety of the fundamental group of an oriented punctured surface Σ with negative Euler characteristic. The main technical tool we use is a set of geometric invariants of a triple of flags in the complex hyperbolic plane H²_{\mathbb C}{\bf H^2_{\mathbb {C}}} . We establish a bijection between a set of decorations of an ideal triangulation of Σ and a subset of the PU(2,1)-representation variety of π ₁(Σ).     

The geodesic flow of a nonpositively curved graph manifold

We consider discrete cocompact isometric actions where X is a locally compact Hadamard space (following [B] we will refer to CAT(0) spaces — complete, simply connected length spaces with nonpositive curvature in the sense of Alexandrov — as Hadamard spaces) and G belongs to a class of groups (“admissible groups”) which includes fundamental groups of 3-dimensional graph manifolds. We identify invariants (“geometric data”) of the action which determine, and are determined by, the equivariant homeomorphism type of the action of G on the ideal boundary of X. Moreover, if are two actions with the same geometric data and is a G-equivariant quasi-isometry, then for every geodesic ray there is a geodesic ray (unique up to equivalence) so that . This work was inspired by (and answers) a question of Gromov in [Gr3, p. 136]. Submitted: May 2001.     

A hierarchy of generalized invariants for linear partial differential operators

We study invariants of linear partial differential operators in two variables under gauge transformations. Using the Beals-Kartashova factorization, we construct a hierarchy of generalized invariants for operators of an arbitrary order. We study the properties of these invariants and give some examples. We also show that the classic Laplace invariants correspond to some particular cases of generalized invariants. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 3, pp. 470–478, June, 2006.     

Finite-order invariants of ornaments

Anornament is a collection of oriented closed curves in a plane, no three of which intersect at the same point. We consider homotopy invariants of ornaments. Thefinite-order invariants of ornaments are a natural analog of the Vassiliev invariants of links. The calculation of them is based on the homological study of the corresponding space of singular objects. We perform the “local” part of these calculations and a part of the “global” one, which allows us to estimate the dimensions of the spaces of invariants of any order. We also construct explicity two large series of such invariants and establish some new algebraic structures in the space of invariants. Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 35, Algebraicheskaya Geometriya-6, 1996.     

The normalized cyclomatic quotient associated with presentations of finitely generated groups

Given a presentation of ann-generated group, we define the normalized cyclomatic quotient (NCQ) of it, which gives a number between 1−n and 1. It is computed through an investigation of the asymptotic behavior of a kind of an “average rank”, or more precisely the quotient of the rank of the fundamental group of a finite subgraph of the corresponding Cayley graph by the size of the subgraph. In many ways (but not always) the NCQ behaves similarly to the behavior of the spectral radius of a symmetric random walk on the graph. In particular, it characterizes amenable groups. For some types of groups, like finite, amenable or free groups, its value equals that of the Euler characteristic of the group. We give bounds for the value of the NCQ for factor groups and subgroups, and formulas for its value on direct and free products. Some other asymptotic invariants are also discussed.     

S.A.G.B.I. bases for rings of formal modular seminvariants

We use the theory of S.A.G.B.I. bases to construct a generating set for the ring of invariants for the four and five dimensional indecomposable modular representations of a cyclic group of prime order. We observe that for the four dimensional representation the ring of invariants is generated in degrees less than or equal to 2p–3, and for the five dimensional representation the ring of invariants is generated in degrees less than or equal to 2p–2. Received: January 22, 1997     

Nonlinear transversely isotropic elastic solids: an alternative representation

A strain energy function which depends on five independent variablesthat have immediate physical interpretation is proposed forfinite strain deformations of transversely isotropic elasticsolids. Three of the five variables (invariants) are the principalstretch ratios and the other two are squares of the dot productbetween the preferred direction and two principal directionsof the right stretch tensor. The set of these five invariantsis a minimal integrity basis. A strain energy function, expressedin terms of these invariants, has a symmetry property similarto that of an isotropic elastic solid written in terms of principalstretches. Ground state and stress–strain relations aregiven. The formulation is applied to several types of deformations,and in these applications, a mathematical simplicity is highlighted.The proposed model is attractive if principal axes techniquesare used in solving boundary-value problems. Experimental advantageis demonstrated by showing that a simple triaxial test can varya single invariant while keeping the remaining invariants fixed.A specific form of strain energy function can be easily obtainedfrom the general form via a triaxial test. Using series expansionsand symmetry, the proposed general strain energy function isrefined to some particular forms. Since the principal stretchesare the invariants of the strain energy function, the Valanis–Landelform can be easily incorporated into the constitutive equation.The sensitivity of response functions to Cauchy stress datais discussed for both isotropic and transversely isotropic materials.Explicit expressions for the weighted Cauchy response functionsare easily obtained since the response function basis is almostmutually orthogonal.     

Chern–Simons Invariants of Torus Links

We compute the vacuum expectation values of torus knot operators in Chern–Simons theory, and we obtain explicit formulae for all classical gauge groups and for arbitrary representations. We reproduce a known formula for the HOMFLY invariants of torus knots and links, and we obtain an analogous formula for Kauffman invariants. We also derive a formula for cable knots. We use our results to test a recently proposed conjecture that relates HOMFLY and Kauffman invariants.     

Hyperbolic balance laws: Riemann invariants and the generalized Riemann problem

The Generalized Riemann Problem (GRP) for a nonlinear hyperbolic system of m balance laws (or alternatively “quasi-conservative” laws) in one space dimension is now well-known and can be formulated as follows: Given initial-data which are analytic on two sides of a discontinuity, determine the time evolution of the solution at the discontinuity. In particular, the GRP numerical scheme (second-order high resolution) is based on an analytical evaluation of the first time derivative. It turns out that this derivative depends only on the first-order spatial derivatives, hence the initial data can be taken as piecewise linear. The analytical solution is readily obtained for a single equation (m = 1) and, more generally, if the system is endowed with a complete (coordinate) set of Riemann invariants. In this case it can be “diagonalized” and reduced to the scalar case. However, most systems with m > 2 do not admit such a set of Riemann invariants. This paper introduces a generalization of this concept: weakly coupled systems (WCS). Such systems have only “partial set” of Riemann invariants, but these sets are weakly coupled in a way which enables a “diagonalized” treatment of the GRP. An important example of a WCS is the Euler system of compressible, nonisentropic fluid flow (m = 3). The solution of the GRP discussed here is based on a careful analysis of rarefaction waves. A “propagation of singularities” argument is applied to appropriate Riemann invariants across the rarefaction fan. It serves to “rotate” initial spatial slopes into “time derivative”. In particular, the case of a “sonic point” is incorporated easily into the general treatment. A GRP scheme based on this solution is derived, and several numerical examples are presented. Special attention is given to the “acoustic approximation” of the analytical solution. It can be viewed as a proper linearization (different from the approach of Roe) of the nonlinear system. The resulting numerical scheme is the simplest (second-order, high-resolution) generalization of the Godunov scheme.     

Infinitely many universally tight torsion free contact structures with vanishing Ozsváth–Szabó contact invariants

Ozsváth–Szabó contact invariants are a powerful way to prove tightness of contact structures but they are known to vanish in the presence of Giroux torsion. In this paper we construct, on infinitely many manifolds, infinitely many isotopy classes of universally tight torsion free contact structures whose Ozsváth–Szabó invariant vanishes. We also discuss the relation between these invariants and an invariant on T³ and construct other examples of new phenomena in Heegaard–Floer theory. Along the way, we prove two conjectures of K. Honda, W. Kazez and G. Matić about their contact topological quantum field theory. Almost all the proofs in this paper rely on their gluing theorem for sutured contact invariants.     

Parametric Bäcklund transformations I: Phenomenology

We begin an exploration of parametric Bäcklund transformations for hyperbolic Monge-Ampère systems. (The appearance of an arbitrary parameter in the transformation is a feature of several well-known completely integrable PDEs.) We compute invariants for such transformations and explore the behavior of four examples, two of which are new, in terms of their invariants, symmetries, and conservation laws. We prove some preliminary results and indicate directions for further research.

     

Approximating L 2-Invariants and Homology Growth

In this paper we consider the asymptotic behavior of invariants such as Betti numbers, minimal numbers of generators of singular homology, the order of the torsion subgroup of singular homology, and torsion invariants. We will show that all these vanish in the limit if the CW-complex under consideration fibers in a specific way. In particular we will show that all these vanish in the limit if one considers an aspherical closed manifold which admits a non-trivial S ¹-action or whose fundamental group contains an infinite normal elementary amenable subgroup. By considering classifying spaces we also get results for groups.     

Topological invariance of intersection homology without sheaves

In (1) Goresky and MacPherson defined intersection homology groups for triangulable pseudomanifolds and showed they were PL invariants. Then in [2] they generalized these groups to any pseudomanifold and showed they were topological invariants. These groups have generated a great deal of interest. However, [2] is difficult for many mathematicians (including this author) because it requires a familiarity with a great deal of hefty sheaf-theoretic machinery. This is too bad, because the basic ideas behind intersection homology (elucidated in [1]) are very geometric.In this paper we give a sheafless definition of intersection homology groups for an arbitrary stratified set and we give an elementary sheafless proof that they are topological invariants, i.e. independent of the stratification.In doing so, we find some new perversities whose intersection homology groups are topological invariants. Unfortunately, these new perverse intersection homology classes do not seem to intersect with anything (which is probably why they were ignored by Goresky and MacPherson). But in any case these groups are invariants of singular spaces which might be of some interest.     

Relation of orbital integrals on SO(5) and PGL(2)

We relate the “Fourier” orbital integrals of corresponding spherical functions on thep-adic groups SO(5) and PGL(2). The correspondence is defined by a “lifting” of representations of these groups. This is a local “fundamental lemma” needed to compare the geometric sides of the global Fourier summation formulae (or relative trace formulae) on these two groups. This comparison leads to conclusions about a well known lifting of representations from PGL(2) to PGSp(4). This lifting produces counter examples to the Ramanujan conjecture.     

On fundamental groups related to the Hirzebruch surface <Emphasis Type="Italic">F</Emphasis><Subscript>1</Subscript>

Given a projective surface and a generic projection to the plane, the braid monodromy factorization (and thus, the braid monodromy type) of the complement of its branch curve is one of the most important topological invariants, stable on deformations. From this factorization, one can compute the fundamental group of the complement of the branch curve, either in ℂ² or in ℂℙ². In this article, we show that these groups, for the Hirzebruch surface F _1,(a,b), are almost-solvable. That is, they are an extension of a solvable group, which strengthen the conjecture on degeneratable surfaces. This work was supported by the Emmy Noether Institute Fellowship (by the Minerva Foundation of Germany) and Israel Science Foundation (Grant No. 8008/02-3)