Poincare Conjecture Research Articles

Neutral Poincare problem in Lp-Sobolev spaces: Regularity and Fredholmness

Neutral Poincare problem in Lp-Sobolev spaces: Regularity and Fredholmness

The 3-manifold recognition problem

We introduce a natural Relative Simplicial Approximation Property for maps from a 2-cell to a generalized 3-manifold and prove that, modulo the Poincare Conjecture, 3-manifolds are precisely the generalized 3-manifolds satisfying this approximation property. The central technical result establishes that every generalized 3-manifold with this Relative Simplicial Approximation Property is the cell-like image of some generalized 3-manifold having just a 0-dimensional set of nonmanifold singularities.

Approximate General Solution Of The Two-Dimensional Duffing Dynamical System

This paper presents the approximate general solution of the triple well, double oscillator non-linear dynamical system. This system is non-integrable and the approximate general solution is calculated by application of the Last Geometric Theorem of Poincare (Birkhoff, 1913, 1925). The original problem, known as the Duffing one, is a 1 degree of freedom system that, besides the conservative force component, includes dumping and external forcing terms (see details in the web site: http://www.uncwil.edu/people/hermanr/chaos/ted/chaos.html). The problem considered here is a 2 degree of freedom, autonomous and conservative one, without dumping, and of axisymmetric potential. The space of permissible motions is scanned for identification of all solutions re-entering after from one to nine oscillations and the precise families of periodic solutions are computed, including their stability parameter, covering all cases with periods T corresponding to 4osc/T. Seven sub-domains of the space of solutions were investigated in detail by zooming, an operation that proved the possibility to advance the accuracy of the approximate general solution to the level permitted by the integration routine. The approximation of the general solution, although impressive, provides clear evidence of the complexity of the problem and the need to proceed to larger period families. Nevertheless, it allows prediction of the areas where chaos and order regions in the Poincare surfaces of section are to be expected. Examples of such surfaces of sections, as well as of types of closed solutions, are given. Two peculiar points of the space of solutions were identified as crossing, or source points from which infinite families of periodic solutions emanate. The morphology and stability of solutions of the problem are studied and discussed.

Poincare Theorem for Difference Equations

Poincare Theorem for Difference Equations

Basis of Local Approach in Classical Statistical Mechanics

An ensemble of classical subsystems interacting with surrounding particles has been considered. In general case, a phase volume of the subsystems ensemble was shown to be a function of time. The evolutional equations of the ensemble are obtained as well as the simplest solution of these equations representing the quasi-local distribution with the temperature pattern being assigned. Unlike the Gibbs's distribution, the energy of interaction with surrounding particles appears in the distribution function, which make possible both evolution in the equilibrium case and fluctuations in the non-equilibrium one. The expression for local entropy is obtained. The derivation of hydrodynamic equations from Boltzmann equation has been analyzed. The hydrodynamic equations obtained from Boltzmann equation were shown to be equations for ideal liquid. Reasons for stochastic description in deterministic Hamilton's systems, conditions of applicability of Poincare's recurrence theorem as well as the problem of irreversibility have been considered.

Recent progress on the Poincaré conjecture and the classification of 3-manifolds

If M is a closed 3-manifold with trivial fundamental group, then is M diffeomorphic to S? The Poincare Conjecture is that the answer to this question is “Yes.” Developing tools to attack this problem formed the basis for much of the work in 3-dimensional topology over the last century, including for example, the proof of Dehn’s lemma and the loop theorem and the study of surgery on knots and links. In the 1980’s Thurston developed another approach to 3-manifolds, see [24] and [4]. He considered 3-manifolds with riemannian metrics of constant negative curvature −1. These manifolds, which are locally isometric to hyperbolic 3-space, are called hyperbolic manifolds. There are fairly obvious obstructions showing that not every 3-manifold can admit such a metric. Thurston formulated a general conjecture that roughly says that the obvious obstructions are the only ones; should they vanish for a particular 3-manifold then that manifold admits such a metric. His proof of various important special cases of this conjecture led him to formulate a more general conjecture about the existence of locally homogeneous metrics, hyperbolic or otherwise, for all manifolds; this is called Thurston’s Geometrization Conjecture for 3-manifolds. The statement of this conjecture is somewhat complicated, so it is deferred until Section 2. An important point is that Thurston’s Geometrization Conjecture includes the Poincare Conjecture as a very special case. In addition, Thurston’s conjecture has two advantages over the Poincare Conjecture: • It applies to all closed oriented 3-manifolds. • It posits a close relationship between topology and geometry in dimension three.

Normalizability, Synchronicity, and Relative Exactness for Vector Fields in C2

In this paper, we study the necessary and su.cient condition under which an orbitally normalizable vector field of saddle or saddle-node type in C2 is analytically conjugate to its formal normal form (i.e., normalizable) by a transformation fixing the leaves of the foliation locally. First, we express this condition in terms of the relative exactness of a certain 1-form derived from comparing the time-form of the vector field with the time-form of the normal form. Then we show that this condition is equivalent to a synchronicity condition: the vanishing of the integral of this 1-form along certain asymptotic cycles de.ned by the vector field. This can be seen as a generalization of the classical Poincare theorem saying that a center is isochronous (i.e., synchronous to the linear center) if and only if it is linearizable. The results, in fact, allow us in many cases to compare any two vector fields which differ by a multiplicative factor. In these cases we show that the two vector fields are analytically conjugate by a transformation fixing the leaves of the foliation locally if and only if their time-forms are synchronous.

Boundary-value problems for linear PDEs with variable coefficients

A new method is introduced for studying boundary-value problems for a class of linear partial differential equations (PDEs) with variable coefficients. This method is based on ideas recently introduced by the author for the study of boundary-value problems for PDEs with constant coefficients. As illustrative examples the following boundary-value problems are solved: a Dirichlet and a Neumann problem on the half line for the time-dependent Schrodinger equation with a space-dependent potential; and a Poincare problem on the quarter plane for a variable coefficient generalization of the Laplace equation.

Inverse approach to study the planar polynomial vector field with algebraic solutions

We construct a polynomial differential system that admits a given set of invariant algebraic curves. For such a system we solve the Darboux problem (the existence of the Darboux first integral), the Poincare problem (the existence of an upper bound for the degree of invariant algebraic curve) and study Hilbert's 16th problem for algebraic limit cycles (the existence of an upper bound for the number of algebraic limit cycles).

Current events sessions

Will the Poincare Conjecture be solved this week? What is the Geometric Langlands Conjecture about? What could one do to analyze a stream of data flowing by too fast to capture? Asking such questions is part of being a mathematician, provoked by our sense that underneath the vastness of mathematics is a fundamental unity that allows us to look into many different corners—though we couldn’t possibly work in all of them—as well as by simple curiosity and common-room gossip. The idea behind the Current Events Session at the Joint Mathematics Meetings, begun last year, was to create an event where the speakers were expected not to report on their own work but to survey some of the most interesting current developments in mathematics, pure and applied. The wonderful tradition of the Bourbaki Seminar was an inspiration, but I hoped for an even wider range of subjects. A broadly constituted advisory committee was charged to select topics and excellent speakers. The programs of the sessions in January 2003 and 2004 will give an idea of the sessions.

Rate of convergence of solutions of rational difference equation of second order

We investigate the rate of convergence of solutions of some special cases of the equation , with positive parameters and nonnegative initial conditions. We give precise results about the rate of convergence of the solutions that converge to the equilibrium or period-two solution by using Poincare's theorem and an improvement of Perron's theorem.

A quasi-periodic Poincar�'s theorem

We study the persistence of invariant tori on resonant surfaces of a nearly integrable Hamiltonian system under the usual Kolmogorov non-degenerate condition. By introducing a quasi-linear iterative scheme to deal with small divisors, we generalize the Poincare theorem on the maximal resonance case (i.e., the periodic case) to the general resonance case (i.e., the quasi-periodic case) by showing the persistence of majority of invariant tori associated to non-degenerate relative equilibria on any resonant surface.

The importance of the P versus NP question

The P versus NP problem is to determine whether every language accepted by some nondeterministic Turing machine in polynomial time is also accepted by some deterministic Turing machine in polynomial time. Unquestionably this problem has caught the interest of the mathematical community. For example, it is the first of seven million-dollar “Millennium Prize Problems” listed by the Clay Mathematics Institute [www.claymath.org]. The Riemann Hypothesis and Poincare Conjecture, both mathematical classics, are farther down the list. On the other hand, Fields Medalist Steve Smale lists P versus NP as problem number three, after Riemann and Poincare, in “Mathematical Problems for the Next Century” [Smale 1998]. But P versus NP is also a problem of central interest in computer science. It was posed thirty years ago [Cook 1971; Levin 1973] as a problem concerned with the fundamental limits of feasible computation. Although this question is front and center in complexity theory, NP-completeness proofs have become pervasive in many other areas of computer science, including artificial intelligence, databases, programming languages, and computer networks (see Garey and Johnson [1979] for 300 early examples). If the question is resolved, what would be the consequences? Consider first a proof of P=NP. It is possible that the proof is nonconstructive, in the sense that it does not yield an algorithm for any NP-complete problem. Or it might give an impractical algorithm, for example, running in time n100. In either of these cases, the proof would probably have few practical consequences other than to disappoint complexity theorists. However, experience has shown that when natural problems are proved to be in P, a feasible algorithm can be found. There are potential counterexamples to this assertion; most famously, the deep results of Robertson and Seymour [1993–1995], who prove that every minor closed family of graphs can be recognized in time O(n3), but their algorithm has such huge constants it is not practical. But practical algorithms are known for some specific minor-closed families (such as planar graphs), and possibly could be found for other examples if sufficient effort is expended. If P=NP is proved by exhibiting a truly feasible algorithm for an NP-complete problem such as SATISFIABILITY (deciding whether a collection of propositional clauses has a satisfying assignment), the practical consequences would be stunning. First, most of the hundreds of problems shown to be NP-complete can be efficiently reduced to SATISFIABILITY, so many of the optimization problems important to industry could be solved. Second, mathematics would be transformed, because computers could find a formal proof of any theorem which has a proof of reasonable length. This is because formal proofs (say in Zermelo–Fraenkel set theory) are

Kaun banega crorepati —A million dollars for a mathematician

I will now embark on explaining as best as I can to the non-mathematician what the Poincare conjecture is all about.

Stephen Smale: The Mathematician Who Broke the Dimension Barrier

One room schoolhouse Marxism and mathematics at Ann Arbor Early mathematical audacity On the beaches of Rio Berkeley to Columbia and back to Berkeley The Lone Ranger of the antiwar movement The summer of 1966 Smale versus the National Science Foundation The aesthetic side: Minerals and photography Adventure and physical risks Other people Smale the mathematician Mathematical appendix A: Smale's thesis Mathematical appendix B: Everting the sphere Mathematical appendix C: Chaos and the horseshoe Mathematical appendix D: The higher dimensional Poincare conjecture Notes Index.

Caratheodory and the Foundations of Thermodynamics and Statistical Physics

Constantin Caratheodory offered the first systematic and contradiction free formulation of thermodynamics on the basis of his mathematical work on Pfaff forms. Moreover, his work on measure theory provided the basis for later improved formulations of thermodynamics and physics of continua where extensive variables are measures and intensive variables are densities. Caratheodory was the first to see that measure theory and not topology is the natural tool to understand the difficulties (ergodicity, approach to equilibrium, irreversibility) in the Foundations of Statistical Physics. He gave a measure-theoretic proof of Poincare's recurrence theorem in 1919. This work paved the way for Birkhoff to identify later ergodicity as metric transitivity and for Koopman and von Neumann to introduce spectral analysis of dynamical systems in Hilbert spaces. Mixing provided an explanation of the approach to equilibrium but not of irreversibility. The recent extension of spectral theory of dynamical systems to locally convex spaces, achieved by the Brussels–Austin groups, gives new nontrivial time asymmetric spectral decompositions for unstable and/or non-integrable systems. In this way irreversibility is resolved in a natural way.

An inverse to the Poincaré conjecture

The notion of Poincare manifolds is introduced, which are roughly 1-connected manifolds which are determined by tem heir homology. The Poincare conjecture says that spheres are Poincare manifolds. The determination of all Poincare manifolds is a sort of inverse to the Poincare conjecture. Besides some information in low dimensions we give some results on generalized Poincare manifolds.

On integrable cases of the poincare problem

The equations of motion for the classical Poincare problem of the rotational motion of a rigid body with an ellipsoidal cavity containing liquid in canonical Andoyer variables have been obtain. Thr...

Wick Power Series Converging to Nonlocal Fields

We consider the infinite series in Wick powers of a generalized free field that are convergent under smoothing with analytic test functions and realize a nonlocal extension of the Borchers equivalence classes. The nonlocal fields to which the Wick power series converge are proved to be asymptotically commuting. This property serves as a natural generalization of the relative locality of the Wick polynomials. The proposed proof is based on exploiting the analytic properties of the vacuum expectation values in the x space and applying the Cauchy–Poincare theorem.

Indices of Toeplitz tuples on pseudoregular domains

This paper proves an index theorem of Toeplitz tuples on pseudoregular domains in Cn. Geometrically, the index of Toeplitz tupleT n is (-1) n time wrapping number of φ n around the origin. As one of the applications of the index theorem, we completely characterize the automorphism groups of Toeplitz algebras on Poincare domain. As another application, it is shown thatC*(Ω)≅C*(Bn) for every Poincare domain Ω inC n (n≠2). It is also noticed thatC*(Ω)≅C*(B 2) if and only if the Poincare conjecture is true for ∂Ω.