a tqft from quantum teichmuller theory by using quantum teichmuller theory we construct a one parameter family of tqfts on the categroid of admissible leveled shaped 3 manifolds

cusp size bounds from singular surfaces in hyperbolic 3 manifolds singular maps of surfaces into a hyperbolic 3 manifold are utilized to find upper bounds on meridian length l curve length and maximal cusp volume for the manifold this allows a proof of the fact that there exist hyperbolic knots with arbitrarily small cusp density and that every closed orientable 3 manifold contains a knot whose complement is hyperbolic with maximal cusp volume less than or equal to 9 we also find particular upper bounds on meridian length l curve length and maximal cusp volume for hyperbolic knots in s 3 depending on crossing number particular improved bounds are obtained for alternating knots

semi invariants and their use for stability analysis of planar systems semi invariants and relative characteristic functions extend to nonlinear systems the concept of eigenvector eigenvalue pair for linear systems and are therefore very useful to depict the behaviour of the system in this article semi invariants are used to construct explicitly lyapunov functions useful for studying the stability of the origin for continuous time systems moreover using well known tools from differential geometry such as orbital symmetries it is shown how semi invariants can be found for several classes of systems important connections with centre manifold theory are pointed out by using the proposed general techniques a new proof of a result claimed to bendixson bendixson i 1901 sur les courbes definies par des equations differentielles acta mathematics 24 1 88 is given

the hyperelliptic limit cycles of the lienard systems abstract for lienard systems x y y f m x y g n x with f m and g n real polynomials of degree m and n respectively in h zoladek algebraic invariant curves for the lienard equation trans amer math soc 350 1998 1681 1701 the author showed that if m 3 and m 1 n 2 m there always exist lienard systems which have a hyperelliptic limit cycle llibre and zhang j llibre xiang zhang on the algebraic limit cycles of lienard systems nonlinearity 21 2008 2011 2022 proved that the lienard systems with m 3 and n 5 have no hyperelliptic limit cycles and that there exist lienard systems with m 4 and 5 n 8 which do have hyperelliptic limit cycles so it is still an open problem to characterize the lienard systems which have an algebraic limit cycle in cases m 4 and m 1 n 2 m in this paper we will prove that there exist lienard systems with m 5 and m 1 n 2 m which have hyperelliptic limit cycles

cusp geometry of fibered 3 manifolds let f be a surface and suppose that varphi colon f to f is a pseudo anosov homeomorphism fixing a puncture p of f the mapping torus m m varphi is hyperbolic and contains a maximal cusp c about the puncture p we show that the area and height of the cusp torus partial c is equal to the stable translation distance of varphi acting on the arc complex scr a f p up to an explicitly bounded multiplicative error our proof relies on elementary facts about the hyperbolic geometry of pleated surfaces in particular the proof of this theorem does not use any deep results from teichm uller theory kleinian group theory or the coarse geometry of scr a f p a similar result holds for quasi fuchsian manifolds n cong f times bbb r in that setting we find a combinatorial estimate for the area and height of the cusp annulus in the convex core of n up to explicitly bounded multiplicative and additive error as an application we show that covers of punctured surfaces induce quasi isometric embeddings of arc complexes

stable maps and branched shadows of 3 manifolds in the early 1990s turaev introduced the notion of shadows as a combinatorial presentation of both 4 and 3 manifolds later costantino thurston revealed a strong relation between the stein factorizations of stable maps of 3 manifolds into the real plane and the shadows of the manifolds in fact a shadow can be seen locally as the stein factorization of a stable map in this paper we define the notion of stable map complexity for a compact orientable 3 manifold bounded by possibly empty tori counting with some weights the minimal number of singular fibers of codimension 2 of stable maps into the real plane and prove that this number equals the minimal number of vertices of its branched shadows in consequence we give a complete characterization of hyperbolic links in the 3 sphere whose exteriors have stable map complexity 1 in terms of dehn surgeries and also give an observation concerning the coincidence of the stable map complexity and shadow complexity using estimations of hyperbolic volumes

an elementary proof for a compact imbedding result in generalized electromagnetic theory propriete de plongement compact et certaines de ses consequences on etudie la validite du plongement suivant r 9 g e q 1 d q g l 2 q loc g

boundary layers on sobolev besov spaces and poisson s equation for the laplacian in lipschitz domains we study inhomogeneous boundary value problems for the laplacian in arbitrary lipschitz domains with data in sobolev besov spaces as such this is a natural continuation of work in jerison and kenig j funct anal 1995 16 219 where the inhomogeneous dirichlet problem is treated via harmonic measure techniques the novelty of our approach resides in the systematic use of boundary integral methods in this regard the key results are establishing the invertibility of the classical layer potential operators on scales of sobolev besov spaces on lipschitz boundaries for optimal ranges of indices applications tolp based helmholtz type decompositions of vector fields in lipschitz domains are also presented

a note on relaxation oscillators with exact limit cycles abstract in this note we give a family of planar polynomial differential systems with a prescribed hyperbolic limit cycle this family constitutes a corrected and wider version of an example given in the work m a abdelkader relaxation oscillators with exact limit cycles j math anal appl 218 1998 308 312 the result given in this note may be used to construct models of lienard differential equations exhibiting a desired limit cycle

on the analyticity of the semigroup generated by the stokes operator with neumann type boundary conditions on lipschitz subdomains of riemannian manifolds we study the analyticity of the semigroup generated by the stokes operator equipped with neumann type boundary conditions on l p spaces in lipschitz domains our strategy is to regularize this operator by considering the hodge laplacian which has the additional property that it commutes with the leray projection

darboux integrability and the inverse integrating factor abstract we mainly study polynomial differential systems of the form dx dt p x y dy dt q x y where p and q are complex polynomials in the dependent complex variables x and y and the independent variable t is either real or complex we assume that the polynomials p and q are relatively prime and that the differential system has a darboux first integral of the form h f 1 1 f p p exp h 1 g 1 n 1 1 exp h q g q n q q where the polynomials f i and g j are irreducible the polynomials g j and h j are coprime and the i and j are complex numbers when i 1 p and j 1 q prelle and singer proved that these systems have a rational integrating factor we improve this result as follows assume that h is a rational function which is not polynomial following to poincare we define the critical remarkable values of h then we prove that the system has a polynomial inverse integrating factor if and only if h has at most two critical remarkable values under some assumptions over the darboux first integral h we show first that the system has a polynomial inverse integrating factor and secondly that if the degree of the system is m the homogeneous part of highest degree of h is a multi valued function and the functions exp h j g j are exponential factors for j 1 q then the system has a polynomial inverse integrating factor of degree m 1 we also present versions of these results for real polynomial differential systems finally we apply these results to real polynomial differential systems having a darboux first integral and limit cycles or foci

potential theory on lipschitz domains in riemannian manifolds sobolev besov space results and the poisson problem abstract we continue a program to develop layer potential techniques for pde on lipschitz domains in riemannian manifolds building on l p and hardy space estimates established in previous papers here we establish sobolev and besov space estimates on solutions to the dirichlet and neumann problems for the laplace operator plus a potential on a lipschitz domain in a riemannian manifold with a metric tensor smooth of class c 1 for some 0 we treat the inhomogeneous problem and extend it to the setting of manifolds results obtained for the constant coefficient laplace operator on a lipschitz domain in euclidean space with the dirichlet boundary condition by d jerison and c kenig

integral estimates for null lagrangians sobolev spaces of differential forms are studied lp projections onto exact forms are introduced as a tool to obtain integral estimates for null lagrangians new results on compensated compactness are given and mean coercive variational integrals are found existence of minima of certain mean coercive functionals is established

the weil petersson and thurston symplectic forms we consider the weil petersson form on the teichmuller space mathscr t s of a surface s of genus at least 2 and we compute it in terms of the shearing coordinates for mathscr t s associated to a geodesic lamination on s in the corresponding expression the weil petersson form coincides with thurston s intersection form on the space of transverse cocycles for

the weber seifert dodecahedral space is non haken in this paper we settle thurston s old question of whether the weber seifert dodecahedral space is non haken a problem that has been a benchmark for progress in computational 3 manifold topology over recent decades we resolve this question by combining recent significant advances in normal surface enumeration new heuristic pruning techniques and a new theoretical test that extends the seminal work of jaco and oertel

an h s p text curl varomega estimate for the maxwell system we derive an h 0 s p text curl varomega estimate for the solutions of the maxwell type equations modeled with anisotropic and w s infty varomega regular coefficients here we obtain the regularity of the solutions for the integrability and smoothness indices p s in a plane domain characterized by the apriori lower upper bounds of a and the apriori upper bound of its holder semi norm of order s the proof relies on a perturbation argument generalizing groger s l p type estimate known for the elliptic problems to the maxwell system

pl equivariant surgery and invariant decompositions of 3 manifolds using normal surface theory h j2 we introduce the notion of least weight normal surfaces the weight of a normal surface is a nonnegative integer invariant of the normal isotopy class of the surface if we focus on a particular class of normal surfaces and choose representatives which minimize the weight over the class then we have least weight normal surfaces it is remarkable how these least weight normal surfaces exhibit many of the same useful properties as least area minimal surfaces they provide a piecewise linear pl environment to obtain the recent topological results coming from the analysis and geometry of least area surfaces in an impressive series of papers meeks and yau m y i m y z m y 3 m y m y meeks simon and yau m s y scott s meeks and scott m s and freedman hass and scott f h s introduced the analysis and geometry of least area surfaces into the study of topological questions about 3 manifolds the consequences have led to the resolution of many outstanding problems in the topology of 3 manifolds since most of the problems being solved with these new techniques are topological in nature it has been felt that there should be a more topological approach to the proofs of these results besides the bulk of the work in the theory of

core curves of triangulated solid tori one of the most powerful tools in 3 manifold topology is normal surface theory many topologically relevant surfaces can be placed into normal form or some variant of this with respect to any triangulation t of the manifold m an important case of this phenomenon is when m is the solid torus and the surface is a meridian disc the existence of a meridian disc in normal form and the fact that this can be algorithmically detected is the key to haken s solution to the problem of recognizing the unknot 1 however normal surface theory suffers from some substantial limitations possibly the most important of which is that every normal meridian disc may have exponential complexity more precisely if n is the number of tetrahedra in t then normal surface theory only produces a meridian disc d with at most 2 normal triangles and squares where k 10 see 2 this is more than just an artifice of the theory because one can find triangulations of the solid torus where every normal meridian disc has exponentially many squares and triangles indeed we will do this explicitly in section 6 however in this paper we show that for a related problem there is a solution with linearly bounded complexity in addition to the meridian disc the solid torus contains another important sub object the core curve c which is defined up to ambient isotopy to be s d s where is a point in the interior of d in this paper we will address the problem of placing c into normal form the surprising conclusion is that this can be achieved with a linear upper bound on the complexity of c

all principal congruence link groups abstract we enumerate all the principal congruence link complements in s 3 thereby answering a question of w thurston

cusp shapes under cone deformation a horospherical torus about a cusp of a hyperbolic manifold inherits a euclidean similarity structure called a cusp shape we bound the change in cusp shape when the hyperbolic structure of the manifold is deformed via cone deformation preserving the cusp the bounds are in terms of the change in structure in a neighborhood of the singular locus alone we then apply this result to provide information on the cusp shape of many hyperbolic knots given only a diagram of the knot specifically we show there is a universal constant c such that if a knot admits a prime twist reduced diagram with at least c crossings per twist region then the length of the second shortest curve on the cusp torus is bounded the bound is linear in the number of twist regions of the diagram

maximum amplitude of limit cycles in lienard systems we establish sufficient criteria for the existence of a limit cycle in the lienard system x over y f x y over x where f x is odd in their simplest form the criteria lead to the result that for all finite nonzero the amplitude of the limit cycle is less than and 0 a u where f a 0 and 0 u f x dx 0 we take the van der pol oscillator as a specific example and establish that for all finite nonzero the amplitude of its limit cycle is less than 2 0672 a value whose precision is limited by the capacity of our symbolic computation software package we show how the criterion for the upper bound can be extended to establish a bound on the amplitude of a limit cycle in systems where f x contains both odd and even components we also show how the criteria can be used to establish bounds for bifurcation sets

explicit non algebraic limit cycles for polynomial systems we consider a system of the form email protected p n x y xr m x y email protected q n x y yr m x y where p n x y q n x y and r m x y are homogeneous polynomials of degrees n n and m respectively with n

heegaard surfaces and measured laminations i the waldhausen conjecture we give a proof of the so called generalized waldhausen conjecture which says that an orientable irreducible atoroidal 3 manifold has only finitely many heegaard splittings in each genus up to isotopy jaco and rubinstein have announced a proof of this conjecture using different methods

euclidean decompositions of noncompact hyperbolic manifolds on introduit une methode pour diviser une variete hyperbolique non compacte de volume fini en morceaux euclidiens canoniques

dehn filling volume and the jones polynomial given a hyperbolic 3 manifold with torus boundary we bound the change in volume under a dehn filling where all slopes have length at least 2 this result is applied to give explicit diagrammatic bounds on the volumes of many knots and links as well as their dehn fillings and branched covers finally we use this result to bound the volumes of knots in terms of the coefficients of their jones polynomials

algorithmic derivation of centre conditions the conditions for a critical point of a polynomial differential system to be a centre are of particular significance because of the frequency with which they are required in applications we demonstrate how computer algebra can be effectively employed in the search for necessary and sufficient conditions for critical points of such systems to be centres we survey recent developments and illustrate our approach by means of examples

the brezis nirenberg problem for the curl curl operator abstract we look for solutions e r 3 of the problem e e e p 2 e in e 0 on on a bounded lipschitz domain r 3 where denotes the curl operator in r 3 the equation describes the propagation of the time harmonic electric field e x e i t in a nonlinear isotropic material with e 2 0 where and e stand for the permeability and the linear part of the permittivity of the material the nonlinear term e p 2 e with p 2 is responsible for the nonlinear polarisation of and the boundary conditions are those for surrounded by a perfect conductor the problem has a variational structure and we deal with the critical value p for instance in convex domains or in domains with c 1 1 boundary p 6 2 is the sobolev critical exponent and we get the quintic nonlinearity in the equation we show that there exist a cylindrically symmetric ground state solution and a finite number of cylindrically symmetric bound states depending on 0 we develop a new critical point theory which allows to solve the problem and which enables us to treat more general anisotropic media as well as other variational problems

besov and triebel lizorkin regularity for the hodge decomposition and applications to magnetic potentials abstract the hodge decomposition is a fundamental result in the theory of differential forms it has been extensively studied from different perspectives and it is the source of several applications in physics and mathematics for example it is important in the analysis of navier stokes and maxwell s equations it can be used to prove results on inverse scattering problems in quantum mechanics and it is useful for the study of magnetic potentials associated to a magnetic field the regularity properties of the hodge decomposition have been studied in several works in the last decades most prominently it has been proved that sobolev regularity holds true in this paper we prove regularity for the hodge decomposition in a much larger scale of function spaces besov and triebel lizorkin spaces on compact riemannian manifolds with boundary we prove regularity in such spaces for the hodge morrey decomposition and the hodge morrey friedrichs decomposition we furthermore state and prove a refinement for the hodge morrey friedrichs decomposition with regularity that allows expressing every differential form as the sum of the differential of a form plus the co differential of another form plus a harmonic field belonging to a finite dimensional space this holds true even if the original form is barely differentiable with sth order of differentiability for every real number s 0 we finally provide an application to the study of magnetic potentials associated to magnetic fields more precisely we prove the existence of a magnetic potential associated to a magnetic field the former being one time more differentiable than the latter in terms of besov and triebel lizorkin regularity this is an important motivation for us since we have the intention to apply our results to further investigations in relativistic inverse time dependent scattering problems in quantum mechanics where besov and triebel lizorkin regularity for magnetic potentials is relevant

z 2 thurston norm and complexity of 3 manifolds a new lower bound on the complexity of a 3 manifold is given using the mathbb z 2 thurston norm this bound is shown to be sharp and the minimal triangulations realising it are characterised using normal surfaces consisting entirely of quadrilateral discs

intersection theory on the moduli space of curves and the matrix airy function we show that two natural approaches to quantum gravity coincide this identity is nontrivial and relies on the equivalence of each approach to kdv equations we also investigate related mathematical problems

a discrete transform and decompositions of distribution spaces abstract we study a representation formula of the form q q q for a distribution on r n this formula is obtained by discretizing and localizing a standard littlewood paley decomposition the map taking to the sequence q q with q running over the dyadic cubes in r n is called the transform the functions q and q have a particularly simple form moreover most of the familiar distribution spaces lp spaces 1 r n we also consider pointwise multipliers for the characteristic function of a domain we obtain boundedness results for a general class of domains which properly includes lipschitz domains several interpolation methods are easily analyzed via the sequence spaces for real interpolation we obtain among other things an extension to the case p 0 this in turn gives a new approach to the traditional atomic decomposition of hardy spaces

characteristic subsurfaces and dehn filling let m be a simple knot manifold using the characteristic submanifold theory and the combinatorics of graphs in surfaces we develop a method for bounding the distance between the boundary slope of an essential surface in m which is not a fiber or a semi fiber and the boundary slope of a certain type of singular surface applications include bounds on the distances between exceptional dehn surgery slopes it is shown that if the fundamental group of m has no non abelian free subgroup and if m is a reducible manifold which is not homeomorphic to s 1 s 2 or p 3 p 3 then 5 under the same condition on m it is shown that if m is seifert fibered then 6 moreover in the latter situation character variety techniques are used to characterize the topological types of m and m in case the bound of 6 is attained

relaxation oscillators with exact limit cycles as an inverse problem for relaxation oscillators modeled by the autonomous lienard differential equation we assume plausible exact expressions for the limit cycles in the phase plane which correspond to periodic solutions toward which all solutions converge or from which they recede as time tends to infinity we illustrate this by two examples one of which concerns an approximation to a van der pol oscillator the results could be utilized for the synthesis problem i e the construction of oscillator circuits yielding desired periodic behavior

the maximal number of exceptional dehn surgeries thurston s hyperbolic dehn surgery theorem is one of the most important results in 3 manifold theory and it has stimulated an enormous amount of research if m is a compact orientable hyperbolic 3 manifold with boundary a single torus then the theorem asserts that for all but finitely many slopes s on m the manifold m s obtained by dehn filling along s also admits a hyperbolic structure the slopes s where m s is not hyperbolic are known as exceptional a major open question has been what is the maximal number of exceptional slopes on such a manifold m when m is the exterior of the figure eight knot the number of exceptional slopes is 10 and this was conjectured by gordon in 19 to be an upper bound that holds for all m in this paper we prove this conjecture

recognizing a relatively hyperbolic group by its dehn fillings dehn fillings for relatively hyperbolic groups generalize the topological dehn surgery on a non compact hyperbolic 3 manifold such as a hyperbolic knot complement we prove a rigidity result saying that if two non elementary relatively hyperbolic groups without suitable splittings have sufficiently many isomorphic dehn fillings then these groups are in fact isomorphic our main application is a solution to the isomorphism problem in the class of non elementary relatively hyperbolic groups with residually finite parabolic groups and with no suitable splittings

inflations of ideal triangulations abstract starting with an ideal triangulation of m the interior of a compact 3 manifold m with boundary no component of which is a 2 sphere we provide a construction called an inflation of the ideal triangulation to obtain a strongly related triangulation of m itself besides a step by step algorithm for such a construction we provide examples of an inflation of the two tetrahedra ideal triangulation of the complement of the figure eight knot in s 3 giving a minimal triangulation having ten tetrahedra of the figure eight knot exterior as another example we provide an inflation of the one tetrahedron gieseking manifold giving a minimal triangulation having seven tetrahedra of a non orientable compact 3 manifold with klein bottle boundary several applications of inflations are discussed

optimizing the double description method for normal surface enumeration many key algorithms in 3 manifold topology involve the enumeration of normal surfaces which is based upon the double description method for finding the vertices of a convex polytope typically we are only interested in a small subset of these vertices many key algorithms in 3 manifold topology involve the enumeration of normal surfaces which is based upon the double description method for finding the vertices of a convex polytope typically we are only interested in a small subset of these vertices thus opening the way for substantial optimization here we give an account of the vertex enumeration problem as it applies to normal surfaces and present new optimizations that yield strong improvements in both running time and memory consumption the resulting algorithms are tested using the freely available software package regina

crosscap numbers and the jones polynomial abstract we give sharp two sided linear bounds of the crosscap number non orientable genus of alternating links in terms of their jones polynomial our estimates are often exact and we use them to calculate the crosscap numbers for several infinite families of alternating links and for several alternating knots with up to twelve crossings we also discuss generalizations of our results for classes of non alternating links

increasing the number of fibered faces of arithmetic hyperbolic 3 manifolds we exhibit a closed hyperbolic 3 manifold which satisfies a very strong form of thurston s virtual fibration conjecture in particular this manifold has finite covers which fiber over the circle in arbitrarily many ways more precisely it has a tower of finite covers where the number of fibered faces of the thurston norm ball goes to infinity in fact faster than any power of the logarithm of the degree of the cover and we give a more precise quantitative lower bound the example manifold m is arithmetic and the proof uses detailed number theoretic information at the level of the hecke eigenvalues to drive a geometric argument based on fried s dynamical characterization of the fibered faces the origin of the basic fibration m s 1 is the modular elliptic curve e x 0 49 which admits multiplication by the ring of integers of q 7 we first base change the holomorphic differential on e to a cusp form on gl 2 over k q 3 and then transfer over to a quaternion algebra d k ramified only at the primes above 7 the fundamental group of m is a quotient of the principal congruence subgroup of o d of level 7 to analyze the topological properties of m we use a new practical method for computing the thurston norm which is of independent interest we also give a noncompact finite volume hyperbolic 3 manifold with the same properties by using a direct topological argument

0 efficient triangulations of 3 manifolds efficient triangulations of 3 manifolds are defined and studied it is shown that any triangulation of a closed orientable irreducible 3 manifold m can be modified to a 0 efficient triangulation orm can be shown to be one of the manifolds s 3 rp 3 or l 3 1 similarly any triangulation of a com pact orientable irreducible irreducible 3 manifold can be modified to a 0 efficient triangulation the notion of a 0 efficient ideal tr is de fined it is shown if m is a compact orientable irreducible irreducible 3 manifold having no essential annuli and distinct from the 3 cell then m admits an ideal triangulation furthermore it is shown that any ideal trian gulation of such a 3 manifold can be modified to a 0 efficient ideal triangula tion a 0 efficient triangulation of a closed manifold has only one vertex or the manifold is s 3 and the triangulation has precisely two vertices 0 efficient tri angulations of 3 manifolds with boundary and distinct from the 3 cell have all their vertices in the boundary and then just one vertex in each bound ary component as tools we introduce the concepts of barrier surface and shrinking as well as the notion of crushing a triangulation along a normal sur face a number of applications are given including an algorithm to construct an irreducible decomposition of a closed orientable 3 manifold an algorithm to construct a maximal collection of pairwise disjoint normal 2 spheres in a closed 3 manifold an alternate algorithm for the 3 sphere recognition prob lem results on edges of low valence in minimal triangulations of 3 manifolds and a construction of irreducible knots in closed 3 manifolds

embeddability in the 3 sphere is decidable we show that the following algorithmic problem is decidable given a 2 dimensional simplicial complex can it be embedded topologically or equivalently piecewise linearly in r3 by a known reduction it suffices to decide the embeddability of a given triangulated 3 manifold x into the 3 sphere s3 the main step which allows us to simplify x and recurse is in proving that if x can be embedded in s3 then there is also an embedding in which x has a short meridian that is an essential curve in the boundary of x bounding a disk in s3 b x with length bounded by a computable function of the number of tetrahedra of x

the regularity of the stokes operator and the fujita kato approach to the navier stokes initial value problem in lipschitz domains abstract we study the regularity of the navier stokes equations in arbitrary lipschitz domains

heegaard splittings the virtually haken conjecture and property τ the behaviour of finite covers of 3 manifolds is a very important but poorly understood topic there are three increasingly strong conjectures in the field that have remained open for over twenty years the virtually haken conjecture the positive virtual b1 conjecture and the virtually fibred conjecture any of these would have profound ramifications for 3 manifold theory in this paper we explore the interaction of these conjectures with the following seemingly unrelated areas eigenvalues of the laplacian and heegaard splittings

perturbative series and the moduli space of riemann surfaces on utilise des techniques de la theorie quantique des champs pour calculer des quantites reliees aux groupes de symetrie de paires f g ou f est une surface et g une epine de f

stability results on interpolation scales of quasi banach spaces and applications we investigate the stability of fredholm properties on interpolation scales of quasi banach spaces this analysis is motivated by problems arising in pde s and several applications are presented

a new criterion for controlling the number of limit cycles of some generalized lienard equations abstract we consider a class of planar differential equations which include the lienard differential equations by applying the bendixson dulac criterion for l connected sets we reduce the study of the number of limit cycles for such equations to the condition that a certain function of just one variable does not change sign as an application this method is used to give a sharp upper bound for the number of limit cycles of some lienard differential equations in particular we present a polynomial lienard system with exactly three limit cycles

volume optimization normal surfaces and thurston s equation on triangulated 3 manifolds we propose a finite dimensional variational principle on triangulated 3 manifolds so that its critical points are related to solutions to thurston s gluing equation and haken s normal surface equation the action functional is the volume

computing trisections of 4 manifolds we describe an algorithm to compute trisections of orientable four manifolds using arbitrary triangulations as input this results in explicit complexity bounds for the trisection genus of a 4 manifold in terms of the number of pentachora 4 simplices in a triangulation

infinitely many hyperbolic 3 manifolds which contain no reebless foliation calegari ca has shown that if m is also atoroidal then or1 m is gromov negatively curved furthermore thurston has proposed an approach to demonstrating geometrization for such m many 3 manifolds contain reebless foliations and it has often been conjectured that all closed hyperbolic 3 manifolds do it is our impression that for many years hatcher provided the sole voice of dissent in this paper we give the first examples of closed hyperbolic 3 manifolds which contain no reebless foliation

sharp hodge decompositions maxwell s equations and vector poisson problems on nonsmooth three dimensional riemannian manifolds we solve three basic potential theoretic problems hodge decompositions for vector fields poisson problems for the hodge laplacian and inhomogeneous maxwell equations in arbitrary lipschitz subdomains of a smooth compact three dimensional riemannian manifold in each case we derive sharp estimates on sobolev besov scales and establish integral representation formulas for the solution the proofs rely on tools from harmonic analysis and algebraic topology such as calderon zygmund theory and de rham theory

universal bounds for hyperbolic dehn surgery this paper gives a quantitative version of thurston s hyperbolic dehn surgery theorem applications include the first universal bounds on the number of nonhyperbolic dehn fillings on a cusped hyperbolic 3 manifold and estimates on the changes in volume and core geodesic length during hyperbolic dehn filling the proofs involve the construction of a family of hyperbolic conemanifold structures using infinitesimal harmonic deformations and analysis of geometric limits

dehn fillings and elementary splittings we consider conditions on relatively hyperbolic groups about the non existence of certain kinds of splittings and show these properties persist in long dehn fillings we deduce that certain connectivity properties of the bowditch boundary persist un

cusp structures of alternating links summary an alternating link lfr is canonically associated with every finite connected planar graph f the natural ideal polyhedral decomposition of the complement of 5 is investigated natural singular geometric structures exist on s3 r with respect to which the geometry of the cusp has a shape reflecting the combinatorics of the underlying link projection for the class of balanced graphs this induces a flat structure on peripheral tori modelled on the tessellation of the plane by equilateral triangles examples of links containing immersed closed zl injective surfaces in their complements are given these surfaces persist after most surgeries on the link the resulting closed 3 manifolds consequently being determined by their fundamental groups

essential laminations in 3 manifolds it is a well established principle in manifold topology that by studying codimension 1 objects and their complementary pieces one obtains a great deal of topological and geometric information about the manifold itself this is particularly true for 3 dimensional manifolds using incompressible surfaces a great many advances have been made during the last 30 years the most spectacular results were obtained by haken ha waldhausen w and thurston t1 more recently work t2 g14 using taut foliations in 3 manifolds has proven fruitful unfortunately in some sense most closed 3 manifolds do not contain incompressible surfaces and it is currently not known exactly which 3 manifolds have taut foliations the purpose of this paper is to study another codimension 1 object the essential lamination which is a generalization of the incompressible surface as well as the taut foliation we will show that the existence of an essential lamination in a 3 manifold m implies that m has useful properties similar to those of manifolds having either an incompressible surface or a taut foliation we will also show that the universal cover of a manifold containing an essential lamination is r3 precise definitions of terms used in the following theorem will be given in section 1 here we give only a rough idea we begin with two alternative approximate definitions of essential lamination i an essential lamination in a 3 manifold m is a lamination satisfying four conditions the inclusion of leaves of the lamination into m should induce an injection on stj the complement of the lamination should be irreducible no leaf should be a sphere and the lamination should be end incompressible to say that a lamination is end incompressible means roughly that a folded leaf can be straightened using an isotopy there are no infinite folds ii alternatively

realization problems for limit cycles of planar polynomial vector fields abstract we show that for any finite configuration of closed curves r 2 one can construct an explicit planar polynomial vector field that realizes up to homeomorphism as the set of its limit cycles with prescribed periods multiplicities and stabilities the only obstruction given on this data is the obvious compatibility relation between the stabilities and the parity of the multiplicities the constructed vector fields are darboux integrable and admit a polynomial inverse integrating factor

cyclic surgery degrees of maps of character curves and volume rigidity for hyperbolic manifolds this paper proves a theorem about dehn surgery using a new theorem about psl2c character varieties confirming a conjecture of boyer and zhang this p aper shows that a small hyperbolic knot in a homotopy sphere having a non trivial cyclic slope r has an incompressible surface with non integer boundary slope strictly between r 1 and r 1 a corollary is that any small knot which has only integer boundary slopes has property p the proof uses connections between the topology of the complement of the knot m and the psl2c character variety of m that were discovered by culler and shalen the key lemma which should be of independent interest is that for certain components of the character variety of m the map on character varieties induced by m m is a birational isomorphism onto its image this in turn depends on a fancy version of mostow rigidity due to gromov thurston and goldman the proof of the theorem about dehn surgery also gives information about the diameter of the set of boundary slopes of m let me start with the theorem about dehn surgery let k be a knot in a compact closed 3manifold that is a tame embedding s 1 the complement of k m is minus an open regular neighborhood of k so m is a compact 3 manifold whose boundary is a torus suppose is a simple closed curve in m we can create a closed manifold m from m by taking a solid torus and gluing its boundary to m in such a way that bounds a disc in the solid torus m depends only on the isotopy class of the new manifold m is called a dehn filling of m or a dehn surgeryon k recently many people have studied what kinds of manifolds arise when you

angled decompositions of arborescent link complements this paper describes a way to subdivide a 3 manifold into angled blocks namely polyhedral pieces that need not be simply connected when the individual blocks carry dihedral angles that fit together in a consistent fashion we prove that a manifold constructed from these blocks must be hyperbolic the main application is a new proof of a classical unpublished theorem of bonahon and siebenmann that all arborescent links except for three simple families of exceptions have hyperbolic complements in the 1990s andrew casson introduced a powerful technique for constructing and studying cusped hyperbolic 3 manifolds his idea was to subdivide a manifold m into angled ideal tetrahedra that is tetrahedra whose vertices are removed and whose edges carry prescribed dihedral angles when the dihedral angles of the tetrahedra add up to 2 around each edge of m the triangulation is called an angled triangulation casson proved that every orientable cusped 3 manifold that admits an angled triangulation must also admit a hyperbolic metric and outlined a possible way to find the hyperbolic metric by studying the volumes of angled tetrahedra an idea also developed by rivin 17 the power of casson s approach lies in the fact that the defining equations of an angled triangulation are both linear and local making angled triangulations relatively easy to find and deform much easier than to study an actual hyperbolic triangulation as in 14 20 or in some aspects of thurston s seminal approach 22 the aim of this paper is to extend this approach to larger and more complicated building blocks these blocks can be ideal polyhedra instead of tetrahedra but they may also have nontrivial topology in general an angled block will be a 3 manifold whose boundary is subdivided into faces looking locally like the faces of an ideal polyhedron in a sense to be defined the edges between adjacent faces carry prescribed dihedral angles in section 2 we will describe the precise combinatorial conditions that the dihedral angles must satisfy these conditions will imply the following generalization of a result by lackenby 11 corollary 4 6

besov spaces on domains in rd we study besov spaces 5 lp i2 0 p q a oo on do mains i2 in rd we show that there is an extension operator w which is a bounded mapping from b lp u onto b lp rd this is then used to derive various properties of the besov spaces such as interpolation theorems for a pair of b lp il atomic decompositions for the elements of 5 lp f2 and a description of the besov spaces by means of spline approximation

angle structures and normal surfaces let m be the interior of a compact 3 manifold with boundary and let t be an ideal triangulation of m this paper describes necessary and sufficient conditions for the existence of angle structures semi angle structures and generalised angle structures on m t respectively in terms of a generalised euler characteristic function on the solution space of the normal surface theory of m t this extends previous work of kang and rubinstein and is itself generalised to a more general setting for 3 dimensional pseudo manifolds

every knot has characterising slopes let k be a knot in the 3 sphere a slope p q is said to be characterising for k if whenever p q surgery on k is homeomorphic via an orientation preserving homeomorphism to p q surgery on another knot k in the 3 sphere then k and k are isotopic it was an old conjecture of gordon proved by kronheimer mrowka ozsvath and szabo that every slope is characterising for the unknot in this paper we show that every knot k has infinitely many characterising slopes confirming a conjecture of baker and motegi in fact p q is characterising for k provided p is at most q and q is sufficiently large

on bogovskiĭ and regularized poincare integral operators for de rham complexes on lipschitz domains we study integral operators related to a regularized version of the classical poincare path integral and the adjoint class generalizing bogovski s integral operator acting on differential forms in mathbb r n we prove that these operators are pseudodifferential operators of order 1 the poincare type operators map polynomials to polynomials and can have applications in finite element analysis for a domain starlike with respect to a ball the special support properties of the operators imply regularity for the de rham complex without boundary conditions using poincare type operators and with full dirichlet boundary conditions using bogovski type operators for bounded lipschitz domains the same regularity results hold and in addition we show that the cohomology spaces can always be represented by fancyscript c infty functions

improving a method for the study of limit cycles of the lienard equation in recent papers we have introduced a method for the study of limit cycles of the li enard system mathrm x ifmmode dot else fi y ensuremath f x mathrm y ifmmode dot else fi ensuremath x where f x is an odd polynomial the method gives a sequence of polynomials r n x whose roots are related to the number and location of the limit cycles and a sequence of algebraic approximations to the bifurcation set of the system in this paper we present a variant of the method that gives very important qualitative and quantitative improvements

three dimensional manifolds kleinian groups and hyperbolic geometry 1 a conjectural picture of 3 manifolds a major thrust of mathematics in the late 19th century in which poincare had a large role was the uniformization theory for riemann surfaces that every conformai structure on a closed oriented surface is represented by a riemannian metric of constant curvature for the typical case of negative euler characteristic genus greater than 1 such a metric gives a hyperbolic structure any small neighborhood in the surface is isometric to a neighborhood in the hyperbolic plane and the surface itself is the quotient of the hyperbolic plane by a discrete group of motions the exceptional cases the sphere and the torus have spherical and euclidean structures three manifolds are greatly more complicated than surfaces and i think it is fair to say that until recently there was little reason to expect any analogous theory for manifolds of dimension 3 or more except perhaps for the fact that so many 3 manifolds are beautiful the situation has changed so that i feel fairly confident in proposing the

boundary layer methods for lipschitz domains in riemannian manifolds abstract we extend to the variable coefficient case boundary layer techniques that have been successful in the treatment of the laplace equation and certain other constant coefficient elliptic partial differential equations on lipschitz domains in euclidean space we treat the laplace operator on lipschitz domains in a manifold with c 1 metric tensor and study the dirichlet neumann and oblique derivative boundary problems

hardy spaces and the neumann problem in l p for laplace s equation in lipschitz domains on donne des resultats optimaux pour la resolubilite du probleme de neumann dans des domaines de lipschitz a donnees dans l p on obtient des resultats du point extreme correspondant pour des espaces de hardy

word hyperbolic dehn surgery in the late 1970 s thurston dramatically changed the nature of 3 manifold theory with the introduction of his geometrisation conjecture and by proving it in the case of haken 3 manifolds 23 the conjecture for general closed orientable 3 manifolds remains perhaps the most important unsolved problem in the subject a weaker form of the conjecture 19 deals with the fundamental group of a closed orientable 3 manifold it proposes that either it contains z z as a subgroup or it is word hyperbolic in the sense of gromov 11 word hyperbolic groups are precisely those groups which satisfy a linear isoperimetric inequality they are of fundamental importance in geometric group theory and have very many useful properties

on a classification of polynomial differential operators with respect to the type of first integrals abstract this paper gives a classification of polynomial differential operators x x 1 x 1 x 2 1 x 2 x 1 x 2 2 i x i the classification is defined through an order derived from x let x x y be the associated differential polynomial the order is defined as the order of a differential ideal that is an essential extension of x the main result shows the order can only be four possible values 0 1 2 3 or furthermore when the order is finite the essential extension x a where a is a differential polynomial with coefficients obtained through a rational solution of a partial differential equation given explicitly by coefficients of x when the order is infinite the extension is identical with x in addition if and only if the order is 0 1 or 2 the associated polynomial differential equation has liouvillian first integrals examples and connections with godbillon vey sequences are also discussed

totally geodesic seifert surfaces in hyperbolic knot and link complements ii the first examples of totally geodesic seifert surfaces are constructed for hyperbolic knots and links including both free and totally knotted surfaces then it is proved that two bridge knot complements cannot contain totally geodesic orientable surfaces