on witten s proof of the positive energy theorem this paper gives a mathematically rigorous proof of the positive energy theorem using spinors this completes and simplifies the original argument presented by edward witten we clarify the geometric aspects of this argument and prove the necessary analytic theorems concerning the relevant dirac operator

categorical formulation of finite dimensional quantum algebras we describe how frobenius monoids give the correct categorical description of certain kinds of finite dimensional quantum algebras we develop the concept of an involution monoid and use it to construct a correspondence between finite dimensional c algebras and certain types of frobenius monoids in the category of hilbert spaces using this technology we recast the spectral theorems for commutative c algebras and for normal operators into an explicitly categorical language and we examine the case that the results of measurements do not form finite sets but rather objects in a finite boolean topos we describe the relevance of these results for topological quantum field theory

deforming three manifolds with positive scalar curvature in this paper we prove that the moduli space of metrics with positive scalar curvature of an orientable compact three manifold is path connected the proof uses the ricci flow with surgery the conformal method and the connected sum construction of gromov and lawson the work of perelman on hamilton s ricci flow is fundamental as one of the applications we prove the path connectedness of the space of trace free asymptotically flat solutions to the vacuum einstein constraint equations on r 3

the geometry of tensor calculus i this paper defines and proves the correctness of the appropriate string diagrams for various kinds of monoidal categories with duals mathematics subject classifications 1991 18d10 52b11 53a45 57m25 68q10 82b23

quantum graph homomorphisms via operator systems abstract we explore the concept of a graph homomorphism through the lens of c algebras and operator systems we start by studying the various notions of a quantum graph homomorphism and examine how they are related to each other we then define and study a c algebra that encodes all the information about these homomorphisms and establish a connection between computational complexity and the representation of these algebras we use this c algebra to define a new quantum chromatic number and establish some basic properties of this number we then suggest a way of studying these quantum graph homomorphisms using certain completely positive maps and describe their structure finally we use these completely positive maps to define the notion of a quantum core of a graph

the reconstruction of a multidimensional sequence from the phase or magnitude of its fourier transform this paper addresses two fundamental issues involved in the reconstruction of a multidimensional sequence from either the phase or magnitude of its fourier transform the first issue relates to the uniqueness of a multidimensional sequence in terms of its phase or magnitude although phase or magnitude information alone is not sufficient in general to uniquely specify a sequence a large class of sequences are shown to be recoverable from their phase or magnitude the second issue which is addressed in this paper concerns the actual reconstruction of a multidimensional sequence from its phase or magnitude for those sequences which are uniquely specified by their phase several practical algorithms are described which may be used to reconstruct a sequence from its phase several examples of phase only reconstruction are also presented unfortunately however even for those sequences which are uniquely defined by their magnitude it appears that a practical algorithm is yet to be developed for reconstructing a sequence from only its magnitude nevertheless an iterative procedure which has been proposed is briefly discussed and evaluated

quantum automorphism groups of homogeneous graphs associated to a finite graph x is its quantum automorphism group g the main problem is to compute the poincare series of g meaning the series f z 1 c1z c2z2 whose coefficients are multiplicities of 1 into tensor powers of the fundamental representation in this paper we find a duality between certain quantum groups and planar algebras which leads to a planar algebra formulation of the problem together with some other results this gives f for all homogeneous graphs having 8 vertices or less

conformal deformation to constant negative scalar curvature on noncompact riemannian manifolds si m g est une variete de riemann complete de courbure scalaire non positive s satisfaisant s x e 0 pour x m m 0 ou m 0 est un ensemble compact alors il y a une metrique conforme complete g de courbure scalaire s 1

on quotients of spaces with ricci curvature bounded below abstract let m g be a smooth riemannian manifold and g a compact lie group acting on m effectively and by isometries it is well known that a lower bound of the sectional curvature of m g is again a bound for the curvature of the quotient space which is an alexandrov space of curvature bounded below moreover the analogous stability property holds for metric foliations and submersions the goal of the paper is to prove the corresponding stability properties for synthetic ricci curvature lower bounds specifically we show that such stability holds for quotients of rcd k n spaces under isomorphic compact group actions and more generally under metric measure foliations and submetries an rcd k n space is a metric measure space with an upper dimension bound n and weighted ricci curvature bounded below by k in a generalized sense in particular this shows that if m g has ricci curvature bounded below by k r and dimension n then the quotient space is an rcd k n space additionally we tackle the same problem for the cd cd and mcp curvature dimension conditions we provide as well geometric applications which include a generalization of kobayashi s classification theorem of homogeneous manifolds to rcd k n spaces with essential minimal dimension n n a structure theorem for rcd k n spaces admitting actions by large compact groups and geometric rigidity results for orbifolds such as cheng s maximal diameter and maximal volume rigidity theorems finally in two appendices we apply the methods of the paper to study quotients by isometric group actions of discrete spaces and of super ricci flows

kochen specker sets and the rank 1 quantum chromatic number the quantum chromatic number of a graph g is sandwiched between its chromatic number and its clique number which are well known np hard quantities we restrict our attention to the rank 1 quantum chromatic number q 1 g which upper bounds the quantum chromatic number but is defined under stronger constraints we study its relation with the chromatic number g and the minimum dimension of orthogonal representations g it is known that g q 1 g g we answer three open questions about these relations we give a necessary and sufficient condition to have g q 1 g we exhibit a class of graphs such that g q 1 g and we give a necessary and sufficient condition to have q 1 g g our main tools are kochen specker sets collections of vectors with a traditionally important role in the study of contextuality of physical theories and more recently in the quantification of quantum zero error capacities finally as a corollary of our results and a result by avis et al on the quantum chromatic number we give a family of kochen specker sets of growing dimension

moduli spaces of critical riemannian metrics with l n 2 norm curvature bounds abstract we consider the moduli space of a class of critical metrics on compact manifolds that includes extremal kahler metrics we show that under the conditions of volume and diameter bounds l n 2 curvature bounds and sobolev constant bounds this moduli space can be compactified by including orbifolds with finitely many orbifold points in dimension 4 these orbifold points are c riemannian orbifold points and in the 4 dimensional extremal kahler case they are also of class c

quantum zero error source channel coding and non commutative graph theory alice and bob receive a bipartite state possibly entangled from some finite collection or from some subspace alice sends a message to bob through a noisy quantum channel such that bob may determine the initial state with zero chance of error this framework encompasses for example teleportation dense coding entanglement assisted quantum channel capacity and one way communication complexity of function evaluation with classical sources and channels this problem can be analyzed using graph homomorphisms we show this quantum version can be analyzed using homomorphisms on non commutative graphs an operator space generalization of graphs previously the lovasz vartheta number has been generalized to non commutative graphs we show this to be a homomorphism monotone thus providing bounds on quantum source channel coding we generalize the schrijver and szegedy numbers and show these to be monotones as well as an application we construct a quantum channel whose entanglement assisted zero error one shot capacity can only be unlocked using a non maximally entangled state these homomorphisms allow definition of a chromatic number for non commutative graphs many open questions are presented regarding the possibility of a more fully developed theory

all teleportation and dense coding schemes we establish a one to one correspondence between 1 quantum teleportation schemes 2 dense coding schemes 3 orthonormal bases of maximally entangled vectors 4 orthonormal bases of unitary operators with respect to the hilbert schmidt scalar product and 5 depolarizing operations whose kraus operators can be chosen to be unitary the teleportation and dense coding schemes are assumed to be tight in the sense that all hilbert spaces involved have the same finite dimension d and the classical channel involved distinguishes d 2 signals a general construction procedure for orthonormal bases of unitaries involving latin squares and complex hadamard matrices is also presented

subcritical approach to conformally invariant extension operators on the upper half space abstract in this work we obtain sharp embedding inequalities for a family of conformally invariant integral extension operators this family includes among others the classical poisson extension operator and the extension operator with riesz kernel we show that the sharp constants in these inequalities are attained and classify the corresponding extremal functions we also compute the limiting behavior at the boundary of the extensions of the extremal functions

novel criteria of uniqueness for signal reconstruction from phase an approach for ascertaining whether a signal is uniquely determined by its fourier transform phase is proposed it is shown that uniqueness corresponds to the nonsingularity of a matrix which can be formed from the finite length real sequence the criterion of uniqueness for reconstructing a one dimensional finite length sequence from its phase is discussed the criterion of uniqueness for reconstructing a multidimensional finite length sequence is presented

quantum symmetry groups of finite spaces we determine the quantum automorphism groups of finite spaces these are compact matrix quantum groups in the sense of woronowicz

invariants of conformal laplacians the conformal laplacian d d d n 2 s 4 n 1 acting on functions on a riemannian manifold mn with scalar curvature s is a conformally invariant operator in this paper we will use d to construct new conformal invariants one of these is a pointwise invariant one is the integral of a local expression and one is a nonlocal spectral invariant derived from functional determinants we begin in 1 by describing the laplacian d and its green function in the context of conformal geometry we then derive a basic formula giving the variation in the heat kernel of d this formula is strikingly simpler than the corresponding formula for the ordinary laplacian given by ray and singer 15 the heat kernel of d has an asymptotic expansion k t x x 4 t n 2 ak x tk in 2 we prove that a n 2 2 is a pointwise conformal invariant of weight 2 i e it satisfies a n 2 2 x 2g 2a n 2 2 x g where g is the metric and is any smooth positive function in particular this shows the existence of a nontrivial locally computable conformally invariant density naturally associated to the conformal structure of an even dimensional manifold the key to the proof is to consider the parametrix of the green s function which is obtained from the heat kernel by an integral transform one finds that a n 2 2 occurs as the coefficient of the first log term in this parametrix and its conformal invariance then follows directly from the conformal invariance of the green s function in 3 we show that an 2 is a global conformal invariant the calculations in 4 show that it is not a pointwise invariant the proof is a direct calculation of the invariant of an 2 using equation 1 10

zero error communication via quantum channels noncommutative graphs and a quantum lovasz number we study the quantum channel version of shannon s zero error capacity problem motivated by recent progress on this question we propose to consider a certain subspace of operators so called operator systems as the quantum generalization of the adjacency matrix in terms of which the zero error capacity of a quantum channel as well as the quantum and entanglement assisted zero error capacities can be formulated and for which we show some new basic properties most importantly we define a quantum version of lovasz famous function on general operator systems as the norm completion or stabilization of a naive generalization of we go on to show that this function upper bounds the number of entanglement assisted zero error messages that it is given by a semidefinite program whose dual we write down explicitly and that it is multiplicative with respect to the tensor product of operator systems corresponding to the tensor product of channels we explore various other properties of the new quantity which reduces to lovasz original in the classical case give several applications and propose to study the operator systems associated with channels as noncommutative graphs using the language of hilbert modules

iterative techniques for minimum phase signal reconstruction from phase or magnitude in this paper we develop iterative algorithms for reconstructing a minimum phase sequence from the phase or magnitude of its fourier transform these iterative solutions involve repeatedly imposing a causality constraint in the time domain and incorporating the known phase or magnitude function in the frequency domain this approach is the basis of a new means of computing the hilbert transform of the log magnitude or phase of the fourier transform of a minimum phase sequence which does not require phase unwrapping finally we discuss the potential use of this iterative computation in determining samples of the unwrapped phase of a mixed phase sequence

the mass of an asymptotically flat manifold we show that the mass of an asymptotically flat n manifold is a geometric invariant the proof is based on harmonic coordinates and to develop a suitable existence theory results about elliptic operators with rough coefficients on weighted sobolev spaces are summarised some relations between the mass scalar curvature and harmonic maps are described and the positive mass theorem for n dimensional spin manifolds is proved

effects of additive noise on signal reconstruction from fourier transform phase the effects of additive noise in the given phase on signal reconstruction from the fourier transform phase are experimentally studied specifically the effects on the sequence reconstruction of different methods of sampling the degraded phase of the number of nonzero points in the sequence and of the noise level are examined a sampling method that significantly reduces the error in the reconstructed sequence is obtained and the error is found to increase as the number of nonzero points in the sequence increases and as the noise level increases in addition an averaging technique is developed which reduces the effects of noise when the continuous phase function is known finally as an illustration of how the results in this paper may be applied in practice fourier transform signal coding is considered coding only the fourier transform phase and reconstructing the signal from the coded phase is found to be considerably less efficient i e a higher bit rate is required for the same mean square error than reconstructing from both the coded phase and magnitude

a compositional approach to quantum functions we introduce a notion of quantum function and develop a compositional framework for finite quantum set theory based on a 2 category of quantum sets and quantum functions we use this framework to formulate a 2 categorical theory of quantum graphs which captures the quantum graphs and quantum graph homomorphisms recently discovered in the study of nonlocal games and zero error communication and relates them to quantum automorphism groups of graphs considered in the setting of compact quantum groups we show that the 2 categories of quantum sets and quantum graphs are semisimple we analyze dualisable and invertible 1 morphisms in these 2 categories and show that they correspond precisely to the existing notions of quantum isomorphism and classical isomorphism between sets and graphs

proof of the positive mass theorem ii the positive mass theorem states that for a nontrivial isolated physical system the total energy which includes contributions from both matter and gravitation is positive this assertion was demonstrated in our previous paper in the important case when the space time admits a maximal slice here this assumption is removed and the general theorem is demonstrated abstracts of the results of this paper appeared in 11 and 13

estimating quantum chromatic numbers abstract we develop further the new versions of quantum chromatic numbers of graphs introduced by the first and fourth authors we prove that the problem of computation of the commuting quantum chromatic number of a graph is solvable by an sdp algorithm and describe an hierarchy of variants of the commuting quantum chromatic number which converge to it we introduce the tracial rank of a graph a parameter that gives a lower bound for the commuting quantum chromatic number and parallels the projective rank and prove that it is multiplicative we describe the tracial rank the projective rank and the fractional chromatic numbers in a unified manner that clarifies their connection with the commuting quantum chromatic number the quantum chromatic number and the classical chromatic number respectively finally we present a new sdp algorithm that yields a parameter larger than the lovasz number and is yet a lower bound for the tracial rank of the graph we determine the precise value of the tracial rank of an odd cycle

extremal metrics of zeta function determinants on 4 manifolds in conformal geometry the sobolev inequality at a critical exponent has received much attention in particular the determination of the best constants has played a crucial role in the yamabe problem in dimension two the analogous problem deals with the moser trudinger inequality on a compact riemann surface m2 there exists a constant c c m so that f e47w2 c m if f ivw12 1 and f w 0 the connection of this inequality with geometry comes through the zeta functional determinant of the laplacian as defined by ray singer for a riemannian metric g let 0 a1 a2 be the spectrum

quantum groups acting on 4 points we classify the compact quantum groups acting on 4 points these are the quantum subgroups of the quantum permutation group q4 our main tool is a new presentation for the algebra c q4 corresponding to an isomorphism of type q4 so 1 3 the quantum subgroups of q4 are subject to a mckay type correspondence that we describe at the level of algebraic invariants

an integral type brezis nirenberg problem on the heisenberg group abstract this paper is devoted to a class of integral type brezis nirenbreg problems on the heisenberg group they are a class of nonlinear integral equations on the bounded domains of heisenberg group and related to the cr yamabe problems on cr manifold based on the sharp hardy littlewood sobolev inequalities on the heisenberg group the nonexistence and existence results are obtained

scalar curvature functions in a conformal class of metrics and conformal transformations on etudie le probleme de prescrire la courbure scalaire dans une classe conforme pour l action du groupe conforme on montre l universalite des conditions d integrabilite dues a j l kazdan et f w warner

holder regularity of solutions for schrodinger operators on stratified spaces we study the regularity properties for solutions of a class of schrodinger equations v u 0 on a stratified space m endowed with an iterated edge metric the focus is on obtaining optimal holder regularity of these solutions assuming fairly minimal conditions on the underlying metric and potential

analytical formulae for reconstruction of certain discrete signals from phase level and line crossings we provide simple and explicit formulae for reconstructing any member of a class of discrete time signals from the frequencies at which its fourier phase crosses any specific level of constant phase or a linear phase line with integer slope provided that the number of crossings equals the length of the signal support unlike previous closed form solutions solution of an ill conditioned system of linear equations is not required the associated uniqueness results reduce in special cases to previous results for reconstruction from fourier transform real and imaginary part zero crossings

a class of fully nonlinear equations arising from conformal geometry in this paper we study the long time existence of solutions for a class of fully nonlinear parabolic equations arising from conformal geometry in particular we prove that every smooth compact n dimensional manifold n ge3 admits a riemannian metric g with its ricci curvature ric and scalar curvature r satisfying det ric r cdot g const

quantum semigroups from synchronous games we show that the c algebras associated with synchronous games give rise to certain quantum families of maps between the input and output sets of the game in particular situations e g for graph endomorphism games these quantum families have a natural quantum semigroup structure and if the condition of preservation of a natural state is added they are in fact compact quantum groups we show that the c algebras associated with synchronous games give rise to certain quantum families of maps between the input and output sets of the game in particular situations e g for graph endomorphism games these quantum families have a natural quantum semigroup structure and if the condition of preservation of a natural state is added they are in fact compact quantum groups

bigalois extensions and the graph isomorphism game we study the graph isomorphism game that arises in quantum information theory we prove that the non commutative algebraic notion of a quantum isomorphism between two graphs is same as the more physically motivated one arising from the existence of a perfect quantum strategy for graph isomorphism game this is achieved by showing that every algebraic quantum isomorphism between a pair of quantum graphs x and y arises from a certain measured bigalois extension for the quantum automorphism groups g x and g y of x and y in particular this implies that the quantum groups g x and g y are monoidally equivalent we also establish a converse to this result which says that a compact quantum group g is monoidally equivalent to the quantum automorphism group g x of a given quantum graph x if and only if g is the quantum automorphism group of a quantum graph that is algebraically quantum isomorphic to x using the notion of equivalence for non local games we apply our results to other synchronous games including the synbcs game and certain related graph homomorphism games

reconstruction of signals from phase efficient algorithms segmentation and generalizations in this paper we develop several new formulations of the phase only reconstruction problem which lead to new efficient algorithms for signal reconstruction we describe a procedure for approximately reconstructing a long duration signal from short segments phase only blind deconvolution for linear phase filters relies on mirror image symmetry in the filter transfer function h z we generalize this deconvolution result to include several other types of symmetries for h z

nonlocal games and quantum permutation groups we present a strong connection between quantum information and quantum permutation groups specifically we define a notion of quantum isomorphisms of graphs based on quantum automorphisms from the theory of quantum groups and then show that this is equivalent to the previously defined notion of quantum isomorphism corresponding to perfect quantum strategies to the isomorphism game moreover we show that two connected graphs x and y are quantum isomorphic if and only if there exists x v x and y v y that are in the same orbit of the quantum automorphism group of the disjoint union of x and y this connection links quantum groups to the more concrete notion of nonlocal games and physically observable quantum behaviours we exploit this link by using ideas and results from quantum information in order to prove new results about quantum automorphism groups and about quantum permutation groups more generally in particular we show that asymptotically almost surely all graphs have trivial quantum automorphism group furthermore we use examples of quantum isomorphic graphs from previous work to construct an infinite family of graphs which are quantum vertex transitive but fail to be vertex transitive answering a question from the quantum group literature our main tool for proving these results is the introduction of orbits and orbitals orbits on ordered pairs of quantum permutation groups we show that the orbitals of a quantum permutation group form a coherent configuration algebra a notion from the field of algebraic graph theory we then prove that the elements of this quantum orbital algebra are exactly the matrices that commute with the magic unitary defining the quantum group we furthermore show that quantum isomorphic graphs admit an isomorphism of their quantum orbital algebras which maps the adjacency matrix of one graph to that of the other

on the positive mass theorem for manifolds with corners we study the positive mass theorem for certain non smooth metrics following p miao s work our approach is to smooth the metric using the ricci flow as well as improving some previous results on the behaviour of the adm mass under the ricci flow we extend the analysis of the zero mass case to higher dimensions

on the singular second order elliptic equation abstract let m g be a smooth and compact riemannian manifold with dimensions n 3 this paper deals with the existence of nontrivial solutions for a nonlinear second order elliptic equation with a singular term and critical sobolev growth we provide a geometric application of this singular equation

an obata singular theorem for stratified spaces consider a stratified space with a positive ricci lower bound on the regular set and no cone angle larger than 2 for such stratified space we know that the first non zero eigenvalue of the laplacian is larger than or equal to the dimension we prove here an obata rigidity result when the equality is attained the lower bound of the spectrum is attained if and only if the stratified space is isometric to a spherical suspension moreover we show that the diameter is at most equal to and it is equivalent for the diameter to be equal to and for the first non zero eigenvalue of the laplacian to be equal to the dimension we finally give a consequence of these results related to the yamabe problem consider an einstein stratified space without cone angles larger than 2 if there is a metric conformal to the einstein metric and with constant scalar curvature then it is an einstein metric as well furthermore if its conformal factor is not a constant then the space is isometric to a spherical suspension

the morita theory of quantum graph isomorphisms we classify instances of quantum pseudo telepathy in the graph isomorphism game exploiting the recently discovered connection between quantum information and the theory of quantum automorphism groups specifically we show that graphs quantum isomorphic to a given graph are in bijective correspondence with morita equivalence classes of certain frobenius algebras in the category of finite dimensional representations of the quantum automorphism algebra of that graph we show that such a frobenius algebra may be constructed from a central type subgroup of the classical automorphism group whose action on the graph has coisotropic vertex stabilisers in particular if the original graph has no quantum symmetries quantum isomorphic graphs are classified by such subgroups we show that all quantum isomorphic graph pairs corresponding to a well known family of binary constraint systems arise from this group theoretical construction we use our classification to show that of the small order vertex transitive graphs with no quantum symmetry none is quantum isomorphic to a non isomorphic graph we show that this is in fact asymptotically almost surely true of all graphs

conformal structures of static vacuum data in the cauchy problem for asymptotically flat vacuum data the solution jets along the cylinder at space like infinity develop in general logarithmic singularities at the critical sets at which the cylinder touches future past null infinity the tendency of these singularities to spread along the null generators of null infinity obstructs the development of a smooth conformal structure at null infinity for the solution jets arising from time reflection symmetric data to extend smoothly to the critical sets it is necessary that the cotton tensor of the initial three metric h satisfies a certain conformally invariant condition at space like infinity it is sufficient that h be asymptotically static at space like infinity the purpose of this article is to characterize the gap between these conditions we show that with the class of metrics which satisfy condition on the cotton tensor and a certain non degeneracy requirement is associated a one form with conformally invariant differential d we provide two criteria if h is real analytic is closed and one of its integrals satisfies a certain equation then h is conformal to static data near space like infinity if h is smooth is asymptotically closed and one of its integrals satisfies a certain equation asymptotically then h is asymptotically conformal to static data at space like infinity

signal reconstruction from fourier transform sign information in this paper we present new results on the reconstruction of signals from only the sign of the real part of the fourier transform specifically we develop new theoretical results which state conditions under which two dimensional signals are uniquely specified to within a scale factor with this information and show that these conditions include a broad class of signals furthermore we apply this result to the problem of reconstructing two dimensional signals from their zero crossings we also present two algorithms for reconstructing a signal from sign information in either the time or frequency domain