eigenfunctions of the laplacian on rotationally symmetric manifolds eigenfunctions of the laplacian on a negatively curved rotationally symmetric manifold m rn ds2 ds2 dr2 f r 2d 2 are constructed explicitly under the assumption that an integral of f r converges this integral is the same one which gives the existence of nonconstant harmonic functions on m

a spectral multiplier theorem for a sublaplacian on su 2 we prove a hormander type spectral multiplier theorem for a sublaplacian on su 2 with critical index determined by the euclidean dimension of the group this result is the analogue for su 2 of the result for the heisenberg group obtained by d muller and e m stein and by w hebisch

controllability cost of conservative systems resolvent condition and transmutation abstract this article concerns the exact controllability of unitary groups on hilbert spaces with unbounded control operator it provides a necessary and sufficient condition not involving time which blends a resolvent estimate and an observability inequality by the transmutation of controls in some time l for the corresponding second order conservative system it is proved that the cost of controls in time t for the unitary group grows at most like exp l 2 t as t tends to 0 in the application to the cost of fast controls for the schrodinger equation l is the length of the longest ray of geometric optics which does not intersect the control region this article also provides observability resolvent estimates implying fast smoothing effect controllability at low cost and underscores that the controllability cost of a system is not changed by taking its tensor product with a conservative system

a poincare inequality on loop spaces we show that the laplacian on the loop space over a class of riemannian manifolds has a spectral gap the laplacian is defined using the levi civita connection the brownian bridge measure and the standard bismut tangent spaces

semiclassical asymptotics gauge fields and quantum chaos a gauge field is given by a connection on a principal bundle p m we consider the semiclassical behavior of a family of schrodinger operators associated with a gauge field in the limit as h 0 we relate the spectral theory of such operators to behavior of the hamiltonian flow on the natural phase space associated to a gauge field examining in particular situations where this flow exhibits chaotic behavior

discretization of riemannian manifolds applied to the hodge laplacian for kappa geq 0 and r 0 0 let bbb m n kappa r 0 be the set of all connected compact n dimensional riemannian manifolds such that k g leq kappa and inj m g geq r 0 we study the relation between the k rm th positive eigenvalue of the hodge laplacian on differential forms and the k rm th positive eigenvalue of the combinatorial laplacian associated to an open cover acting on v c ech cochains we show that for a fixed sufficiently small varepsilon 0 there exist positive constants c 1 and c 2 depending only on n kappa and varepsilon such that for any m in bbb m n kappa r 0 and for any varepsilon discretization x of m we have c 1 lambda k p x leq lambda k p m leq c 2 lambda k p x for any k leq k k depends on x moreover we find a lower bound for the spectrum of the combinatorial laplacian and a lower bound for the spectrum of the hodge laplacian

collapsing irreducible 3 manifolds with nontrivial fundamental group let m be a closed orientable irreducible non simply connected 3 manifold we prove that if m admits a sequence of riemannian metrics which volume collapses and whose sectional curvature is locally controlled then m is a graph manifold this is the last step in perelman s proof of thurston s geometrisation conjecture

role of sparsity and structure in the optimization landscape of non convex matrix sensing in this work we study the optimization landscape of the non convex matrix sensing problem that is known to have many local minima in the worst case since the existing results are related to the notion of restricted isometry property rip that cannot directly capture the underlying structure of a given problem they can hardly be applied to real world problems where the amount of data is not exorbitantly high to address this issue we develop the notion of kernel structure property to obtain necessary and sufficient conditions for the inexistence of spurious local solutions for any class of matrix sensing problems over a given search space this notion precisely captures the underlying sparsity and structure of the problem based on tools in conic optimization we simplify the conditions for a certain class of problems to show their satisfaction and apply them to data analytics for power systems

keller osserman conditions for diffusion type operators on riemannian manifolds abstract in this paper we obtain essentially sharp generalized keller osserman conditions for wide classes of differential inequalities of the form l u b x f u l u and l u b x f u l u g u h u on weighted riemannian manifolds where l is a non linear diffusion type operator prototypical examples of these operators are the p laplacian and the mean curvature operator the geometry of the underlying manifold is reflected via bounds for the modified bakry emery ricci curvature by growth conditions for the functions b and l a weak maximum principle which extends and improves previous results valid for the laplacian is also obtained geometric comparison results valid even in the case of integral bounds for the modified bakry emery ricci tensor are presented

phase retrieval via wirtinger flow theory and algorithms we study the problem of recovering the phase from magnitude measurements specifically we wish to reconstruct a complex valued signal boldsymbol x in mathbb c n about which we have phaseless samples of the form y r left langle boldsymbol a r boldsymbol x rangle right 2 r 1 ldots m knowledge of the phase of these samples would yield a linear system this paper develops a nonconvex formulation of the phase retrieval problem as well as a concrete solution algorithm in a nutshell this algorithm starts with a careful initialization obtained by means of a spectral method and then refines this initial estimate by iteratively applying novel update rules which have low computational complexity much like in a gradient descent scheme the main contribution is that this algorithm is shown to rigorously allow the exact retrieval of phase information from a nearly minimal number of random measurements indeed the sequence of successive iterates provably converges to the solution at a geometric rate so that the proposed scheme is efficient both in terms of computational and data resources in theory a variation on this scheme leads to a near linear time algorithm for a physically realizable model based on coded diffraction patterns we illustrate the effectiveness of our methods with various experiments on image data underlying our analysis are insights for the analysis of nonconvex optimization schemes that may have implications for computational problems beyond phase retrieval

collapsing with a lower bound on the curvature operator cheeger and gromov showed that f structures are related to collapse with a double sided curvature bound we define fibered f structures and extend some of the cheeger gromov results to the setting of collapse with a lower bound on the curvature operator

ollivier ricci curvature for general graph laplacians heat equation laplacian comparison non explosion and diameter bounds abstract discrete time random walks on a finite set naturally translate via a one to one correspondence to discrete laplace operators typically ollivier curvature has been investigated via random walks we first extend the definition of ollivier curvature to general weighted graphs and then give a strikingly simple representation of ollivier curvature using the graph laplacian using the laplacian as a generator of a continuous time markov chain we connect ollivier curvature with the heat equation which is strongly related to continuous time random walks in particular we prove that a lower bound on the ollivier curvature is equivalent to a certain lipschitz decay of solutions to the heat equation this is a discrete analogue to a celebrated ricci curvature lower bound characterization by renesse and sturm our representation of ollivier curvature via the laplacian allows us to deduce a laplacian comparison principle by which we prove non explosion and improved diameter bounds

on the upper estimate of the heat kernel of a complete riemannian manifold let m be a complete non compact riemannian manifold whose sectional curvature is bounded between two constants k and k then one expects that the heat diffusion in such a manifold behaves like the heat diffusion in euclidean space the purpose of this paper is to give a justification of such a statement in 5 j cheeger and the third author found a lower estimate of the heat kernel by comparing it with the heat kernel of the space form whose curvature is the lower bound of the curvature of the manifold this lower estimate is sharp if we insist the dependence should be on the lower bound of the ricci curvature alone it remains to give an upper estimate of the heat kernel one does not however expect to have a comparison theorem for the upper bound because it is more sensitive to the geometry of the manifold in fact the heat kernel of the upper hialf space and the heat kernel of the complete hyperbolic manifold with finite volume have quite different behavior this is reflected by the fact that the laplacian has no discrete spectrum in the first case while infinite number of discrete eigenvalues may exist in the latter case what we will prove here is that in any case the heat kernel has to decay in a manner similar to the euclidean heat kernel thus we will prove that for any constant c 4 there exists c1 depending on c t the bound of the curvature of m and x so that for all t e 0 t

gaussian upper bounds for the heat kernels of some second order operators on riemannian manifolds abstract we describe a method of obtaining gaussian upper bounds on heat kernels which unifies and improves recent results for hypoelliptic operators in divergence form and for laplace beltrami operators on complete riemannian manifolds

essential spectrum and heat kernel abstract upper bounds are derived for the large time behavior of the heat kernel on riemannian manifolds whose essential spectrum has a positive lower bound applications include a relative signature theorem and the construction of bounded harmonic functions

existence and non existence of area minimizing hypersurfaces in manifolds of non negative ricci curvature we study minimal hypersurfaces in manifolds of non negative ricci curvature euclidean volume growth and quadratic curvature decay at infinity by comparison with capped spherical cones we identify a precise borderline for the ricci curvature decay above this value no complete area minimizing hypersurfaces exist below this value in contrast we construct examples

heat equation derivative formulas for vector bundles we use martingale methods to give bismut type derivative formulas for differentials and co differentials of heat semigroups on forms and more generally for sections of vector bundles the formulas are mainly in terms of weitzenbock curvature terms in most cases derivatives of the curvature are not involved in particular our results improve b k driver s formula in 1997 j math pures appl 9 76 703 737 for logarithmic derivatives of the heat kernel measure on a riemannian manifold our formulas also include the formulas of k d elworthy and x m li 1998 c r acad sci paris ser i math 327 87 92

integrated harnack inequalities on lie groups we show that the logarithmic derivatives of the convolution heat kernels on a uni modular lie group are exponentially integrable this result is then used to prove an integrated harnack inequality for these heat kernels it is shown that this integrated harnack inequality is equivalent to a version of wang s harnack inequality a key feature of all of these inequalities is that they are dimension independent finally we show these inequalities imply quasi invariance properties of heat kernel measures for two classes of infinite dimensional lie groups

the e 2 a sections of mathrm sl 2 a in this paper we characterize the normal subgroups of sl 2 a for commutative rings a in a class which includes the arithmetic dedekind domains with infinitely many units and all rings satisfying the first stable range condition the characterization works as follows given an ideal j of a a group u of units in a j and an additive subgroup p of a closed under certain operations a structure we have dubbed a radix there is a canonically defined normal subgroup g j u p of sl 2 a these subgroups are generic in the sense that the commutators sl 2 a g j u p are all of the commutators of sl 2 a with normal subgroups of sl 2 a and a subgroup n is normal in sl 2 a if and only if sl 2 a g j u p c n c g j u p for one of these generic groups this is proved in the main theorem 3 1 3 the proof is accomplished by using results of bass milnor and serre 4 vaserstein 37 and liehl 17 in conjunction with the theory developed in sections 1 and 2 also invaluable is a refinement of the method of reduction introduced in 8 and 9 it appears in the proof of theorem 3 1 2 and is what makes it possible to reduce the question for arithmetic dedekind domains to the case of proper homomorphic images all of which are finite rings and satisfy the

heat kernel on smooth metric measure spaces with nonnegative curvature we derive a local gaussian upper bound for the f heat kernel on complete smooth metric measure space m g e f dv with nonnegative bakry emery ricci curvature as applications we obtain a sharp l f 1 liouville theorem for f subharmonic functions and an l f 1 uniqueness property for nonnegative solutions of the f heat equation assuming f is of at most quadratic growth in particular any l f 1 integrable f subharmonic function on gradient shrinking and steady ricci solitons must be constant we also provide explicit f heat kernel for gaussian solitons

eigenvalues estimates on complete noncompact riemannian manifolds and applications we obtain some sharp estimates on the first eigenvalues of complete noncom pact riemannian manifolds under assumptions of volume growth using these estimates we study hypersurfaces with constant mean curvature and give some estimates on the mean curvatures

good wannier bases in hilbert modules associated to topological insulators for a large class of physically relevant operators on a manifold with discrete group action we prove general results on the non existence of a basis of well localized wannier functions for their spectral subspaces this turns out to be equivalent to the freeness of a certain hilbert module over the group c algebra canonically associated with the spectral subspace this brings into play k theoretic methods and justifies their importance as invariants of topological insulators in physics

on poincare type inequalities using estimates of the heat kernel we prove a poincare inequality for star shape domains on a complete manifold the method also gives a lower bound for the gap of the first two neumann eigenvalues of a schrodinger operator

minimizing effective resistance of a graph the effective resistance between two nodes of a weighted graph is the electrical resistance seen between the nodes of a resistor network with branch conductances given by the edge weights the effective resistance comes up in many applications and fields in addition to electrical network analysis including for example markov chains and continuous time averaging networks in this paper we study the problem of allocating edge weights on a given graph in order to minimize the total effective resistance i e the sum of the resistances between all pairs of nodes we show that this is a convex optimization problem and can be solved efficiently either numerically or in some cases analytically we show that optimal allocation of the edge weights can reduce the total effective resistance of the graph compared to uniform weights by a factor that grows unboundedly with the size of the graph we show that among all graphs with n nodes the path has the largest value of optimal total effective resistance and the complete graph has the least

heat extensions optimal atomic decompositions and sobolev embeddings in presence of symmetries on manifolds our purpose is twofold we give the heat semi group characterization of spaces of besov and triebel lizorkin types defined on riemannian manifolds of bounded geometry moreover we study boundedness and compactness of sobolev embeddings of the spaces in presence of symmetries to make it possible we first construct an optimal atomic decomposition for the above function spaces

local index theory over etale groupoids we give a superconnection proof of connes index theorem for proper cocompact actions of etale groupoids this includes connes general foliation index theo rem for foliations with hausdorholonomy groupoid

matrix completion from noisy entries given a matrix m of low rank we consider the problem of reconstructing it from noisy observations of a small random subset of its entries the problem arises in a variety of applications from collaborative filtering the netflix problem to structure from motion and positioning we study a low complexity algorithm introduced by keshavan montanari and oh 2010 based on a combination of spectral techniques and manifold optimization that we call here optspace we prove performance guarantees that are order optimal in a number of circumstances

riesz transforms associated to schrodinger operators on weighted hardy spaces let w be some ap weight and enjoy reverse holder inequality and let l v be a schrodinger operator on rn where v lloc1 rn is a non negative function on rn in this article we introduce weighted hardy spaces hl w1 rn associated to l in terms of the area function characterization and prove their atomic characters we show that the riesz transform l 1 2 associated to l is bounded on lwp rn for 1 p 2 and bounded from hl w1 rn to the classical weighted hardy space hw1 rn

the longitudinal cocycle and the index of toeplitz operators abstract let d be a self adjoint leafwise elliptic operator on a foliated manifold compressing multiplication operators to the range of the positive spectral projection for d yields the class of leafwise toeplitz operators the extension generated by these operators is constructed a topological formula for the index of a toeplitz operator with invertible symbol is given this index can also be obtained by pairing the k theory class of the symbol with a certain cyclic cocycle if one lifts an elliptic operator on a closed manifold to a leafwise elliptic operator on an associated flat foliated principal bundle then this cocycle can be used to obtain refined invariants of the original operator

matrix completion from a few entries let m be an n n matrix of rank r and assume that a uniformly random subset e of its entries is observed we describe an efficient algorithm which we call optspace that reconstructs m from e o rn observed entries with relative root mean square error 1 2 rmse c nr e 1 2 with probability larger than 1 1 n3 further if r o 1 and m is sufficiently unstructured then optspace reconstructs it exactly from e o n log n entries with probability larger than 1 1 n3 this settles in the case of bounded rank a question left open by candes and recht and improves over the guarantees for their reconstruction algorithm the complexity of our algorithm is o e r log n which opens the way to its use for massive data sets in the process of proving these statements we obtain a generalization of a celebrated result by friedman kahn szemeredi and feige ofek on the spectrum of sparse random matrices

twistor spaces and the general adiabatic expansions abstract we investigate the behavior of derivatives of the fundamental solution of a parabolic equation for the square of dirac operator on a twistor space when the metric is blown up in the base space direction such a blowing up operation is expected to be an effective method for extracting some intrinsic values from various geometric invariants most of whose cores consist of some derivatives of the fundamental solution

strong p completeness of stochastic differential equations and the existence of smooth flows on noncompact manifolds here we discuss the regularity of solutions of sde s and obtain conditions under which a sde on a complete riemannian manifoldm has a global smooth solution flow in particular improving the usual global lipschitz hypothesis whenm r n there are also results on non explosion of diffusions

stochastic differential equations on noncompact manifolds moment stability and its topological consequences in this paper we discuss the stability of stochastic differential equations and the interplay between the moment stability of a sde and the topology of the underlying manifold sufficient and necessary conditions are given for the moment stability of a sde in terms of the coefficients finally we prove a vanishing result for the fundamental group of a complete riemannian manifold in terms of purely geometrical quantities

bar partial torsion for complex manifolds and the adiabatic limit torsion for complex manifolds and the adiabatic limit we consider a complex fibration f m l b and pull back bundles e 1 and e 2 over m using the adiabatic limit idea we compute the metric invariant tp e1 tp e2 where tp e denotes the complex ray singer torsion

a newton based method for nonconvex optimization with fast evasion of saddle points machine learning problems such as neural network training tensor decomposition and matrix factorization require local minimization of a nonconvex function this local minimization is challenged b

sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative ricci curvature in this paper we consider complete noncompact riemannian manifolds m g with nonnegative ricci curvature and euclidean volume growth of dimension n ge 3 for every bounded open subset omega subset m with smooth boundary we prove that begin aligned int limits partial omega left frac mathrm h n 1 right n 1 mathrm d sigma ge mathrm avr g big mathbb s n 1 big end aligned where mathrm h is the mean curvature of partial omega and mathrm avr g is the asymptotic volume ratio of m g moreover the equality holds true if and only if m setminus omega g is isometric to a truncated cone over partial omega an optimal version of huisken s isoperimetric inequality for 3 manifolds is obtained using this result finally exploiting a natural extension of our techniques to the case of parabolic manifolds we also deduce an enhanced version of kasue s non existence result for closed minimal hypersurfaces in manifolds with nonnegative ricci curvature

symplectic commutator subgroups abstract this paper expands on the work of douglas costa and gordon keller costa and keller used a structure called the commutator subgroup to find the normal subgroups of special linear groups and symplectic groups it is often difficult to determine which matrices are in the commutator subgroup they create a homomorphism on the special linear group which provides us with a formula on the entries of a matrix in that group in order to tell us whether or not a matrix is in the commutator subgroup this information is not only useful in finding normal subgroups but in studying power residue symbols this paper creates a homomorphism on the symplectic group sp 2 m 2 a for a commutative ring a that provides us with a formula on the entries of an element of that group which tells us whether or not the matrix is in the commutator subgroup this homomorphism is also extended to one on ep 2 m 2 a s the elementary group generated by elementaries whose off daigonal entries come from s for s a particular kind of ideal and jordan ideal

extremal of log sobolev inequality and w entropy on noncompact manifolds abstract let m be a complete connected noncompact manifold with bounded geometry under a condition near infinity we prove that the log sobolev functional 1 1 has an extremal function decaying exponentially near infinity we also prove that an extremal function may not exist if the condition is violated this result has the following consequences 1 it seems to give the first example of connected complete manifolds with bounded geometry where a standard log sobolev inequality does not have an extremal 2 it gives a negative answer to the open question on the existence of extremal of perelman s w entropy in the noncompact case which was stipulated by perelman 2002 22 p 9 3 2 remark 3 it helps to prove in some cases that noncompact shrinking breathers of ricci flow are gradient shrinking solitons

solving semidefinite quadratic linear programs using sdpt3 this paper discusses computational experiments with linear optimization problems involving semidefinite quadratic and linear cone constraints sqlps many test problems of this type are solved using a new release of sdpt3 a matlab implementation of infeasible primal dual path following algorithms the software developed by the authors uses mehrotra type predictor corrector variants of interior point methods and two types of search directions the hkm and nt directions a discussion of implementation details is provided and computational results on problems from the sdplib and dimacs challenge collections are reported

surfaces with boundary their uniformizations determinants of laplacians and isospectrality let be a compact surface of type g n n 0 obtained by re moving n disjoint disks from a closed surface of genus g assuming 0 while osgood phillips and sarnak o p s3 showed the properness when g 0

space of ricci flows i based on the compactness of the moduli of non collapsed calabi yau spaces with mild singularities we set up a structure theory for polarized kahler ricci flows with proper geo metric bounds our theory is a generalization of the structure theory of non collapsed kahler einstein manifolds as applications we prove the hamilton tian conjecture and the partial c 0 conjecture of tian

plancherel type estimates and sharp spectral multipliers abstract we study general spectral multiplier theorems for self adjoint positive definite operators on l2 x where x is any open subset of a space of homogeneous type we show that the sharp hormander type spectral multiplier theorems follow from the appropriate estimates of the l2 norm of the kernel of spectral multipliers and the gaussian bounds for the corresponding heat kernel the sharp hormander type spectral multiplier theorems are motivated and connected with sharp estimates for the critical exponent for the riesz means summability which we also study here we discuss several examples which include sharp spectral multiplier theorems for a class of scattering operators on r3 and new spectral multiplier theorems for the laguerre and hermite expansions

properties of the probability density function of the non central chi squared distribution abstract in this paper we consider the probability density function pdf of a non central 2 distribution with arbitrary number of degrees of freedom for this function we prove that can be represented as a finite sum and we deduce a partial derivative formula moreover we show that the pdf is log concave when the degrees of freedom is greater or equal than 2 at the end of this paper we present some turan type inequalities for this function and an elegant application of the monotone form of l hospital s rule in probability theory is given

boundedness of spectral multipliers of generalized laplacians on compact manifolds with boundary consider a second order strongly elliptic negative semidefinite differential operator l may be a system on a compact riemannian manifold overline m with smooth boundary where the domain of l is defined by a coercive boundary condition classically known results and also recent work in duong et al j funct anal 196 443 485 2002 and duong and mcintosh rev math iberoam 15 233 265 1999 establish sufficient conditions for l infty text bmo l continuity of varphi sqrt l where varphi in s 0 1 mathbb r and l is a suitable elliptic operator using a variant of the cheeger gromov taylor functional calculus due to mauceri et al math res lett 16 861 879 2009 and short time upper bounds on the integral kernel of e tl due to greiner arch ration mech anal 41 168 218 1971 we prove that a variant of such sufficient conditions holds for our operator l

the atomic decomposition of weighted hardy spaces associated to self adjoint operators on product spaces abstract let l be a non negative self adjoint operator acting on l 2 r n satisfying a pointwise gaussian estimate for its heat kernel let w be an a p weight on r n r n 1 p in this article we obtain a weighted q atomic decomposition with q p for the weighted hardy space h l w 1 r n r n associated to l

well posedness and regularity for the elasticity equation with mixed boundary conditions on polyhedral domains and domains with cracks we prove a regularity result for the anisotropic linear elasticity equation p u rm div left boldmath mathsf c cdot nabla u right f with mixed displacement and traction boundary conditions on a curved polyhedral domain omega subset mathbb r 3 in weighted sobolev spaces mathcal k m 1 a 1 omega for which the weight is given by the distance to the set of edges in particular we show that there is no loss of mathcal k m a regularity our curved polyhedral domains are allowed to have cracks we establish a well posedness result when there are no neighboring traction boundary conditions and a 0 that depends on p on the boundary conditions and on the domain our results extend to other strongly elliptic systems and higher dimensions

gradient descent with random initialization fast global convergence for nonconvex phase retrieval this paper considers the problem of solving systems of quadratic equations namely recovering an object of interest varvec x natural in mathbb r n from m quadratic equations samples y i varvec a i top varvec x natural 2 1 le i le m this problem also dubbed as phase retrieval spans multiple domains including physical sciences and machine learning we investigate the efficacy of gradient descent or wirtinger flow designed for the nonconvex least squares problem we prove that under gaussian designs gradient descent when randomly initialized yields an epsilon accurate solution in o big log n log 1 epsilon big iterations given nearly minimal samples thus achieving near optimal computational and sample complexities at once this provides the first global convergence guarantee concerning vanilla gradient descent for phase retrieval without the need of i carefully designed initialization ii sample splitting or iii sophisticated saddle point escaping schemes all of these are achieved by exploiting the statistical models in analyzing optimization algorithms via a leave one out approach that enables the decoupling of certain statistical dependency between the gradient descent iterates and the data

uniqueness of l 1 solutions for the laplace equation and the heat equation on riemannian manifolds on deduit une hypothese geometrique optimale sur une variete de riemann complete qui assure des proprietes d unicite pour des solutions l 1 de l equation de laplace et de la chaleur

low rank phase retrieval we develop two iterative algorithms for solving the low rank phase retrieval lrpr problem lrpr refers to recovering a low rank matrix boldsymbol x from magnitude only phaseless measurements of random linear projections of its columns both methods consist of a spectral initialization step followed by an iterative algorithm to maximize the observed data likelihood we obtain sample complexity bounds for our proposed initialization approach to provide a good approximation of the true boldsymbol x when the rank is low enough these bounds are significantly lower than what existing single vector phase retrieval algorithms need via extensive experiments we show that the same is also true for the proposed complete algorithms