a cotlar type maximal function associated with fourier multipliers abstract we prove the l p boundedness of a maximal operator associated with a dyadic frequency decomposition of a fourier multiplier under a weak regularity assumption

a new method optimizing the subgraph centrality of large networks abstract since many realistic networks such as wireless sensor ad hoc networks usually do not agree very well with the basic network models such as small word and scale free models we often need to obtain some expected structural features such as a small average path length and a regular degree distribution while optimizing the connectivity of these networks although a minor addition of links for optimizing network connectivity is not likely to change the structural properties of a network it is necessary to investigate the impact of link addition on network properties as the number of the added links increases however to the best of our knowledge the study of that problem has not been found so far furthermore two closely related questions to that problem i e how to measure and how to improve network connectivity have not been studied carefully enough yet to address the three problems above the authors derive a better measure of network connectivity for large networks and a new strategy that can increase decrease network connectivity the most and propose a spectral density algorithm optimizing the connectivity of large networks which is able to indicate the impact on the structural properties of a network while increasing decreasing its connectivity providing us a guided optimization of network connectivity in other words our algorithm can optimize not only the connectivity of a large network but also its structural features meanwhile our new findings about spectral density are also concluded in this paper in addition we may also apply this algorithm to solve all eigenvalues of an n n matrix with a low complexity of o n 2 at most

decreasing the spectral radius of a graph by link removals the decrease of the spectral radius an important characterizer of network dynamics by removing links is investigated the minimization of the spectral radius by removing m links is shown to be an np complete problem which suggests considering heuristic strategies several greedy strategies are compared and several bounds on the decrease of the spectral radius are derived the strategy that removes that link l i j with largest product x1 i x1 j of the components of the eigenvector x1 belonging to the largest adjacency eigenvalue is shown to be superior to other strategies in most cases furthermore a scaling law where the decrease in spectral radius is inversely proportional to the number of nodes n in the graph is deduced another sublinear scaling law of the decrease in spectral radius versus the number m of removed links is conjectured

subvarieties of suc 2 and 2theta divisors in the jacobian let suc 2 l denote the projective moduli variety of semistable rank 2 vector bundles with determinant l pic c on a smooth curve c of genus g 2 and suppose that deg l is even it is well known that on the one hand the singular locus of suc 2 l is isomorphic to the kummer variety of the jacobian and on the other hand that when c is nonhyperelliptic suc 2 o has an injective morphism into the linear series 2 on the jacobian jg 1 c which restricts to the kummer embedding a 7 a a on the singular locus dually suc 2 k injects into the linear series l on j c where l o 2 for any theta characteristic and again this map restricts to the kummer map jg 1 c 2 l on the singular locus this map to projective space the two cases are of course isomorphic comes from the complete series on the ample generator of the picard group and at least for a generic curve is an embedding of the moduli space moreover its image contains much of the geometry studied in connection with the schottky problem notably the configuration of prym kummer varieties in this paper we explore a little of the interplay via this embedding between the geometry of vector bundles and the geometry of 2 divisors on the vector bundle side we are principally concerned with the brill noether loci wr suc 2 k defined by the condition h e r on stable bundes e these are analogous to the very classical varieties w r g 1 jg 1 c unlike the line bundle theory however general results connectedness dimension smoothness and so on are not known for the varieties wr see 6 on the 2 side we shall consider the fay trisecants of the 2 embedded kummer variety and the subseries p 00 l consisting of divisors having multiplicity 4 at the origin this subseries is known to be important in the study of principally polarised abelian varieties 10 in the jacobian of a curve its base locus is the surface c c j c plus a pair of isolated points

synchronizability in complex ad hoc dynamical networks with accelerated growth recent research works have been pursued in connection with network synchronizability under various constraints for different topological structures and evolving mechanisms however the fundamental question of how the synchronizability of the networks relates to accelerated growth and ad hoc property in the evolving processes remains underexplored here we study the ad hoc dynamical models with accelerated growth where the total number of edges increases faster than linearly with network size by adopting three attachment mechanisms random rewired and preferential attachment we investigate the second largest eigenvalues and the ratios of the extremal eigenvalues of coupled matrices in the accelerated models with different evolving parameters for the new models we demonstrate the robustness and fragility of synchronization against random and specific attacks by numerical simulations we find that accelerated growth represents a convenient tool for improving the synchronizability of an evolving network furthermore we show that not only network synchronization but also topological periodicity has robustness and fragility against random failures and specifical removal of the nodes respectively in particular when ad hoc property is suggested in the evolving networks we find that the deletion of nodes is easier to change network synchronizability and robustness compared to the addition of nodes

enhancing the spectral gap of networks by node removal dynamics on networks are often characterized by the second smallest eigenvalue of the laplacian matrix of the network which is called the spectral gap examples include the threshold coupling strength for synchronization and the relaxation time of a random walk a large spectral gap is usually associated with high network performance such as facilitated synchronization and rapid convergence in this study we seek to enhance the spectral gap of undirected and unweighted networks by removing nodes because practically the removal of nodes often costs less than the addition of nodes addition of links and rewiring of links in particular we develop a perturbative method to achieve this goal the proposed method realizes better performance than other heuristic methods on various model and real networks the spectral gap increases as we remove up to half the nodes in most of these networks

enhancing synchronizability of diffusively coupled dynamical networks a survey in this paper we review the literature on enhancing synchronizability of diffusively coupled dynamical networks with identical nodes the last decade has witnessed intensive investigations on the collective behavior over complex networks and synchronization of dynamical systems is the most common form of collective behavior for many applications it is desired that the synchronizability the ability of networks in synchronizing activity of their individual dynamical units is enhanced there are a number of methods for improving the synchronization properties of dynamical networks through structural perturbation in this paper we survey such methods including adding removing nodes and or edges rewiring the links and graph weighting these methods often try to enhance the synchronizability through minimizing the eigenratio of the laplacian matrix of the connection graph a synchronizability measure based on the master stability function formalism we also assess the performance of the methods by numerical simulations on a number of real world networks as well as those generated through models such as preferential attachment watts strogatz and erdos renyi

on the consequences of a mihlin hormander functional calculus maximal and square function estimates we prove that the existence of a mihlin hormander functional calculus for an operator l implies the boundedness on l p of both the maximal multiplier operators and the continuous square functions build on spectral multipliers of l the considered multiplier functions are finitely smooth and satisfy an integral condition at infinity in particular multipliers of compact support are admitted

a conic bundle degenerating on the kummer surface let c be a genus 2 curve and mathcal su c 2 the moduli space of semi stable rank 2 vector bundles on c with trivial determinant in bolognesi adv geom 7 1 113 144 2007 we described the parameter space of non stable extension classes of the canonical sheaf of c by 1 in this paper we study the classifying rational map varphi mathbb p ext 1 omega omega 1 cong mathbb p 4 dashrightarrow mathcal su c 2 cong mathbb p 3 that sends an extension class to the corresponding rank two vector bundle moreover we prove that if we blow up mathbb p 4 along a certain cubic surface s and mathcal su c 2 at the point p corresponding to the bundle mathcal o oplus mathcal o then the induced morphism tilde varphi bl s rightarrow bl p mathcal su c 2 defines a conic bundle that degenerates on the blow up at p of the kummer surface naturally contained in mathcal su c 2 furthermore we construct the mathbb p 2 bundle that contains the conic bundle and we discuss the stability and deformations of one of its components

gromov invariants for holomorphic maps from riemann surfaces to grassmannians two compactifications of the space of holomorphic maps of fixed degree from a compact riemann surface to a grassmannian are studied it is shown that the uhlenbeck compactification has the structure of a projective variety and is dominated by the algebraic compactification coming from the grothendieck quot scheme the latter may be embedded into the moduli space of solutions to a generalized version of the vortex equations studied by bradlow this gives an effective way of computing certain intersection numbers known as gromov invariants on the space of holomorphic maps into grassmannians we carry out these computations in the case where the riemann surface has genus one department of mathematics university of utah salt lake city utah 84112 e mail address bertram math utah edu department of mathematics brown university providence rhode island 02912 e mail address daskal gauss math brown edu department of mathematics university of california irvine california 92717 e mail address raw math uci edu license or copyright restrictions may apply to redistribution see http www ams org journal terms of use

on the ideals of secant varieties to certain rational varieties abstract if x p n is a reduced and irreducible projective variety it is interesting to find the equations describing the higher secant varieties of x in this paper we find those equations in the following cases x p n 1 p n t p n is the segre embedding of the product and n is large with respect to the n i theorem 2 4 the x are some unbalanced segre veronese embeddings x is a del pezzo surface

regularity and normality of the secant variety to a projective curve abstract for a smooth curve of genus g embedded by a line bundle of degree at least 2 g 3 we show that the ideal sheaf of the secant variety is 5 regular this bound is sharp with respect to both the degree of the embedding and the bound on the regularity further we show that the secant variety is projectively normal for the generic embedding of degree at least 2 g 3

the generic green lazarsfeld secant conjecture using lattice theory on special k3 surfaces calculations on moduli stacks of pointed curves and voisin s proof of green s conjecture on syzygies of canonical curves we prove the prym green conjecture on the naturality of the resolution of a general prym canonical curve of odd genus as well as many cases of the green lazarsfeld secant conjecture on syzygies of non special line bundles on general curves

koszul homology and syzygies of veronese subalgebras a graded k algebra r has property np if it is generated in degree 1 has relations in degree 2 and the syzygies of order p on the relations are linear the green lazarsfeld index of r is the largest p such that it satisfies the property np our main results assert that under a mild assumption on the base field the cth veronese subring of a polynomial ring has green lazarsfeld index c 1 the same conclusion also holds for an arbitrary standard graded algebra provided c gg 0

secant varieties and birational geometry we show how to use information about the equations defining secant varieties to smooth projective varieties in order to construct a natural collection of birational transformations these were first constructed as flips in the case of curves by m thaddeus via geometric invariant theory and the first flip in the sequence was constructed by the author for varieties of arbitrary dimension in an earlier paper we expose the finer structure of a second flip again for varieties of arbitrary dimension we also prove a result on the cubic generation of the secant variety and give some conjectures on the behavior of equations defining the higher secant varieties

singularities and syzygies of secant varieties of nonsingular projective curves in recent years the equations defining secant varieties and their syzygies have attracted considerable attention the purpose of the present paper is to conduct a thorough study on secant varieties of curves by settling several conjectures and revealing interaction between singularities and syzygies the main results assert that if the degree of the embedding line bundle of a nonsingular curve of genus g is greater than 2g 2k p for nonnegative integers k and p then the k th secant variety of the curve has normal du bois singularities is arithmetically cohen macaulay and satisfies the property n k 2 p in addition the singularities of the secant varieties are further classified according to the genus of the curve and the castelnuovo mumford regularities are also obtained as well as one of the main technical ingredients we establish a vanishing theorem on the cartesian products of the curve which may have independent interests and may find applications elsewhere

groupe de picard des varietes de modules de fibres semi stables sur les courbes algebriques soient x une courbe algebrique projective lisse de genre g 2 sur r d des entiers avec r 2 on note u r d la variete de modules des fibres algebriques semi stables sur x de rang r et de degre d us r d l ouvert de u r d correspondant aux fibres stables on sait que u r d est une variete algebrique projective irreductible et normale si r et d ne sont pas premiers entre eux u r d n est pas lisse sauf dans une exception le cas ou g 2 r 2 et d est pair on supposera dans cet article qu on n est pas dans ce cas on a alors codimu r d u r d us r d 2 et u r d us r d est le lieu des points singuliers de u r d si l est un fibre en droites de degre d sur x on note u r l la sous variete fermee de u r d correspondant aux fibres vectoriels de determinant isomorphe a l on etudie dans cet article les groupes de picard pic u r d et pic u r l on montre d abord que les varietes u r d et u r l sont localement factorielles on calcule ensuite pic u r d et pic u r l qui sont decrits au moyen des diviseurs thetas generalises

properties and applications of laplacian spectra for koch networks for a network knowledge of its laplacian eigenvalues is central to understanding its structure and dynamics in this paper we study the laplacian spectra of a family of koch networks with scale free and small world properties we derive some recursive relations between the laplacian characteristic polynomials of koch networks and their subgraphs at different iterations based on the obtained recurrence relations we determine explicitly the product of all nonzero laplacian eigenvalues as well as the sum of the reciprocals of these eigenvalues then using these results we further evaluate the number of spanning trees kirchhoff index global mean first passage time and average path length of the family of koch networks finally we determine the number of spanning forests under certain conditions we expect that our method can be adapted to other types of self similar networks

on maximal functions for mikhlin hormander multipliers abstract given mikhlin hormander multipliers m i i 1 n with uniform estimates we prove an optimal log n 1 bound in l p for the maximal function sup i f 1 m i f and related bounds for maximal functions generated by dilations these improve the results in m christ l grafakos p honzik a seeger maximal functions associated with multipliers of mikhlin hormander type math z 249 2005 223 240

on the vanishing of higher syzygies of curves ii we verify green s conjecture for a generic k gonal curve of genus g for g geq k k 1 2

on the normality of secant varieties abstract in this paper we show that the secant variety to a smooth projective variety embedded by a sufficiently positive line bundle is normal as an application we deduce that the secant variety to a general canonical curve of genus at least 7 is normal

su 2 wzw theory at higher genera we compute by free field techniques the scalar product of thesu 2 chern simons states on genus 1 surfaces the result is a finite dimensional integral over positions of screening charges and one complex modular parameter it uses an effective description of the cs states closely related to the one worked out by bertram 1 the scalar product formula allows to express the higher genus partition functions of the wzw conformal field theory by finite dimensional integrals it should provide the hermitian metric preserved by the knizhnik zamolodchikov bernard connection describing the variations of the cs states under the change of the complex structure of the surface

ideals of bounded rank symmetric tensors are generated in bounded degree we study the syzygies of secant ideals of veronese subrings of a fixed commutative graded algebra over a field of characteristic 0 one corollary is that the degrees of the minimal generators of the ith syzygy module of the coordinate ring of the rth secant variety of any veronese embedding of a projective scheme x can be bounded by a constant that only depends on i r and x and not on the choice of the veronese embedding

on syzygies of calabi yau varieties and varieties of general type abstract we establish projective normality property n p and effective very ampleness for line bundles on a calabi yau variety as well as for pluricanonical bundles of a variety of general type we also prove a result on the degrees of generators of canonical rings under the condition that the image of the canonical map has minimal degree

geometry and the complexity of matrix multiplication it has been known for some time that algebraic geometry and representation theory are useful for proving lower bounds on the complexity of the matrix multiplication operator in this talk i will explain how geometry can be used to prove both upper and lower bounds after a discussion of the problem and its history i will present very recent work with m michalek on border rank algorithms faculty host eric allender

on the projective normality of line bundles of extremal degree we propose a new method using deformation theory to study the maximal rank conjecture for line bundles of extremal degree which can be viewed as the first case to test the conjecture we prove that maximal rank conjecture holds by our new method

subadditivity of syzygies of koszul algebras estimates are obtained for the degrees of minimal syzygies of quotient algebras of polynomial rings for a class that includes koszul algebras in almost all characteristics these degrees are shown to increase by at most two from one syzygy module to the next one even slower growth is proved if in addition the algebra satisfies green and lazarsfeld s condition n q with q ge 2

on the hitchin system 1 1 what is known as the hitchin system is a completely integrable hamiltonian system cihs involving vector bundles over algebraic curves identified by hitchin in h1 h2 it was recently generalized by faltings f in this paper we only consider the case of ranktwo vector bundles with trivial determinant in that case the hitchin system corresponding to a curve c of genus g is obtained as follows let

2 1 d topological quantum field theory and 2 d conformal field theory we describe the relation between three dimensional topological quantum field theory and two dimensional conformal field theory some applications to quantum knot invariants leading to the equivalence of chern simons witten and kohno s approaches are outlined

maximal functions associated with fourier multipliers of mikhlin hormander type we show that maximal operators formed by dilations of mikhlin hormander multipliers are typically not bounded on lp d we also give rather weak conditions in terms of the decay of such multipliers under which lp boundedness of the maximal operators holds