noncommutative ricci flow in a matrix geometry we study noncommutative ricci flow in a finite dimensional representation of a noncommutative torus it is shown that the flow exists and converges to the flat metric we also consider the evolution of entropy and a definition of scalar curvature in terms of the ricci flow

runge kutta solutions of a hyperbolic conservation law with source term spurious long term solutions of a finite difference method for a hyperbolic conservation law with a general nonlinear source term are studied results are contrasted with those that have been established for nonlinear ordinary differential equations various types of spurious behavior are examined including spatially uniform equilibria that exist for arbitrarily small time steps nonsmooth steady states with profiles that jump between fixed levels and solutions with oscillations that arise from nonnormality and exist only in finite precision arithmetic it appears that spurious behavior is associated in general with insufficient spatial resolution the potential for curbing spuriosity by using adaptivity in space or time is also considered

parabolic monge ampere equations on riemannian manifolds on a compact riemannian manifold m g we consider the parabolic monge ampere equation t x t logdet g x hess x t detg x x t f x x 0 phiv 0 x here is a real parameter andf 0 m r are smooth functions we show existence of for all timestindependent of if 0 then t t converges exponentially towards a solution of the stationary problem ast in the special case 0 f 0 one has 0 and we determine the convergence rate t in thel2 norm more precisely

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

semistable bundles over an elliptic curve abstract we identify the moduli space m n d of semistable bundles of rank n and degree d on an elliptic curve c as a symmetric product of the curve its dimension being the greatest common divisor of n and d under this identification the determinant map det m n d corresponds to the abel jacobi map shc jh so that the fixed determinant moduli space is identified with a projective space we then describe the brill noether loci theta divisors and picard groups and compute the space of theta functions for both the full moduli space and the fixed determinant moduli space of any rank and any degree we extend to elliptic curves a theorem of drezet and narasimhan on the picard group of moduli spaces as well as two formuulas of beauville narasimhan and ramanan on theta functions of level one our results verify in particular for an elliptic curve a formula of e verlinde and an equality first noticed by bott and szenes relating theta functions over the full moduli space and over the fixed determinant moduli space

compactified picard stacks over the moduli stack of stable curves with marked points abstract in this paper we give a construction of algebraic artin stacks endowed with a modular map onto the moduli stack of stable curves of genus g with n marked points the stacks we construct are smooth irreducible and have dimension 4 g 3 n yielding a geometrically meaningful compactification of the universal picard stack parametrizing n pointed smooth curves together with a line bundle

the second homology group of the mapping class group of an orientable surface in i 7 mumford shows that the picard group p ic is isomorphic to h2 f 2 and conjectures the latter is rank one g 3 we prove this below for g 5 another interpretation of this theorem may be obtained by identifying h2 f as bordism classes of fiber bundles f w4 t where t is a closed oriented surface sect 0 when f is closed every such bundle is bordant to f w t a bundle admitting a section s t w the theorem then says that

poincare families of g bundles on a curve let g be a reductive group over an algebraically closed field k consider the moduli space of stable principal g bundles on a smooth projective curve c over k we give necessary and sufficient conditions for the existence of poincare bundles over open subsets of this moduli space and compute the orders of the corresponding obstruction classes this generalizes the previous results of newstead ramanan and balaji biswas nagaraj newstead to all reductive groups to all topological types of bundles and also to all characteristics

a gieseker type degeneration of moduli stacks of vector bundles on curves we construct a new degeneration of the moduli stack of vector bundles over a smooth curve when the curve degenerates to a singular curve which is irreducible with one double point we prove that the total space of the degeneration is smooth and its special fibre is a divisor with normal crossings furthermore we give a precise description of how the normalization of the special fibre of the degeneration is related to the moduli space of vector bundles over the desingularized curve

perelman s entropy functional at type i singularities of the ricci flow we study blow ups around fixed points at type i singularities of the ricci flow on closed manifolds using perelman s w functional first we give an alternative proof of the result obtained by naber 18 and enders muller topping 8 that blow up limits are non flat gradient shrinking ricci solitons our second and main result relates a limit w density at a type i singular point to the entropy of the limit gradient shrinking soliton obtained by blowing up at this point in particular we show that no entropy is lost at infinity during the blow up process

classification of manifolds with weakly 1 4 pinched curvatures we show that a compact riemannian manifold with weakly pointwise 1 4 pinched sectional curvatures is either locally symmetric or diffeomorphic to a space form more generally we classify all compact locally irreducible riemannian manifolds m with the property that m r 2 has non negative isotropic curvature

pseudospectra of the convection diffusion operator the spectrum of the simplest 1d convection diffusion operator is a discrete subset of the negative real axis but the pseudospectra are regions in the complex plane that approximate parabolas put another way the norm of the resolvent is exponentially large as a function of the peclet number throughout a certain parabolic region these observations have a simple physical basis and suggest that conventional spectral analysis for convection diffusion operators may be of limited value in some applications

a modified kahler ricci flow in this note we study a kahler ricci flow modified from the classic version in the non degenerate case strong convergence at infinite time is achieved the main focus should be on degenerate case where some partial results are presented

nonnormality effects in a discretised nonlinear reaction convection diffusion equation what is the long time effect of adding convention to a discretised reaction diffusion equation for linear problems it is well known that convection may denormalise the process and in particular eigenvalue based stability predictions may be overoptimistic this work deals with a related issue with a nonlinear reaction term the nonnormality can greatly influence the long time dynamics for a nonlinear model problem with dirichlet boundary conditions it is shown that the basin of attraction of the correct steady state can be shrunk in a directionally biased manner a normwise analysis provides lower bounds on the basin of attraction and a more revealing picture is provided by pseudo eigenvalues in extreme cases the computed solution can converge to a spurious bounded steady state that exists only in finite precision arithmetic the impact of convection on the existence and stability of spurious periodic solutions is also quantified

coupling of brownian motions and perelman s l functional abstract we show that on a manifold whose riemannian metric evolves under backwards ricci flow two brownian motions can be coupled in such a way that their normalized l distance is a supermartingale as a corollary we obtain the monotonicity of the transportation cost between two solutions of the heat equation in the case that the cost function is the composition of a concave non decreasing function and the normalized l distance in particular it provides a new proof of a recent result of topping p topping l optimal transportation for ricci flow j reine angew math 636 2009 93 122

the local structure of compactified jacobians this paper studies the local geometry of compactified jacobians the main result is a presentation of the completed local ring of the compactified jacobian of a nodal curve as an explicit ring of invariants described in terms of the dual graph of the curve the authors have investigated the geometric and combinatorial properties of these rings in previous work and consequences for compactified jacobians are presented in this paper similar results are given for the local structure of the universal compactified jacobian over the moduli space of stable curves

curvature contraction of convex hypersurfaces by nonsmooth speeds the research of the second author was supported by an australian postgraduate award the research of the first third fourth and fifth authors was supported by discovery project grant dp120100097 of the australian research council

the zero section of the universal semiabelian variety and the double ramification cycle we study the chow ring of the boundary of the partial compactification of the universal family of principally polarized abelian varieties ppav we describe the subring generated by divisor classes and we compute the class of the partial compactification of the universal zero section which turns out to lie in this subring our formula extends the results for the zero section of the universal uncompactified family the partial compactification of the universal family of ppav can be thought of as the first two boundary strata in any toroidal compactification of ag our formula provides a first step in a program to understand the chow groups of a g especially of the perfect cone compactification by induction on genus by restricting to the image of mg under the torelli map our results extend the results of hain on the double ramification cycle answering eliashberg s question

complete gradient shrinking ricci solitons with pinched curvature we prove that any n dimensional complete gradient shrinking ricci soliton with pinched weyl curvature is a finite quotient of mathbb r n mathbb r times mathbb s n 1 or mathbb s n in particular we do not need to assume the metric to be locally conformally flat

manifolds with positive curvature operators are space forms the ricci flow was introduced by hamilton in 1982 h1 in order to prove that a compact three manifold admitting a riemannian metric of positive ricci curvature is a spherical space form in dimension four hamilton showed that compact four manifolds with positive curvature operators are spherical space forms as well h2 more generally the same conclusion holds for compact four manifolds with 2 positive curvature operators che recall that a curvature operator is called 2 positive if the sum of its two smallest eigenvalues is positive in arbitrary dimensions huisken hu described an explicit open cone in the space of curvature operators such that the normalized ricci flow evolves metrics whose curvature operators are contained in that cone into metrics of constant positive sectional curvature hamilton conjectured that in all dimensions compact riemannian manifolds with positive curvature operators must be space forms in this paper we confirm this conjecture more generally we show the following

the second twisted betti number and the convergence of collapsing riemannian manifolds let mi overset d gh longrightarrow x denote a sequence of n manifolds converging to a compact metric space x in the gromov hausdorff topology such that the sectional curvature is bounded in absolute value and dim x n we prove the following stability result if the fundamental groups of mi are torsion groups of uniformly bounded exponents and the second twisted betti numbers of mi vanish then there is a manifold m and a sequence of diffeomorphisms from m to a subsequence of mi such that the distance functions of the pullback metrics converge to a pseudo metric in c0 norm furthermore m admits a foliation with leaves diffeomorphic to flat manifolds not necessarily compact such that a vector is tangent to a leaf if and only if its norm converges to zero with respect to the pullback metrics these results lead to a few interesting applications

convergence of kahler ricci flow in this paper we prove a theorem on convergence of kahler ricci flow on a compact kahler manifold which admits a kahler ricci soliton

the stability space of compactified universal jacobians in this paper we describe compactified universal jacobians i e compactifications of the moduli space of line bundles on smooth curves obtained as moduli spaces of rank 1 torsion free sheaves on stable curves using an approach due to oda seshadri we focus on the combinatorics of the stability conditions used to define compactified universal jacobians we explicitly describe an affine space the stability space with a decomposition into polytopes such that each polytope corresponds to a proper deligne mumford stack that compactifies the moduli space of line bundles we apply this description to describe the set of isomorphism classes of compactified universal jacobians answering a question of melo and to resolve the indeterminacy of the abel jacobi sections addressing a problem raised by grushevsky zakharov

conformal deformations of integral pinched 3 manifolds abstract in this paper we prove that under an explicit integral pinching assumption between the l 2 norm of the ricci curvature and the l 2 norm of the scalar curvature a closed 3 manifold with positive scalar curvature admits a conformal metric of positive ricci curvature in particular using a result of hamilton this implies that the manifold is diffeomorphic to a quotient of s 3 the proof of the main result of the paper is based on ideas developed in an article by m gursky and j viaclovsky

flow of pinched convex hypersurfaces by powers of curvature functions in hyperbolic space abstract this paper concerns closed h convex hypersurfaces of dimension n 2 in the hyperbolic space h n 1 of constant sectional curvature evolving in direction of its normal vector where the speed equals a power 1 of a curvature function f which is monotone symmetric homogeneous of degree 1 it is shown that if the initial h convex hypersurface is pinched then this is maintained under the flow and the hypersurfaces shrink to a round point in h n 1 in finite time as a consequence when rescaling appropriately the evolving hypersurfaces converge smoothly and exponentially to the unit sphere of r n 1

neron models and compactified picard schemes over the moduli stack of stable curves we construct modular deligne mumford stacks representable over parametrizing neron models of jacobians as follows let b be a smooth curve and k its function field let be a smooth genus g curve over k admitting stable minimal model over b the neron model b is then the base change of via the moduli map b of f i e b moreover is compactified by a deligne mumford stack over giving a completion of neron models naturally stratified in terms of neron models

ricci flow on quasi projective manifolds we consider the kahler ricci flow on complete finite volume metrics that live on the complement of a divisor in a compact kahler manifold x assuming certain spatial asymptotics on the initial metric we compute the singularity time in terms of cohomological data on x we also give a sufficient condition for the singularity if there is one to be type ii

ancient solutions of the ricci flow on bundles abstract we generalize the circle bundle examples of ancient solutions of the ricci flow discovered by bakas kong and ni to a class of principal torus bundles over an arbitrary finite product of fano kahler einstein manifolds studied by wang and ziller in the context of einstein geometry as a result continuous families of collapsed and noncollapsed ancient solutions of type i are obtained on circle bundles for all odd dimensions 7 in dimension 7 such examples moreover exist on pairs of homeomorphic but not diffeomorphic manifolds continuous families of collapsed ancient solutions of type i are also obtained on torus bundles for all dimensions 8

stability of the method of lines it is well known that a necessary condition for the lax stability of the method of lines is that the eigenvalues of the spatial discretization operator scaled by the time stepk lie within a distanceo k of the stability region of the time integration formula ask 0 in this paper we show that a necessary and sufficient condition for stability except for an algebraic factor is that the pseudo eigenvalues of the same operator lie within a distanceo o k of the stability region ask 0 our results generalize those of an earlier paper by considering a runge kutta and other one step formulas b implicit as well as explicit linear multistep formulas c weighted norms d algebraic stability e finite and infinite time intervals and f stability regions with cusps in summary the theory presented in this paper amounts to a transplantation of the kreiss matrix theorem from the unit disk for simple power iterations to an arbitrary stability region for method of lines calculations

inverse mean curvature flow with forced term abstract in this paper we study the evolution of convex hypersurfaces by inverse mean curvature minus an external force field c we prove that the flow will preserve the convexity for any c if c 1 h on the initial surface we prove that the flow will expand the hypersurface for all time and after rescaling the hypersurface will converge to a sphere if c 1 h on the initial surface we show that the maximum existence time of the flow is finite and the hypersurface will contract to a point when approaching the final time

a compactification over overline _ of the universal moduli space of slope semistable vector bundles 0 introduction 425 1 the quotient construction 428 2 the fiberwise g i t problem 433 3 cohomology bounds 435 4 slope unstable torsion free sheaves 437 5 special torsion bounded sheaves 440 6 slope semistable torsion free sheaves 443 7 two results in geometric invariant theory 449 8 the construction of ug e r 454 9 basic properties of ug e r 458 10 the isomorphism between ug e 1 and pg e 464 references 470

a unified approach to spurious solutions introduced by time discretization part i basic theory the asymptotic states of numerical methods for initial value problems are examined in particular spurious steady solutions solutions with period 2 in the timestep and spurious invariant curves

chamber behavior of double hurwitz numbers in genus 0 we study double hurwitz numbers in genus zero counting the number of covers cp 1 cp 1 with two branching points with a given branching behavior by the recent result due to goulden jackson and vakil these numbers are piecewise polynomials in the multiplicities of the preimages of the branching points we describe the partition of the parameter space into polynomiality domains called chambers and provide an expression for the difference of two such polynomials for two neighboring chambers besides we provide an explicit formula for the polynomial in a certain chamber called totally negative which enables us to calculate double hurwitz numbers in any given chamber as the polynomial for the totally negative chamber plus the sum of the differences between the neighboring polynomials along a path connecting the totally negative chamber with the given one

on the biregular geometry of the fulton macpherson compactification abstract let x n be the fulton macpherson compactification of the configuration space of n ordered points on a smooth projective variety x we prove that if either n 2 or dim x 2 then the connected component of the identity of aut x n is isomorphic to the connected component of the identity of aut x when x c is a curve of genus g c 1 we classify the dominant morphisms c n c r and thanks to this we manage to compute the whole automorphism group of c n namely aut c n s n aut c for any n 2 while aut c 2 s 2 aut c aut c furthermore we extend these results on the automorphisms to the case where x c 1 c r is a product of curves of genus g c i 2 finally using the techniques developed to deal with fulton macpherson spaces we study the automorphism groups of some kontsevich moduli spaces m 0 n p n d

new li yau hamilton inequalities for the ricci flow via the space time approach we prove li yau hamilton inequalties that extend hamilton s matrix inequality for solutions of the ricci flow with nonnegative curvature operators to obtain our extensions we apply the space time formalism of s c chu and one of the authors to solutions of the ricci flow modified by a cosmological constant then we adjoin to the ricci flow the evolution of a 1 form and a 2 form flowing by a system of heat type equations by a rescaling argument the inequalities we obtain in this manner yield new inequalities which are reminiscent of the linear trace inequality of hamilton and one of the authors

gluing eguchi hanson metrics and a question of page in 1978 gibbons pope and page proposed a physical picture for the ricci flat kahler metrics on the k3 surface based on a gluing construction in this construction one starts from a flat torus with 16 orbifold points and resolves the orbifold singularities by gluing in 16 small eguchi hanson manifolds that all have the same orientation this construction was carried out rigorously by topiwala lebrun singer and donaldson in 1981 page asked whether the above construction can be modified by reversing the orientations of some of the eguchi hanson manifolds this is a subtle question if successful this construction would produce einstein metrics that are neither kahler nor self dual in this paper we focus on a configuration of maximal symmetry involving eight small eguchi hanson manifolds of each orientation that are arranged according to a chessboard pattern by analyzing the interactions between eguchi hanson manifolds with opposite orientation we identify a nonvanishing obstruction to the gluing problem thereby destroying any hope of producing a metric of zero ricci curvature in this way using this obstruction we are able to understand the dynamics of such metrics under ricci flow as long as the eguchi hanson manifolds remain small in particular for the configuration described above we obtain an ancient solution to the ricci flow with the property that the maximum of the riemann curvature tensor blows up at a rate of t 1 2 while the maximum of the ricci curvature converges to 0 2016 wiley periodicals inc

extensions of the universal theta divisor abstract the jacobian varieties of smooth curves fit together to form a family the universal jacobian over the moduli space of smooth pointed curves and the theta divisors of these curves form a divisor in the universal jacobian in this paper we describe how to extend these families over the moduli space of stable pointed curves using a stability parameter we then prove a wall crossing formula describing how the theta divisor varies with this parameter we use this result to analyze divisors on the moduli space of smooth pointed curves that have recently been studied by grushevsky zakharov hain and muller finally we compute the pullback of the theta divisor studied in alexeev s work on stable semiabelic varieties and in caporaso s work on theta divisors of compactified jacobians

compactified picard stacks over overline mathcal m _g we study algebraic artin stacks over overline mathcal m g giving a functorial way of compactifying the relative degree d picard variety for families of stable curves we also describe for every d the locus of genus g stable curves over which we get deligne mumford stacks strongly representable over overline mathcal m g

the calabi flow on kahler surfaces with bounded sobolev constant i we consider the formation of singularities along the calabi flow by assuming the uniformly bounded sobolev constants on kahler surfaces we prove that if curvature tensor is not uniformly bounded then one can form a singular model called deepest bubble such deepest bubble has to be a scalar flat ale kahler metric in certain kahler classes on toric fano surfaces the sobolev constants are a priori bounded along the calabi flow with small calabi energy we can also show in certain cases no deepest bubble can form along the flow it follows that the curvature tensor is uniformly bounded and the flow exists for all time and converges to an extremal metric subsequently to illustrate our results more clearly we focus on an example on mathbb cp 2 blown up three points at generic position our result also implies existence of constant scalar curvature metrics on mathbb cp 2 blown up three points at generic position in the kahler classes where the exceptional divisors have the same area

mean curvature flow of pinched submanifolds to spheres the evolution of hypersurfaces by their mean curvature has been studied by many authors since the appearance of gerhard huisken s seminal paper hu1 more recently mean curvature flow of higher codimension submanifolds has also received attention in this paper we prove a result analogous to that of hu1 for submanifolds of any codimension let f0 n rn k be a smooth immersion of a compact manifold the mean curvature flow with initial condition f0 is a smooth family of immersions f 0 t rn k satisfying

the picard group of the universal moduli space of vector bundles on stable curves abstract we construct the moduli stack of properly balanced vector bundles on semistable curves and we determine explicitly its picard group as a consequence we obtain an explicit description of the picard groups of the universal moduli stack of vector bundles on smooth curves and of the schmitt s compactification over the stack of stable curves we prove some results about the gerbe structure of the universal moduli stack over its rigidification by the natural action of the multiplicative group in particular we give necessary and sufficient conditions for the existence of poincare bundles over the universal curve of an open substack of the rigidification generalizing a result of mestrano ramanan

wall crossings for double hurwitz numbers abstract double hurwitz numbers count covers of p 1 by genus g curves with assigned ramification profiles over 0 and and simple ramification over a fixed branch divisor goulden jackson and vakil have shown double hurwitz numbers are piecewise polynomial in the orders of ramification goulden et al 2005 10 and shadrin shapiro and vainshtein have determined the chamber structure and wall crossing formulas for g 0 shadrin et al 2008 15 this paper gives a unified approach to these results and strengthens them in several ways the most important being the extension of the results of shadrin et al 2008 15 to arbitrary genus the main tool is the authors previous work cavalieri et al 2010 6 expressing double hurwitz number as a sum over certain labeled graphs we identify the labels of the graphs with lattice points in the chambers of certain hyperplane arrangements which give rise to piecewise polynomial functions our understanding of the wall crossing for these functions builds on the work of varchenko 1987 17 and could have broader applications

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

foncteur de picard d un champ algebrique resume in this article we study the picard functor and the picard stack of an algebraic stack we give a new and direct proof of the representability of the picard stack we prove that it is quasi separated and that the connected component of the identity is proper when the fibers of x are geometrically normal we study some examples of picard functors of classical stacks in an appendix we review the lisse etale cohomology of abelian sheaves on an algebraic stack

ancient solutions to the ricci flow with pinched curvature we show that any ancient solution to the ricci flow which satisfies a suitable curvature pinching condition must have constant sectional curvature

finiteness theorems for the picard objects of an algebraic stack abstract we prove some finiteness theorems for the picard functor of an algebraic stack in the spirit of sga 6 exp xii and xiii in particular we give a stacky version of raynaud s relative representability theorem we give sufficient conditions for the existence of the torsion component of the picard functor and for the finite generation of the neron severi groups or of the picard group itself we give some examples and applications in appendix a we prove the semicontinuity theorem for a non necessarily tame algebraic stack

pseudospectra of waveform relaxation operators abstract the performance of the waveform relaxation method for solving systems of odes depends largely on the choices that are made for splitting size of time window and preconditioning although it is known that superlinear convergence is obtained on finite time windows the convergence may be slow in the first few iterations we propose the use of pseudospectra to analyze the convergence ratio of the first few iterations when waveform relaxation is applied to linear systems of odes through pseudospectral radii we can examine the effect of preconditioning and overlapping on the rate of convergence we may also use this to estimate a suitable size of the time window numerical experiments performed on a system of odes arising from the discretization of an advection diffusion equation confirm the validity of the obtained estimates

lifespan theorem for simple constrained surface diffusion flows abstract we consider closed immersed hypersurfaces in r 3 and r 4 evolving by a special class of constrained surface diffusion flows this class of constrained flows includes the classical surface diffusion flow in this paper we present a lifespan theorem for these flows which gives a positive lower bound on the time for which a smooth solution exists and a small upper bound on the total curvature during this time the hypothesis of the theorem is that the surface is not already singular in terms of concentration of curvature this turns out to be a deep property of the initial manifold as the lower bound on maximal time obtained depends precisely upon the concentration of curvature of the initial manifold in l 2 for m 2 immersed in r 3 and additionally on the concentration in l 3 for m 3 immersed in r 4 this is stronger than a previous result on a different class of constrained surface diffusion flows as here we obtain an improved lower bound on maximal time a better estimate during this period and eliminate any assumption on the area of the evolving hypersurface

picard group of moduli of hyperelliptic curves the main subject of this work is the difference between the coarse moduli space and the stack of hyperelliptic curves we compute their picard groups giving explicit description of the generators we get an application to the non existence of a tautological family over the coarse moduli space