transport and diffusion of material quantities on propagating interfaces via level set methods we develop theory and numerical algorithms to apply level set methods to problems involving the transport and diffusion of material quantities in a level set framework level set methods are computational techniques for tracking moving interfaces they work by embedding the propagating interface as the zero level set of a higher dimensional function and then approximate the solution of the resulting initial value partial differential equation using upwind finite difference schemes the traditional level set method works in the trace space of the evolving interface and hence disregards any parameterization in the interface description consequently material quantities on the interface which themselves are transported under the interface motion are not easily handled in this framework we develop model equations and algorithmic techniques to extend the level set method to include these problems we demonstrate the accuracy of our approach through a series of test examples and convergence studies

solving eigenvalue problems on curved surfaces using the closest point method eigenvalue problems are fundamental to mathematics and science we present a simple algorithm for determining eigenvalues and eigenfunctions of the laplace beltrami operator on rather general curved surfaces our algorithm which is based on the closest point method relies on an embedding of the surface in a higher dimensional space where standard cartesian finite difference and interpolation schemes can be easily applied we show that there is a one to one correspondence between a problem defined in the embedding space and the original surface problem for open surfaces we present a simple way to impose dirichlet and neumann boundary conditions while maintaining second order accuracy convergence studies and a series of examples demonstrate the effectiveness and generality of our approach

degeneration of l adic weight spectral sequences the degeneracy of the l adic weight spectral sequence over any trait with no assumptions on its residue field is proved by the method of log geometry in the sense of fontaine illusie

a cell based particle method for modeling dynamic interfaces abstract we propose several modifications to the grid based particle method gbpm 21 for moving interface modeling there are several nice features of the proposed algorithm the new method can significantly improve the distribution of sampling particles on the evolving interface unlike the original gbpm where footpoints sampling points tend to cluster to each other the sampling points in the new method tend to be better separated on the interface moreover by replacing the grid based discretization using the cell based discretization we naturally decompose the interface into segments so that we can easily approximate surface integrals as a possible alternative to the local polynomial least square approximation we also study a geometric basis for local reconstruction in the resampling step we will show that such modification can simplify the overall implementations numerical examples in two and three dimensions will show that the algorithm is computationally efficient and accurate

a grid based particle method for solving partial differential equations on evolving surfaces and modeling high order geometrical motion we develop numerical methods for solving partial differential equations pde defined on an evolving interface represented by the grid based particle method gbpm recently proposed in s leung h k zhao a grid based particle method for moving interface problems j comput phys 228 2009 7706 7728 in particular we develop implicit time discretization methods for the advection diffusion equation where the time step is restricted solely by the advection part of the equation we also generalize the gbpm to solve high order geometrical flows including surface diffusion and willmore type flows the resulting algorithm can be easily implemented since the method is based on meshless particles quasi uniformly sampled on the interface furthermore without any computational mesh or triangulation defined on the interface we do not require remeshing or reparametrization in the case of highly distorted motion or when there are topological changes as an interesting application we study locally inextensible flows governed by energy minimization we introduce tension force via a lagrange multiplier determined by the solution to a helmholtz equation defined on the evolving interface extensive numerical examples are also given to demonstrate the efficiency of the proposed approach

numerical simulation of dealloying by surface dissolution via the evolving surface finite element method in this article we formulate a numerical method for the simulation of dealloying of a binary alloy by the selective removal of one component via electrochemical dissolution such that there is phase separation of the other component at the solid electrolyte interface the evolution of the interface is modelled by a forced mean curvature flow strongly coupled to the solution of a cahn hilliard equation modelling surface phase separation the method is based on a triangulated hypersurface whose evolution is calculated as well as the solution of the cahn hilliard equation by the evolving surface finite element method esfem the numerical experiments simulate complex morphology and concentration patterns providing evidence that the mathematical model may describe the formation of nanoporosity

semiregularity and obstructions of complete intersections abstract we prove that on a smooth projective variety over an algebraically closed field of characteristic 0 the semiregularity map annihilates every obstruction to embedded deformations of a local complete intersection subvariety with extendable normal bundle the proof is based on the theory of l algebras and tamarkin tsygan calculus on the de rham complex of dg schemes

an rbf fd closest point method for solving pdes on surfaces abstract partial differential equations pdes on surfaces appear in many applications throughout the natural and applied sciences the classical closest point method ruuth and merriman 2008 17 is an embedding method for solving pdes on surfaces using standard finite difference schemes in this paper we formulate an explicit closest point method using finite difference schemes derived from radial basis functions rbf fd unlike the orthogonal gradients method piret 2012 22 our proposed method uses rbf centers on regular grid nodes this formulation not only reduces the computational cost but also avoids the ill conditioning from point clustering on the surface and is more natural to couple with a grid based manifold evolution algorithm leung and zhao 2009 26 when compared to the standard finite difference discretization of the closest point method the proposed method requires a smaller computational domain surrounding the surface resulting in a decrease in the number of sampling points on the surface in addition higher order schemes can easily be constructed by increasing the number of points in the rbf fd stencil applications to a variety of examples are provided to illustrate the numerical convergence of the method

the decomposition theorem perverse sheaves and the topology of algebraic maps we give a motivated introduction to the theory of perverse sheaves culminating in the decomposition theorem of beilinson bernstein deligne and gabber a goal of this survey is to show how the theory develops naturally from classical constructions used in the study of topological properties of algebraic varieties while most proofs are omitted we discuss several approaches to the decomposition theorem indicate some important applications and examples

finely harmonic functions with finite dirichlet integral with respect to the green measure we consider finely harmonic functions h on a fine greenian domain v c rd with finite dirichlet integral wrt gm i e f ivh y 2 g x y dm y oo for x c v where m denotes the lebesgue measure g x y the green function we use brownian motion and stochastic calculus to prove that such functions h always have boundary values h along a a brownian paths this partially extends results by doob brelot and godefroid who considered ordinary harmonic functions with finite dirichlet integral wrt m and green lines instead of brownian paths as a consequence of theorem 1 we obtain several properties equivalent to one of these being that h is the harmonic extension to v of a random boundary function h of a certain type i e h x ex h for all x c v another application is that the polar sets are removable singularity sets for finely harmonic functions satisfying this is in contrast with the situation for finely harmonic functions with finite dirichlet integral wrt m

modeling and computation of two phase geometric biomembranes using surface finite elements biomembranes consisting of multiple lipids may involve phase separation phenomena leading to coexisting domains of different lipid compositions the modeling of such biomembranes involves an elastic or bending energy together with a line energy associated with the phase interfaces this leads to a free boundary problem for the phase interface on the unknown equilibrium surface which minimizes an energy functional subject to volume and area constraints in this paper we propose a new computational tool for computing equilibria based on an l 2 relaxation flow for the total energy in which the line energy is approximated by a surface ginzburg landau phase field functional the relaxation dynamics couple a nonlinear fourth order geometric evolution equation of willmore flow type for the membrane with a surface allen cahn equation describing the lateral decomposition a novel system is derived involving second order elliptic operators where the field variables are the positions of material points of the surface the mean curvature vector and the surface phase field function the resulting variational formulation uses h 1 spaces and we employ triangulated surfaces and h 1 conforming quadratic surface finite elements for approximating solutions together with a semi implicit time discretization of the evolution equations an iterative scheme is obtained essentially requiring linear solvers only numerical experiments are presented which exhibit convergence and the power of this new method for two component geometric biomembranes by computing equilibria such as dumbbells discocytes and starfishes with lateral phase separation

a grid based particle method for moving interface problems we propose a novel algorithm for modeling interface motions the interface is represented and is tracked using quasi uniform meshless particles these particles are sampled according to an underlying grid such that each particle is associated to a grid point which is in the neighborhood of the interface the underlying grid provides an eulerian reference and local sampling rate for particles on the interface it also renders neighborhood information among the meshless particles for local reconstruction of the interface the resulting algorithm which is based on lagrangian tracking using meshless particles with eulerian reference grid can naturally handle control topological changes moreover adaptive sampling of the interface can be achieved easily through local grid refinement with simple quad oct tree data structure extensive numerical examples are presented to demonstrate the capability of our new algorithm

schwarz solvers and preconditioners for the closest point method the discretization of surface intrinsic elliptic partial differential equations pdes poses interesting challenges not seen in flat spaces the discretization of these pdes typically proceeds by e

tangent bundle of a complete intersection let x be a fano variety of picard number one defined over an algebraically closed field we give conditions under which the tangent bundle of a complete intersection on x is stable or strongly stable

vanishing via lifting to second witt vectors and a proof of an isotriviality result abstract a proof based on reduction to finite fields of esnault and viehweg s stronger version of the sommese vanishing theorem for k ample line bundles is given this result is used to give different proofs of isotriviality results of parshin and migliorini

toric degenerations of fano threefolds giving weak landau ginzburg models abstract we show that every picard rank one smooth fano threefold has a weak landau ginzburg model coming from a toric degeneration the fibers of these landau ginzburg models can be compactified to k3 surfaces with picard lattice of rank 19 we also show that any smooth fano variety of arbitrary dimension which is a complete intersection of cartier divisors in weighted projective space has a very weak landau ginzburg model coming from a toric degeneration

a localized meshless method for diffusion on folded surfaces partial differential equations pdes on surfaces arise in a variety of application areas including biological systems medical imaging fluid dynamics mathematical physics image processing and computer graphics in this paper we propose a radial basis function rbf discretization of the closest point method the corresponding localized meshless method may be used to approximate diffusion on smooth or folded surfaces our method has the benefit of having an a priori error bound in terms of percentage of the norm of the solution a stable solver is used to avoid the ill conditioning that arises when the radial basis functions rbfs become flat

a grid based particle method for evolution of open curves and surfaces we propose a new numerical method for modeling motion of open curves in two dimensions and open surfaces in three dimensions following the grid based particle method we proposed in s leung h k zhao a grid based particle method for moving interface problems j comput phys 228 2009 2993 3024 we represent the open curve or the open surface by meshless lagrangian particles sampled according to an underlying fixed eulerian mesh the underlying grid is used to provide a quasi uniform sampling and neighboring information for meshless particles the key idea in the current paper is to represent and to track end points of the open curve and boundary points of the open surface explicitly and consistently with interior particles we apply our algorithms to several applications including spiral crystal growth modeling and image segmentation using active contours

pdes on moving surfaces via the closest point method and a modified grid based particle method partial differential equations pdes on surfaces arise in a wide range of applications the closest point method ruuth and merriman 2008 20 is a recent embedding method that has been used to solve a variety of pdes on smooth surfaces using a closest point representation of the surface and standard cartesian grid methods in the embedding space the original closest point method cpm was designed for problems posed on static surfaces however the solution of pdes on moving surfaces is of considerable interest as well here we propose solving pdes on moving surfaces using a combination of the cpm and a modification of the grid based particle method leung and zhao 2009 12 the grid based particle method gbpm represents and tracks surfaces using meshless particles and an eulerian reference grid our modification of the gbpm introduces a reconstruction step into the original method to ensure that all the grid points within a computational tube surrounding the surface are active we present a number of examples to illustrate the numerical convergence properties of our combined method experiments for advection diffusion equations that are strongly coupled to the velocity of the surface are also presented

the implicit closest point method for the numerical solution of partial differential equations on surfaces many applications in the natural and applied sciences require the solutions of partial differential equations pdes on surfaces or more general manifolds the closest point method is a simple and accurate embedding method for numerically approximating pdes on rather general smooth surfaces however the original formulation is designed to use explicit time stepping this may lead to a strict time step restriction for some important pdes such as those involving the laplace beltrami operator or higher order derivative operators to achieve improved stability and efficiency we introduce a new implicit closest point method for surface pdes the method allows for large stable time steps while retaining the principal benefits of the original method in particular it maintains the order of accuracy of the discretization of the underlying embedding pde it works on sharply defined bands without degrading the accuracy of the method and it applies to general smooth surfaces it also is very simple and may be applied to a rather general class of surface pdes convergence studies for the in surface heat equation and a fourth order biharmonic problem are given to illustrate the accuracy of the method we demonstrate the flexibility and generality of the method by treating flows involving diffusion reaction diffusion and fourth order spatial derivatives on a variety of interesting surfaces including surfaces of mixed codimension

quasi frobenius splitting and lifting of calabi yau varieties in characteristic p generalizing the notion of frobenius splitting we prove that every finite height calabi yau variety defined over an algebraically closed field of positive characteristic can be lifted to the ring of witt vectors of length two

mirror symmetry for del pezzo surfaces vanishing cycles and coherent sheaves we study homological mirror symmetry for del pezzo surfaces and their mirror landau ginzburg models in particular we show that the derived category of coherent sheaves on a del pezzo surface x k obtained by blowing up 2 at k points is equivalent to the derived category of vanishing cycles of a certain elliptic fibration w k m k with k 3 singular fibers equipped with a suitable symplectic form moreover we also show that this mirror correspondence between derived categories can be extended to noncommutative deformations of x k and give an explicit correspondence between the deformation parameters for x k and the cohomology class b i h 2 m k

the f pure threshold of a calabi yau hypersurface we compute the f pure threshold of the affine cone over a calabi yau hypersurface and relate it to the order of vanishing of the hasse invariant on the versal deformation space of the hypersurface

gv sheaves fourier mukai transform and generic vanishing we prove a formal criterion for generic vanishing in the sense originated by green and lazarsfeld and pursued further by hacon but in the context of an arbitrary fourier mukai correspondence for smooth projective varieties we apply this to deduce a kodaira type generic vanishing theorem for adjoint bundles associated to nef line bundles and in fact a more general generic nadel type vanishing theorem for multiplier ideal sheaves still in the context of the picard variety the same method gives various other generic vanishing results by reduction to standard vanishing theorems we further use our criterion in order to address some examples related to generic vanishing on higher rank moduli spaces

towards mirror symmetry for varieties of general type abstract the goal of this paper is to propose a theory of mirror symmetry for varieties of general type using landau ginzburg mirrors as motivation we describe the mirror of a hypersurface of general type and more generally varieties of non negative kodaira dimension as the critical locus of the zero fibre of a certain landau ginzburg potential the critical locus carries a perverse sheaf of vanishing cycles our main result shows that one obtains the interchange of hodge numbers expected in mirror symmetry this exchange is between the hodge numbers of the hypersurface and certain hodge numbers defined using a mixed hodge structure on the hypercohomology of the perverse sheaf

moduli of vector bundles frobenius splitting and invariant theory let x be an irreducible projective curve of genus g over an algebraically closed field of positive characteristic 2 3 in part i we prove adapting the degeneration arguments of n tr that moduli spaces of rank two parabolic bundles on x are frobenius split m r for generic smooth x a similar result holds for x nodal a consequence is the verlinde formula in positive characteristic in part ii we give a direct proof of the fact that the local rings of the moduli spaces are f split and further that they are cohen macaulay this involves showing that the ring of invariants of g copies of 2 x 2 matrices under the adjoint action of sl 2 is f split and cohen macaulay

when is the self intersection of a subvariety a fibration abstract we provide a necessary and sufficient condition for the derived self intersection of a smooth subscheme inside a smooth scheme to be a fibration over the subscheme as a consequence we deduce a generalized hkr isomorphism we also investigate the relationship of our result to path spaces in homotopy theory buchweitz flenner formality in algebraic geometry and draw parallels with similar results in lie theory and symplectic geometry

landau ginzburg hodge numbers for mirrors of del pezzo surfaces abstract we consider the conjectures from 10 about landau ginzburg hodge numbers associated to tamely compactifiable landau ginzburg models we test these conjectures in case of dimension two verifying some and giving a counterexample to the other

on the characteristic polynomial of the frobenius on etale cohomology let x be a smooth projective variety of even dimension d over a finite field then we establish a restriction on the value at 1 of the characteristic polynomial of the frobenius on the middle dimensional etale cohomology of x with coefficients in ql d 2