on the existence of local observables in theories with a factorizing s matrix the scaling limit of the two dimensional ising model above the critical temperature is considered as an example for relativistic quantum theories on two dimensional minkowski space exhibiting a factorizing s matrix in this model a recently proposed criterion for the existence of local quantum field theories with a prescribed factorizing scattering matrix is verified thereby establishing a new constructive approach to two dimensional quantum field theory in a particular example the existence proof is accomplished by analysing the nuclearity properties of certain specific subsets of fermionic fock spaces and yields as a byproduct also a verification of the energy nuclearity condition of buchholz and wichmann in models of free fermions in four spacetime dimensions

on quantum fields that generate local algebras the properties of quantum fields generating local nets of von neumann algebras are discussed for fields that satisfy certain energy bounds necessary and sufficient conditions for the existence of such a net and for the existence of product states are given in terms of the vacuum expectation values

monodromy group for a strongly semistable principal bundle over a curve let x be a geometrically irreducible smooth projective curve defined over a field k assume that x has a k rational point fix a k rational point x e x from these data we construct an affine group scheme x defined over the field k as well as a principal x bundle s1755069607000151inline1 over the curve x the group scheme x is given by a xs0211a graded neutral tannakian category built out of all strongly semistable vector bundles over x the principal bundle s1755069607000151inline1 is tautological let g be a linear algebraic group defined over k that does not admit any nontrivial character which is trivial on the connected component containing the identity element of the reduced center of g let e g be a strongly semistable principal g bundle over x we associate to e g a group scheme m defined over k which we call the monodromy group scheme of e g and a principal m bundle e m over x which we call the monodromy bundle of e g the group scheme m is canonically a quotient of x and e m is the extension of structure group of s1755069607000151inline1 the group scheme m is also canonically embedded in the fiber ad e g x over x of the adjoint bundle

embeddings of spherical homogeneous spaces in characteristic p let g be a reductive group over an algebraically closed field of characteristic p 0 we study properties of embeddings of spherical homogeneous g spaces we look at frobenius splittings canonical or by a p 1 th power compatible with certain subvarieties we show the existence of rational g equivariant resolutions by toroidal embeddings and give results about cohomology vanishing and surjectivity of restriction maps of global sections of line bundles we show that the class of homogeneous spaces for which our results hold contains the symmetric homogeneous spaces in characteristic ne 2 and is closed under parabolic induction

complex classical fields a framework for reflection positivity we explore a framework for complex classical fields appropriate for describing quantum field theories our fields are linear transformations on a hilbert space so they are more general than random variables for a probability measure our method generalizes osterwalder and schrader s construction of euclidean fields we allow complex valued classical fields in the case of quantum field theories that describe neutral particles

translationally invariant states and the spectrum ideal in the algebra of test functions for quantum fields a class of states on borchers tensor algebra is constructed these states are invariant under the translation group and fulfill the spectrum condition this leads to a characterization of the linear span of all such states in terms of a simple continuity property

global f regularity of schubert varieties with applications to d modules a projective algebraic variety x over an algebraically closed field k of positive characteristic is called globally f regular if the section ring s c 0n o h x cn of an ample line bundle c on x is strongly f regular in the sense of h chster and huneke 9 cf definition 1 1 this important notion was introduced by karen smith in 18 in this paper we prove that schubert varieties are globally f regular an im mediate consequence is that local rings of schubert varieties are strongly f regular and thereby f rational another consequence is that local rings of varieties like de terminantal varieties that can be identified with open subsets of schubert varieties cf 13 are strongly f regular let x denote a flag variety and y c x a schubert variety over k then the local cohomology sheaves hjy ox are equivariant for the action of the borel subgroup and holonomic in the sense of 4 x x niodules as an application of ir rationality of schubert varieties we apply recent results of blickle cf 2 to prove that the simple objects in the category of equivariant and holonomic px modules are precisely the local cohomology sheaves hy ox where c is the codimension of y in x using a local grothendieck cousin complex from 11 we prove that the decomposition of the local cohomology modules with support in bruhat cells is multiplicity free see 4 2 in characteristic zero the local cohomology modules with support in bruhat cells correspond to dual verma modules in this setting the decomposition behavior and the simple t x modules arise from intersection cohomology complexes of schubert varieties by the riemann hilbert correspondence if we pick the singular codi mension one schubert variety y in the full flag variety z for sl4 in characteristic zero computations in kazhdan lusztig theory show that hy oz is not a simple z module see 4 1

frobenius splitting of schubert varieties and linear syzygies introduction let g be a connected simply connected semi simple alge braic group let b be a borel subgroup of g for an ample line bundle l on g b the questions about projective normality of the embedding of g b in the complete linear system of l and the equations defining the embedding were extensively studied ramanathan in 13 proved that the ideal defining the embedding of g b in the complete linear system of l is generated by quadratic equations kempf and ramanathan proved that the equations defining the embedding of g b in the complete linear system of l lr are quadratic and that a schubert variety x is linearly defined with respect to l lr ramanathan had conjectured that the method of frobenius splitting used in 13 can be used to study higher syzygies let r bh g b ljj where the vector mj ranges in nr be the multigraded coordinate ring of g b with respect to l lr in this paper we use frobenius splittings to study the higher syzygies of the irrelevant maximal ideal m of r we prove that m as a module has linear syzygies our methods also give simpler proofs of the results obtained in 6 and 13 let z be a projective variety defined over an algebraically closed field let l l be ample line bundles on z and let m denote the irrelevant maximal ideal of the multi graded ring r h z ljj in the first section we give a criterion for m to have linear syzigies over r we show that if for every n 2 the first cohomology of the n fold product zn with values in certain ideal sheaves tensored with powers of lj vanishes then m has linear syzygies propositions 1 9 and 1 10 ye also give a similar criterion for a homogeneous quotient ring of r to have linear syzygies over r proposition 1 13 in the second section we show that any r ample line bundles l l on g b and their restrictions to a schubert variety x c g b over an algebracally closed field of positive characteristic satisfy the above criteria to prove this result we use the method of frobenius splittings if z is a frobenius split variety and if l is an ample line bundle on z then we have hl z l 0 for 1 further if y is a compatibly split subvariety of z then the restriction

frobenius amplitude and strong vanishing theorems for vector bundles the primary goal of this paper is to systematically exploit the method of deligne illusie to obtain kodaira type vanishing theorems for vector bundles and more generally coherent sheaves on algebraic varieties the key idea is to introduce a number which provides a cohomological measure of the positivity of a coherent sheaf called the frobenius or f amplitude the f amplitude enters into the statement of the basic vanishing theorem and this leads to the problem of calculating or at least estimating this number most of the work in this paper is devoted to doing this various situations

f regularity of large schubert varieties let g denote a connected reductive algebraic group over an algebraically closed field k and let x denote a projective g g equivariant embedding of g the large schubert varieties in x are the closures of the double cosets bgb where b denotes a borel subgroup of g and g g we prove that these varieties are globally f regular in positive characteristic resp of globally f regular type in characteristic 0 as a consequence the large schubert varieties are normal and cohen macaulay 1 introduction the class of globally f regular varieties was introduced by smith in 21 these are projective algebraic varieties in positive character istics such that all the ideals in their homogeneous coordinate rings are tightly closed the globally f regular varieties and their analogues in characteristic 0 the varieties of globally f regular type have remarkable properties e g they are normal and cohen macaulay and the higher cohomology groups of all nef invertible sheaves are trivial examples of globally f regular varieties include the projective toric varieties prop 6 4 in 21 in this note we obtain the global f regularity of a wider class of varieties with algebraic group action the g g equivariant projective embeddings of any connected reductive group g and the closures in any such embedding of the double cosets bgb where b denotes a borel subgroup of g and g g is arbitrary by the bruhat decomposition these large schubert varieties are parametrized by the weyl group of g examples are the closures of parabolic subgroups we also show that the large schubert varieties are of globally f regular type in characteristic 0 for this we exploit the close relation between global f regularity and frobe nius splitting established in 21 and the frobenius splitting properties of the large schubert varieties proved in 5 and 19 another key ingredient is the global f regularity of the flag varieties and their schubert varieties 12 note that unlike for schubert varieties no desingularization of large schubert varieties is known in general this makes our arguments somewhat indirect as a consequence of our result the large schubert varieties in any equivariant embedding of g are normal and cohen macaulay this was first proved in the case of the canonical compactification of a semisimple adjoint group by frobenius

on the coherence conjecture of pappas and rapoport we prove the generalized coherence conjecture proposed by pappas and rapoport as a corollary one of their theorems which describes the geometry of the special bers of the local models for ramied unitary groups holds unconditionally our proof is based on the study of the geometry in particular certain line bundles and adic sheaves of the global schubert varieties which are the equal characteristic counterparts of the local models

on the flatness of local models for the symplectic group abstract we investigate the bad reduction of certain shimura varieties associated to the symplectic group more precisely we look at a model of the shimura variety at a prime p with parahoric level structure at p we show that this model is flat as conjectured by rapoport and zink ann of math stud 141 1996 and that its special fibre is reduced a crucial ingredient is faltings theorem on the normality of schubert varieties in the affine flag variety

singular loci of ladder determinantal varieties and schubert varieties abstract we relate certain ladder determinantal varieties associated to one sided ladders to certain schubert varieties in sl n q for a suitable n and a suitable parabolic subgroup q and we determine the singular loci of these varieties we state a conjecture on the irreducible components of the singular locus of a schubert variety in the flag variety which is a refinement of the conjecture of lakshmibai and sandhya proc indian acad sci math sci 100 1990 45 52 we prove the conjecture for a certain class of schubert varieties

hilbert series of modules over lie algebroids we consider modules m over lie algebroids ga which are of finite type over a local noetherian ring a using ideals j a such that ga j j and the length lga m jm we can define in a natu

frobenius splitting and cohomology vanishing for schubert varieties let k be an algebraically closed field of characteristic p 0 and x be a projective variety over k we then have the absolute frobenius f x x and an injection ox f 6x given by f fp f e o9x if this makes ox a direct summand in f ox as an x module we call x a frobenius split variety for such a variety the vanishing theorem for ample line bundles follows trivially from serre s vanishing theorem for tensoring ox f ox by an ample line bundle l and noting that l f x f f l f lp projection formula we get that the map in cohomology h x l h x f lp h x lp is an injection iterating this we see that h x l injects into h x lp for every p but for large v the latter is zero thus frobenius split varieties have quite pleasant properties it also turns out that using duality for the frobenius morphism or equivalently the cartier operator one can give a manageable criterion for frobenius splitting the point here is the local nature of duality the relevance of the compatibility of local and global duality was suggested to us by grothendieck s proof of h x y 0 for a noncomplete variety x of dimension n 4 theorem 6 9 and by kempf s paper 10 by this criterion and the bott samelson demazure desingularisation of schubert varieties it follows very easily that schubert varieties in characteristic p are frobenius split the following vanishing theorem is then an immediate consequence let g be a reductive group over the field k of arbitrary characteristic zero or positive let q be a parabolic subgroup and x c g q a schubert variety let l be an ample line bundle on g q then hi x l 0 for i 0 and the restriction map h0 g q l h0 x l is surjective if char k 0 this is a consequence of the compatible frobenius splitting of x in g q and the char k 0 case is handled by semicontinuity the above result for special schubert varieties has been proved by seshadrimusili lakshmibai 12 13 and by kempf 9 by using characteristic free methods for x g q andersen 1 and haboush 5 have given simple proofs using characteristic p methods

on the flatness of models of certain shimura varieties of pel type consider a pel shimura variety associated to a unitary group that splits over an unramified extension of q p rapoport and zink have defined a model of the shimura variety over the ring of integers of the completion of the reflex field at a place lying over p with parahoric level structures at p we show that this model is flat as conjectured by rapoport and zink and that its special fibre is reduced

scattering theory for quantum fields with indefinite metric in this work we discuss the scattering theory of local relativistic quantum fields with indefinite metric since the results of haag ruelle theory do not carry over to the case of indefinite metric 4 we propose an axiomatic framework for the construction of in and out states such that the lsz asymptotic condition can be derived from the assumptions the central mathematical object for this construction is the collection of mixed vacuum expectation values of local in and out fields called the form factor functional which is required to fulfill a hilbert space structure condition given a scattering matrix with polynomial transfer functions we then construct interpolating local relativistic quantum fields with indefinite metric which fit into the given scattering framework

the ring of sections of a complete symmetric variety we study the ring of sections a x of a complete symmetric variety x that is of the wonderful completion of g h where g is an adjoint semisimple group and h is the fixed subgroup for an involutorial automorphism of g we find generators for pic x we generalize the prv conjecture to complete symmetric varieties and construct a standard monomial theory for a x that is compatible with g orbit closures in x this gives a degeneration result and the rational singularityness for a x 2003 elsevier science usa all rights reserved

string localized quantum fields and modular localization we study free covariant quantum bose fields that are associated with irreducible representations of the poincare group and localized in semi infinite strings extending to spacelike infinity among these are fields that generate the irreducible representations of mass zero and infinite spin that are known to be incompatible with point like localized fields for the massive representations and the massless representations of finite helicity all string localized free fields can be written as an integral along the string of point localized tensor or spinor fields as a special case we discuss the string localized vector fields associated with the point like electromagnetic field and their relation to the axial gauge condition in the usual setting

string and brane localized causal fields in a strongly nonlocal model we study a weakly local but nonlocal model in spacetime dimension d 2 and prove that it is maximally nonlocal in a certain specific quantitative sense nevertheless depending on the number of dimensions d it has string localized or brane localized operators which commute at spatial distances in two spacetime dimensions the model even comprises a covariant and local subnet of operators localized in bounded subsets of minkowski space which has a nontrivial scattering matrix the model thus exemplifies the algebraic construction of local operators from algebras associated with nonlocal fields

lines in g p our goal is reached as follows fix a maximal torus t in g and a borel subgroup b t so that we can consider the corresponding root lattice q weight lattice set of simple roots 1 n and set of fundamental weights 1 n we then choose p b so that p corresponds to a subset s of or equivalently or also equivalently of the dynkin diagram of g recall that the picard group of g p can be identified with a sublattice of taken an ample l pic g p and setting

multi cones over schubert varieties let q 1 2 be invertible sheaves on a variety x we may form the multigraded ring a f x where the vector m ranges in n then c spec a is a multi cone if x is a projective cohen macaulay variety n l and l l is very ample serre has given a criterion for c to be cohen macaulay in terms of the cohomology of sheaves on x we know no reasonable analog of this result when n 1 on the other hand we will give such a criterion for c to have rational singularity when x does this is a stronger restriction on the singularities of c then we will give an application to multi cones based on schubert varieties let g be a reductive algebraic group b a borel subgroup and x g b a schubert variety let s176 5q be the line bundles on g b corresponding to the fundamental weights an arbitrary line bundle corresponding to a dominant weight has the form

on the connection between quantum fields and von neumann algebras of local operators the relationship between a standard local quantum field and a net of local von neumann algebras is discussed two natural possibilities for such an association are identified and conditions for these to obtain are found it is shown that the local net can naturally be so chosen that it satisfies the special condition of duality the notion of an intrinsically local field operator is introduced and it is shown that such an operator defines a local net with which the field is locally associated a regularity condition on the field is formulated and it is shown that if this condition holds then there exists a unique local net with which the field is locally associated if and only if the field algebra contains at least one intrinsically local operator conditions under which a field and other fields in its borchers class are associated with the same local net are found in terms of the regularity condition mentioned

frobenius splitting of cotangent bundles of flag varieties let g be a semisimple and simply connected algebraic group over an algebraically closed field of characteristic p 0 let u be the unipotent radical of a borel subgroup b g and u the lie algebra of u springer 16 has shown for good primes that there is a b equivariant isomorphism u u where b acts through conjugation on u and through the adjoint action on u for g sln one has the well known equivariant isomorphism a 7 a i between unipotent and nilpotent upper triangular matrices let p be a good prime for g then there is an isomorphism of homogeneous bundles x g b u g b u where the latter can be identified with the cotangent bundle t g b of g b motivated in part by 12 we establish a link between the g invariant form on the steinberg module st h0 g b p 1 cf 1 8 and frobenius splittings 15 of the cotangent bundle of g b the representation h0 g b 2 p 1 is a quotient of the space of functions h x ox on x here h0 g b m denotes the g module induced from the b module m and half the sum of the roots r opposite to the roots

construction of quantum field theories with factorizing s matrices the subject of this thesis is a novel construction method for interacting relativistic quantum field theories on two dimensional minkowski space employing the algebraic framework of quantum field theory it is shown under which conditions an algebra of observables localized in a wedge shaped region of spacetime can be used to construct model theories a crucial input in this context is the modular nuclearity condition for wedge algebras which implies the existence of local observables as an application of the new method a rigorous construction of a large family of models with factorizing s matrices is obtained in an inverse scattering approach a given factorizing scattering operator is used to define certain semi localized wightman fields associated to it with the help of these fields a wedge algebra can be defined which determines the local observable content of a well defined quantum field theory in this approach the modular nuclearity condition translates to certain analyticity and boundedness conditions on the formfactors of wedge local observables these conditions are shown to hold for a large class of underlying s matrices including the scattering operators of the sinh gordon model and the scaling ising model as special examples the so constructed models are investigated with respect to their scattering properties they are shown to solve the inverse scattering problem for the underlying s matrices and a proof of asymptotic completeness for these models is given

discreteness and rationality of f jumping numbers on singular varieties we prove that the f jumping numbers of the test ideal tau x delta mathfrak a t are discrete and rational under the assumptions that x is a normal and f finite scheme over a field of positive characteristic p kx is mathbb q cartier of index not divisible p and either x is essentially of finite type over a field or the sheaf of ideals mathfrak a is locally principal this is the largest generality for which discreteness and rationality are known for the jumping numbers of multiplier ideals in characteristic zero

image staining and differential diagnosis of ultrasound scans based on the mahalanobis distance the covariance matrices associated with each state of health or disease from a previous study are used as the basis of an image staining display technique for aid in quantitative differential diagnosis a state of health or disease is chosen by the clinician this selects the covariance matrix from the data base a region of interest roi is then scrolled through an abdominal b scan for each position of the roi a point in the four dimensional feature space is calculated a natural measure of the distance of this point from the center of mass multivariate mean of the disease class is calculated in terms of the covariance matrix of this class this measure is the mahalanobis distance the confidence level for acceptance or rejection of the hypothesized disease class is obtained from the probability distribution of this distance the t sup 2 probability law this confidence level is color coded and used as a color stain that overlays the original scan at that position the variability of the calculated features is studied as a function of roi size or the spatial resolution of the color coded image and it is found that for an roi in the neighborhood of 4 cm sup 2 most of the variability due to the finite number of independent samples speckles is averaged out leaving the noise floor associated with inter and intra patient variability rois on the order of 1 cm sup 2 may result with technical advances in b scan resolution a small number of points on organ boundaries are entered by the user to fit with arcs of ellipses to be used to switch between organ liver and kidney data bases as the roi encounters the boundary by selecting in turn various state of health or state of disease databases such images of confidence levels may be used for quantitative differential diagnosis the method is not limited to ultrasound being applicable in principle to features obtained from any modality or multimodality combination

centers of f purity in this paper we study a positive characteristic analogue of the centers of log canonicity of a pair r we call these analogues centers of f purity we prove positive characteristic analogues of subadjunction like results prove new stronger subadjunction like results and in some cases lift these new results to characteristic zero using a generalization of centers of f purity which we call uniformly f compatible ideals we give a characterization of the test ideal which unifies several previous characterizations finally in the case that 0 we show that uniformly f compatible ideals coincide with the annihilators of the mathcal f e r k submodules of e r k as defined by lyubeznik and smith

borel weil theorem for configuration varieties and schur modules we present a generalization of the classical schur modules ofgl n exhibiting the same interplay among algebra geometry and combinatorics a generalized young diagramdis an arbitrary finite subset ofn n for eachd we define the schur modulesdofgl n we introduce a projective variety fdand a line bundle ld and describe the schur module in terms of sections of ld for diagrams with the northeast property i1 j1 i2 j2 d min i1 i2 max j1 j2 d which includes the skew diagrams we resolve the singularities of fdand show analogues of bott s and kempf s vanishing theorems finally we apply the atiyah bott fixed point theorem to establish a weyl type character formula of the form charsd x txwt t i j 1 xix 1j dij t wheretruns over certain standard tableaux ofd our results are valid over fields of arbitrary characteristic

positivity of wightman functionals and the existence of local nets the paper is concerned with the existence of a local net of von neumann algebras associated with a given wightman field for fields satisfying a generalizedh bound the existence of such a net is shown to be equivalent to a certain positivity property of the wightman distributions

frobenius splitting and geometry of g schubert varieties let x be an equivariant embedding of a connected reductive group g over an algebraically closed field k of positive characteristic let b denote a borel subgroup of g a g schubert variety in x is a subvariety of the form diag g v where v is a b b orbit closure in x in the case where x is the wonderful compactification of a group of adjoint type the g schubert varieties are the closures of lusztig s g stable pieces we prove that x admits a frobenius splitting which is compatible with all g schubert varieties moreover when x is smooth projective and toroidal then any g schubert variety in x admits a stable frobenius splitting along an ample divisors although this indicates that g schubert varieties have nice singularities we present an example of a nonnormal g schubert variety in the wonderful compactification of a group of type g2 finally we also extend the frobenius splitting results to the more general class of schubert varieties

generic singularities of certain schubert varieties let g be a connected semisimple algebraic group b a borel subgroup t a maximal torus in b with weyl group w and q a subgroup containing b for w in w let x wq denote the schubert variety overline bwq q for y in w such that x yq subseteq x wq one knows that byq q admits a t stable transversal in x wq which we denote by cal n yq wq we prove that under certain hypotheses cal n yq wq is isomorphic to the orbit closure of a highest weight vector in a certain weyl module we also obtain a generalisation of this result under slightly weaker hypotheses further we prove that our hypotheses are satisfied when q is a maximal parabolic subgroup corresponding to a minuscule or cominuscule fundamental weight and x yq is an irreducible component of the boundary of x wq that is the complement of the open orbit of the stabiliser in g of x wq as a consequence we describe the singularity of x wq along byq q and obtain that the boundary of x wq equals its singular locus

globally f regular and log fano varieties abstract we prove that every globally f regular variety is log fano in other words if a prime characteristic variety x is globally f regular then it admits an effective q divisor such that k x is ample and x has controlled kawamata log terminal in fact globally f regular singularities a weak form of this result can be viewed as a prime characteristic analog of de fernex and hacon s new point of view on kawamata log terminal singularities in the non q gorenstein case we also prove a converse statement in characteristic zero every log fano variety has globally f regular type our techniques apply also to f split varieties which we show to satisfy a log calabi yau condition we also prove a kawamata viehweg vanishing theorem for globally f regular pairs

degenerate flag varieties of type a frobenius splitting and bw theorem let mathcal f a lambda be the pbw degeneration of the flag varieties of type a n 1 these varieties are singular and are acted upon with the degenerate lie group sl n a we prove that mathcal f a lambda have rational singularities are normal and locally complete intersections and construct a desingularization r lambda of mathcal f a lambda the varieties r lambda can be viewed as towers of successive mathbb p 1 fibrations thus providing an analogue of the classical bott samelson demazure hansen desingularization we prove that the varieties r lambda are frobenius split this gives us frobenius splitting for the degenerate flag varieties and allows to prove the borel weil type theorem for mathcal f a lambda using the atiyah bott lefschetz formula for r lambda we compute the q characters of the highest weight mathfrak sl n modules

on compactifications of the steinberg zero fiber abstract let g be a connected semisimple linear algebraic group over an algebraically closed field k of positive characteristic and let x denote an equivariant embedding of g we define a distinguished steinberg fiber n in g called the zero fiber and prove that the closure of n within x is normal and cohen macaulay furthermore when x is smooth we prove that the closure of n is a local complete intersection

an algebraic construction of boundary quantum field theory we build up local time translation covariant boundary quantum field theory nets of von neumann algebras mathcal a v on the minkowski half plane m starting with a local conformal net mathcal a of von neumann algebras on mathbb r and an element v of a unitary semigroup mathcal e mathcal a associated with mathcal a the case v 1 reduces to the net mathcal a considered by rehren and one of the authors if the vacuum character of mathcal a is summable mathcal a v is locally isomorphic to mathcal a we discuss the structure of the semigroup mathcal e mathcal a by using a one particle version of borchers theorem and standard subspace analysis we provide an abstract analog of the beurling lax theorem that allows us to describe in particular all unitaries on the one particle hilbert space whose second quantization promotion belongs to mathcal e mathcal a 0 with mathcal a 0 the u 1 current net each such unitary is attached to a scattering function or more generally to a symmetric inner function we then obtain families of models via any buchholz mack todorov extension of mathcal a 0 a further family of models comes from the ising model

euclidean invariant integral representations for schwinger functionals it is shown that if the schwinger functions of a wightman theory with a mass gap can be represented as moments of a complex measure there is also such a representation by a measure that is invariant under the euclidean group a necessary and sufficient condition is also given for the exponential of a quadratic form on the test function space to be the characteristic functional of a complex measure

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

local algebras of observables and pointlike localized fields we present a method to recover wightman fields from a haag kastler theory of local observables this may provide a basis for the comparison of different theories and for an algebraic description of high energy behaviour

a class of nontrivial weakly local massive wightman fields with interpolating properties it is shown that in a quantum field theory satisfying wightman s axioms with locality replaced by weak locality and cyclicity by a weak irreducibility every unitary poincare invariant and cpt invariant operator is a scattering operator in the lsz sense the proof is given by explicit construction of a corresponding class of nontrivial weakly local massive wightman fields this result implies jost s conjecture that only locality leads to nontrivial restrictions for the scattering operator and extends corresponding results of schneider

affine demazure modules and t fixed point subschemes in the affine grassmannian let g be a simple algebraic group defined over c and t be a maximal torus of g for a dominant coweight of g the t fixed point subscheme formula of the schubert variety formula in the affine grassmannian grg is a finite scheme we prove that for all such if g is of type a or d and for many of them if g is of type e there is a natural isomorphism between the dual of the level one affine demazure module corresponding to and the ring of functions twisted by certain line bundle on grg of formula we use this fact to give a geometrical proof of the frenkel kac segal isomorphism between basic representations of affine algebras of a d e type and lattice vertex algebras

a remark on ulrich and acm bundles abstract i show that on any smooth projective ordinary curve of genus at least two and a projective embedding there is a natural example of a stable ulrich bundle for this embedding namely the sheaf b x 1 of locally exact differentials twisted by o x 1 given by this embedding and in particular there exist ordinary varieties of any dimension which carry ulrich bundles in higher dimensions assuming x is frobenius split variety i show that b x 1 is an acm bundle and if x is also a calabi yau variety and p 2 then b x 1 is not a direct sum of line bundles in particular i show that b x 1 is an acm bundle on any ordinary calabi yau variety i also prove a characterization of projective varieties with trivial canonical bundle such that b x 1 is acm for some projective embedding datum all such varieties are frobenius split with trivial canonical bundle

another strongly exceptional collection of coherent sheaves on a grassmannian abstract on the grassmannian of 2 dimensional subspaces in a finite dimensional linear space we construct a karoubian complete strongly exceptional po set of coherent sheaves parametrized by the cosets of the weyl group of the general linear group of the linear space modulo the weyl group of the parabolic subgroup stabilizing the subspace from subquotients of the frobenius direct image of the structure sheaf of the grassmannian defined over a field of large positive characteristic our collection diverges from the one discovered by kapranov we also show in the general setting over any field of positive characteristic that the sheaf corresponding to the longest element of the cosets is a direct summand of the frobenius direct image of the structure sheaf of the homogeneous space

deformations of quantum field theories and integrable models deformations of quantum field theories which preserve poincare covariance and localization in wedges are a novel tool in the analysis and construction of model theories here a general scenario for such deformations is discussed and an infinite class of explicit examples is constructed on the borchers uhlmann algebra underlying wightman quantum field theory these deformations exist independently of the space time dimension and contain the recently studied warped convolution deformation as a special case in the special case of two dimensional minkowski space they can be used to deform free field theories to integrable models with non trivial s matrix

on the separation property of orbits in representation spaces a subset x of a vector space v is said to have the separation property if it separates linear forms in the following sense given a pair of linearly independent linear forms on v there is a point x on x such that x 0 and x 0 a more geometric way to express this is the following every linear subspace h v of codimension 1 is linearly spanned by its intersection with x the separation property was first asked for conjugacy classes in simple lie algebras in connection with some classification problems we give a general answer for orbits in representation spaces of algebraic groups and discuss in detail some special cases we also introduce a strong and a weak separation property which come up very naturally in our setting it turns out that these separation properties have a number of very nice features for example we discovered the surprising fact that in an irreducible representation of a connected semisimple group every linear hyperplane meets every orbit and we show that a generic orbit always has the separation property

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

rees algebras of f regular type abstract we study the f regularity of rees algebras r i a it in terms of the global f regularity of the blowing up x proj r i of spec a as it reads global f regularity is a global analog of strong f regularity defined via splitting of frobenius maps in prime characteristic and these notions are extended to characteristic zero by reduction modulo p 0 we study in detail the case where a m is a two dimensional local ring and i is an m primary ideal in characteristic zero the condition for r i to have f regular type is described in terms of the dual graph of a resolution x on which i o x is invertible we also prove some miscellaneous results concerning singularities of rees algebras and extended rees algebras of higher dimension

polytopes associated to demazure modules of symmetrizable kac moody algebras of rank two let 1 2 be the two fundamental weights of a symmetrizable kac moody algebra g of rank two hence necessarily affine or finite and an element of the weyl group in this paper we construct polytopes p 1 p 2 rl and a linear map rl h such that for any dominant weight k1 1 k2 2 we have char e e e x where the sum is over all the integral points x of the polytope k1p 1 k2p 2 furthermore we show that there exists a flat deformation of the schubert variety s into the toric variety defined by p 1 p 2

hankel determinantal rings have rational singularities abstract hankel determinantal rings i e determinantal rings defined by minors of hankel matrices of indeterminates arise as homogeneous coordinate rings of higher order secant varieties of rational normal curves they may also be viewed as linear specializations of generic determinantal rings we prove that over fields of characteristic zero hankel determinantal rings have rational singularities in the case of positive prime characteristic we prove that they are f pure independent of the characteristic we give a complete description of the divisor class groups of these rings and show that each divisor class group element is the class of a maximal cohen macaulay module

on the b canonical splittings of flag varieties abstract let g be a semisimple algebraic group over an algebraically closed field of positive characteristic in this note we show that an irreducible closed subvariety of the flag variety of g is compatibly split by the unique canonical frobenius splitting if and only if it is a richardson variety i e an intersection of a schubert and an opposite schubert variety

the space of triangles vanishing theorems and combinatorics abstract we consider compactifications of pn 3 ij the space of triples of distinct points in projective space one such space is a singular variety of configurations of points and lines another is the smooth compactification of fulton and macpherson and a third is the triangle space of schubert and semple we compute the sections of line bundles on these spaces and show that they are equal as gl n representations to the generalized schur modules associated to bad generalized young diagrams with three rows borel weil theorem on the one hand this yields weyl type character and dimension formulas for the schur modules on the other a combinatorial picture of the space of sections cohomology vanishing theorems play a key role in our analysis

frobenius splitting of hilbert schemes of points on surfaces let x be a quasiprojective smooth surface defined over an algebraically closed field of positive characteristic in this note we show that if x is frobenius split then so is the hilbert scheme hilb n x of n points inx in particular we get the higher cohomology vanishing for ample line bundles on hilb n x when x is projective and frobenius split

f rational rings have rational singularities it is proved that an excellent local ring of prime characteristic in which a single ideal generated by any system of parameters is tightly closed must be pseudorational a key point in the proof is a characterization of f rational local rings as those cohen macaulay local rings r m in which the local cohomology module h d m r where d is the dimension of r have no submodules stable under the natural action of the frobenius map an analog for finitely generated algebras over a field of characteristic zero is developed which yields a reasonably checkable tight closure test for rational singularities of an algebraic variety over inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i without reference to a desingularization

algebraic frobenius splitting of cotangent bundles of flag varieties abstract following the program of algebraic frobenius splitting begun by kumar and littelmann we use representation theoretic techniques to construct a frobenius splitting of the cotangent bundle of the flag variety of a semisimple algebraic group over an algebraically closed field of positive characteristic we also show that this splitting is the same as one of the splittings constructed by kumar lauritzen and thomsen

unsupervised segmentation of ultrasonic liver images by multiresolution fractal feature vector the feasibility of selecting fractal feature vector based on multiresolution analysis to segment suspicious abnormal regions of ultrasonic liver images is described in this paper the proposed feature extraction algorithm is based on the spatial frequency decomposition and fractal geometry segmentation of various liver diseases reveals that the fractal feature vector based on multiresolution analysis is trustworthy a quantitative characterization based on the proposed unsupervised segmentation algorithm can be utilized to establish an automatic computer aided diagnostic system as well to increase the visual interpretation capability of ultrasonic liver image for junior physicians off line learning software is developed to investigate the visual criteria

on the causal structure of minkowski spacetime the causal structure of minkowski spacetime m is discussed in terms of the notions of causal complementation and causal completion these geometric notions are relevant for quantum field theory and the theory of the klein gordon equation particular attention is given to closed convex causally complete subsets of m and the properties of such sets are discussed the study of such sets is motivated by potential applications to the theory of local nets of von neumann algebras the notion of the envelope of uniqueness of a subset of m familiar from the theory of the wave equation is discussed and some results about the relation of this envelope to the causal completion of the set are presented

detection of lines and boundaries in speckle images application to medical ultrasound describes an approach to boundary detection in ultrasound speckle based on an image enhancement technique the enhancement algorithm works by filtering the image with sticks short line segments which are varied in orientation to achieve the maximum projected value at each point in this paper we present three significant extensions to improve the performance of the basic method first we investigate the effect of varying the size and shape of the sticks we show that these variations affect the performance of the algorithm in very fundamental ways for example by making it more or less sensitive to thinner or more tightly curving boundaries second we present a means of improving the performance of this technique by estimating the distribution function of the orientation of the line passing through each point finally we show that images can be stained for easier visual interpretation by applying to each pixel a false color whose hue is related to the orientation of the most prominent line segment at that point examples are given to illustrate the performance of the different settings on a single image