stratification for module categories of finite group schemes the second author was partly supported by nsf grant dms 1503044 the fourth author was partly supported by nsf grant dms 0953011

derived localisation of algebras and modules for any dg algebra a not necessarily commutative and a subset s in h a the homology of a we construct its derived localisation l s a together with a map a l s a well defined in the homotopy category of dg algebras which possesses a universal property similar to that of the ordinary localisation but formulated in homotopy invariant terms even if a is an ordinary ring l s a may have non trivial homology unlike the commutative case the localisation functor does not commute in general with homology but instead there is a spectral sequence relating h l s a and l s h a this spectral sequence collapses when e g s is an ore set or when a is a free ring we prove that l s a could also be regarded as a bousfield localisation of a viewed as a left or right dg module over itself combined with the results of dwyer kan on simplicial localisation this leads to a simple and conceptual proof of the topological group completion theorem further applications include algebraic k theory cyclic and hochschild homology strictification of homotopy unital algebras idempotent ideals the stable homology of various mapping class groups and kontsevich s graph homology

cohen macaulay modules co torsion pairs and virtually gorenstein algebras abstract we use torsion pairs in stable categories and cotorsion pairs in modules categories to study in general infinitely generated cohen macaulay modules and a generalization of modules of finite projective or injective dimension over an artin algebra we concentrate our investigation to the study of virtually gorenstein algebras which provide a common generalization of gorenstein algebras and algebras of finite representation or cohen macaulay type this class of algebras on the one hand has rich homological structure and satisfies several representation torsion theoretic finiteness conditions and on the other hand it is closed under various operations for instance derived equivalences and stable equivalences of morita type in addition virtual gorensteinness provides a useful tool for the study of the gorenstein symmetry conjecture and modified versions of the telescope conjecture for module or stable categories

comodules and landweber exact homology theories abstract we show that if e is a commutative mu algebra spectrum such that e is landweber exact over mu then the category of e e comodules is equivalent to a localization of the category of mu mu comodules this localization depends only on the heights of e at the integer primes p it follows for example that the category of e n e n comodules is equivalent to the category of v n 1 bp v n 1 bp comodules these equivalences give simple proofs and generalizations of the miller ravenel and morava change of rings theorems we also deduce structural results about the category of e e comodules we prove that every e e comodule has a primitive we give a classification of invariant prime ideals in e and we give a version of the landweber filtration theorem

convergence rate for a curse of dimensionality free method for a class of hjb pdes in previous work of the first author and others max plus methods have been explored for solution of first order nonlinear hamilton jacobi bellman partial differential equations hjb pdes and corresponding nonlinear control problems although max plus basis expansion and max plus finite element methods can provide substantial computational speed advantages they still generally suffer from the curse of dimensionality here we consider hjb pdes where the hamiltonian takes the form of a pointwise maximum of linear quadratic forms the approach to solution will be rather general but in order to ground the work we consider only constituent hamiltonians corresponding to long run average cost per unit time optimal control problems for the development we consider a previously obtained numerical method not subject to the curse of dimensionality the method is based on construction of the dual space semigroup corresponding to the hjb pde this dual space semigroup is constructed from the dual space semigroups corresponding to the constituent linear quadratic hamiltonians the dual space semigroup is particularly useful due to its form as a max plus integral operator with kernel obtained from the originating semigroup one considers repeated application of the dual space semigroup to obtain the solution although previous work indicated that the method was not subject to the curse of dimensionality it did not indicate any error bounds or convergence rate here we obtain specific error bounds

a curse of dimensionality free numerical method for solution of certain hjb pdes in previous works of the author and others max plus methods have been explored for the solution of first order nonlinear hamilton jacobi bellman partial differential equations hjb pdes and corresponding nonlinear control problems these methods exploit the max plus linearity of the associated semigroups in particular although the problems are nonlinear the semigroups are linear in the max plus sense these methods have been used successfully to compute solutions although they provide certain computational speed advantages they still generally suffer from the curse of dimensionality here we consider hjb pdes in which the hamiltonian takes the form of a pointwise maximum of linear quadratic forms the approach to the solution will be rather general but in order to ground the work we consider only constituent hamiltonians corresponding to long run average cost per unit time optimal control problems for the development we obtain a numerical method not subject to the curse of dimensionality the method is based on construction of the dual space semigroup corresponding to the hjb pde this dual space semigroup is constructed from the dual space semigroups corresponding to the constituent linear quadratic hamiltonians the dual space semigroup is particularly useful due to its form as a max plus integral operator with a kernel obtained from the originating semigroup one considers repeated application of the dual space semigroup to obtain the solution

semistable models for modular curves and power operations for morava e theories of height 2 abstract we construct an integral model for lubin tate curves these curves arise as moduli of finite subgroups of deformations of formal groups in particular they are p adic completions of the modular curves x 0 p at a mod p supersingular point our model is semistable in the sense that the only singularities of its special fiber are normal crossings given this model we obtain a uniform presentation for the dyer lashof algebras for morava e theories of height 2 these algebras are local moduli of power operations in elliptic cohomology

the algebraic chromatic splitting conjecture for noetherian ring spectra we formulate a version of hopkins chromatic splitting conjecture for an arbitrary structured ring spectrum r and prove it whenever pi r is noetherian as an application these results provide a new local to global principle in the modular representation theory of finite groups

min max characterization of a small noise limit on risk sensitive control stochastic control problems on a finite horizon with exponential cost criteria are considered by taking a kind of singular limit a hamilton jacobi isaacs equation is obtained its solution is characterized as the lower value function of a deterministic differential game related to robust control of nonlinear systems

buildings elliptic curves and the k 2 local sphere we investigate a dense subgroup of the second morava stabilizer group given by a certain group of quasi isogenies of a supersingular elliptic curve in characteristic p the group acts on the bruhat tits building for gl2 through its action on the adic tate module this action has finite stabilizers giving a small resolution for the homotopy fixed point spectrum eh 2 hgal by spectra of topological modular forms here e2 is a version of morava e theory and gal gal

smashing localizations of rings of weak global dimension at most one abstract we show for a ring r of weak global dimension at most one that there is a bijection between the smashing subcategories of its derived category and the equivalence classes of homological epimorphisms starting in r if moreover r is commutative we prove that the compactly generated localizing subcategories correspond precisely to flat epimorphisms we also classify smashing localizations of the derived category of any valuation domain and provide an easy criterion for the telescope conjecture tc for any commutative ring of weak global dimension at most one as a consequence we show that the tc holds for any commutative von neumann regular ring r and it holds precisely for those prufer domains which are strongly discrete

tmf0 3 characteristic classes for string bundles we compute the completed tmf 0 3 cohomology of the 7 connected cover bstring of bo we use cubical structures on line bundles over elliptic curves to construct an explicit class which together with the pontryagin classes freely generates the cohomology ring

towards the finiteness of p lk n s0 let g be a closed subgroup of the nth morava stabilizer group sn n2 and let denote the continuous homotopy fixed point spectrum of devinatz and hopkins if g z the subgroup topologically generated by an element z in the p sylow subgroup of sn and z is non torsion in the quotient of by its center we prove that the homology of any k n 2 acyclic finite spectrum annihilated by p is of essentially finite rank we also show that the units in en fixed by z are just the units in the witt vectors with coefficients in the field of pn elements if n 2 and p5 we show that if g is a closed subgroup of not contained in the center then g contains an open subnormal subgroup u such that the mod p homotopy of is of essentially finite rank

two approximations of solutions of hamilton jacobi equations abstract equations of hamilton jacobi type arise in many areas of application including the calculus of variations control theory and differential games the associated initial value problems almost never have global time classical solutions and one must deal with suitable generalized solutions the correct class of generalized solutions has only recently been established by the authors this article establishes the convergence of a class of difference approximations to these solutions by obtaining explicit error estimates analogous results are proved by similar means for the method of vanishing viscosity author

locating the first nodal set in higher dimensions we extend the two dimensional results of jerison 2000 on the location of the nodal set of the first neumann eigenfunction of a convex domain to higher dimensions if a convex domain in n is contained in a long and thin cylinder 0 n x b 0 with nonempty intersections with x 1 0 and x 1 n then the first nonzero eigenvalue is well approximated by the eigenvalue of an ordinary differential equation by a bound proportional to whose coefficients are expressed in terms of the volume of the cross sections of the domain also the first nodal set is located within a distance comparable to e near the zero of the corresponding ordinary differential equation

morava modules and brown comenetz duality fix a prime number p and let x be a finite spectrum whose n 1 st morava k theory is trivial but whose n th morava k theory is nontrivial n 0 we prove following a method outlined to us by m j hopkins that if 2 p n 2 n 2 the morava module of the brown comenetz dual of the e n localization of x is isomorphic to a suspension of the pontryagin dual of the morava module of x to complete this proof we found it necessary to develop a more canonical construction of certain modified adams spectral sequences this construction should be of independent interest

localization and periodicity in unstable homotopy theory in this paper we develop a hierarchy of natural localizations of spaces called the vn periodizations for n 0 which may be used to expose and study periodic phenomena in unstable homotopy theory these vn periodizations act less radically than the corresponding homological localizations 2 and respect fibrations to a very considerable extent a major part of this paper is devoted to developing the general theory of periodizations of spaces thereby providing a foundation for the study of the vn periodization and many others some of this general theory has been developed independently by dror farjoun 13 14 and we refer the reader to his work for an alternative approach with other interesting general results during the past decade remarkable progress was made by ravenel hopkins devinatz and smith 12 16 17 33 toward a global understanding of stable periodic phenomena and we hope that the present paper will help to prepare the way for a similar understanding of unstable periodic phenomena an excellent exposition of localization and periodicity in stable homotopy theory is now available in ravenel s book 35 major features of stable homotopy are understood chromatically as manifestations of more basic periodic phenomena these phenomena belong to a hierarchy starting with those detected rationally followed by those detected in classical k theory and in the successive morava k theories most fundamentally each finite cw spectrum has an intrinsic periodicity given by a vn self map which becomes a selfequivalence after suitable localization for simplicity we describe our results in the pointed homotopy category hoo of connected cw complexes for spaces w y e hoo we call y w periodic or w local when the pointed mapping space map w y is contractible or equivalently when xt w y for t 0 as shown more generally in 3 corollary 7 2 10 or 13 there is a natural initial example x pwx of a map from x to a w periodic space pwx in hoo and we call this the w periodization or w localization of x our notation is derived from the classical example where w sn l and pwx is the n th postnikov section

tate cohomology and periodic localization of polynomial functors in this paper we show that goodwillie calculus as applied to functors from stable homotopy to itself interacts in striking ways with chromatic aspects of the stable category localized at a fixed prime p let t n be the telescope of a v n self map of a finite s module of type n the periodicity theorem of hopkins and smith implies that the bousfield localization functor associated to t n is independent of choices goodwillie s general theory says that to any homotopy functor f from s modules to s modules there is an associated tower under f p d f such that f p d f is the universal arrow to a d excisive functor our first main theorem says that p d f pd 1f always admits a homotopy section after localization with respect to t n and so also after localization with respect to morava k theory k n thus after periodic localization polynomial functors split as the product of their homogeneous factors this theorem follows from our second main theorem which is equivalent to the following for any finite group g the tate spectrum mathcal t g t n is weakly contractible this strengthens and extends previous theorems of greenlees sadofsky hovey sadofsky and mahowald shick the periodicity theorem is used in an essential way in our proof the connection between the two theorems is via a reformulation of a result of mccarthy on dual calculus

roots of unity and the adams novikov spectral sequence for formal a modules the cohomology of a hopf algebroid related to the adams novikov spectral sequence for formal a modules is studied in the special case in which a is the ring of integers in the field obtained by adjoining pth roots of unity to 2p the p adic numbers information about these cohomology groups is used to give new proofs of results about the e2 term of the adams spectral sequence based on 2 local complex k theory and about the odd primary kervaire invariant elements in the usual adams novikov spectral sequence one of the most powerful tools used in the computation of stable homotopy groups is the adams novikov spectral sequence the e2 term of this spectral sequence is a certain ext group derived from a universal formal group law in r3 the corresponding ext group for a universal formal a module for a the ring of algebraic integers in an algebraic number field k or its padic completion was introduced and certain conjectures about these groups were formulated one of these conjectures concerning the value of extli was confirmed in j using a hopf algebroid i e a generalized hopf algebra in which the left and right units need not agree eat which generalizes the hopf algebroid k k of stable cooperations for complex k theory the present paper is concerned with the cohomology of eat in the special case of a p where c is a pth root of unity and zp denotes the p adic integers we will show that in this case eat is contained in an extension of hopf algebroids ea ea t eat and that the cohomology of eat can be completely described this provides us with information about the cohomology of eat via the cartan eilenberg spectral sequence associated to this extension two applications of this result are presented in the case p 2 eat can be identified with the 2 adic completion of the hopf algebroid k k 2 of stable cooperations for 2 primary complex k theory in this case the cohomology received by the editors february 1 1989 1980 mathematics subject classification 1985 revision primary 55t25 secondary 55n22 144l05 1991 american mathematical society 0002 9947 91 1 00 25 per page

cohomological quotients and smashing localizations the quotient of a triangulated category modulo a subcategory was defined by verdier motivated by the failure of the telescope conjecture we introduce a new type of quotients for any triangulated category which generalizes verdier s construction slightly simplifying this concept the cohomological quotients are flat epimorphisms whereas the verdier quotients are ore localizations for any compactly generated triangulated category s a bijective correspondence between the smashing localizations of s and the cohomological quotients of the category of compact objects in s is established we discuss some applications of this theory for instance the problem of lifting chain complexes along a ring homomorphism this is motivated by some consequences in algebraic k theory and demonstrates the relevance of the telescope conjecture for derived categories another application leads to a derived analogue of an almost module category in the sense of gabber ramero it is shown that the derived category of an almost ring is of this form

phantom maps and chromatic phantom maps in the first part we determine conditions on spectra x and y under which either every map from x to y is phantom or no nonzero maps are we also address the question of whether such all or nothing behavior is preserved when x is replaced with v a x for v finite in the second part we introduce chromatic phantom maps a map is n phantom if it is null when restricted to finite spectra of type at least n we define divisibility and finite type conditions which are suitable for studying phantom maps we show that the duality functor wn defined by mahowald and rezk is the analog of brown comenetz duality for chromatic phantom maps and give conditions under which the natural map y w x y is an isomorphism 1 introduction this paper takes place in the stable homotopy category a map x y of spectra is said to be phantom if for every finite spectrum v and every map v x the composite v x y is zero we write ph x y for the group of all phantom maps from x to y the first half of the paper section 2 deals with some results of a general nature the theme is to determine conditions on spectra x and y under which either every map from x to y is phantom or no nonzero maps are we also address the question of whether such all or nothing behavior is preserved when x is replaced with v a x for v finite our main tool is the observation that for y of finite type a map x y is phantom if and only if it is a divisible element of the group x y we also make use of the connection between phantom maps and brown comenetz duality in the second half of the paper section 3 we work in the 7 local stable homotopy category and introduce generalizations of phantom maps called chro matic phantom maps a map x y is n phantom if for every finite spectrum v of type at least n and every map v x the composite v x y is zero the results of this section which require more background of the reader gener alize some of the results of the first half of the paper we define analogs of the divisibility and finite type conditions which are suitable for studying chromatic phantom maps the significance of chromatic phantom maps is demonstrated by the fact that the kernel in homotopy of the important map lnsfi lx n sp is pre cisely the n phantom homotopy classes we show that the duality functor wn defined by mahowald and rezk 11 is the analog of brown comenetz duality our theorem 3 13 gives conditions under which the natural map y w2 xy is

on the telescopic homotopy theory of spaces in telescopic homotopy theory a space or spectrum x is approximated by a tower of localizations lnx n 0 taking account of vn periodic homotopy groups for progressively higher n for each n 1 we construct a telescopic kuhn functor n carrying a space to a spectrum with the same vn periodic homotopy groups and we construct a new functor n left adjoint to n using these functors we show that the nth stable monocular homotopy category comprising the nth fibers of stable telescopic towers embeds as a retract of the nth unstable monocular homotopy category in two ways one giving infinite loop spaces and the other giving infinite ln suspension spaces we deduce that ravenel s stable telescope conjectures are equivalent to unstable telescope conjectures in particular we show that the failure of ravenel s nth stable telescope conjecture implies the existence of highly connected infinite loop spaces with trivial johnson wilson e n homology but nontrivial vn periodic homotopy groups showing a fundamental difference between the unstable chromatic and telescopic theories as a stable chromatic application we show that each spectrum is k n equivalent to a suspension spectrum as an unstable chromatic application we determine the e n localizations and k n localizations of infinite loop spaces in terms of e n localizations of spectra under suitable conditions we also determine the e n localizations and k n localizations of arbitrary postnikov h spaces

the brown peterson homology of mahowald s _ spectra we compute the brown peterson homology of mahowald s xk spectrum which is the thom spectrum induced from q j2k 1 s2 q2s3 bo and the edge homomorphism of the adams novikov spectral sequence for 7r xka we then compute the nonnilpotent elements of 7r xk the main theorem of the beautiful paper nilpotence and stable homotopy theory i 12 states that for an arbitrary ring spectrum x the kernel of the brown peterson hurewicz homomorphism consists of nilpotent elements the more standard way to describe the brown peterson hurewicz homomorphism is that firstly one needs to describe the brown peterson homology of x bp x as well as the comodule structure of bp x over quillen s algebra bp bp secondly one needs to compute the edge homomorphism of the adams novikov differentials in the present paper we will describe bp xk and the bp hurewicz homomorphism explicitly for mahowald s noncommutative ring spectra xk let n s1 bo represent the generator of 7r i bo z 2 since bo is a double loop space there is an induced map r q2 3 bo then one takes the composite map j2k i q25 3 a bo where ji is the james construction these maps result in thom spectra which will be denoted by xk due to mahowald 8 17 18 19 the interest of this xk spectrum is that this xk spectrum realizes part of the dual steenrod algebra that is h xk z 2 z 2 i 42 43 k the basic references for brown peterson theory are 1 26 31 let bp be the 2 primary brown peterson spectrum then the coefficient ring is bp z 2 vl v2 v3 ivii 2 2i 1 quillen s algebra is bp bp bp t t2 t3 jt j 2 2i1 we now state the main results of this paper theorem a let bp be the 2 primary brown peterson spectrum and the invariant ideal ik 2 vl v2 v3 vkl1 then bp xk bp ik tl t2 t3 tk received by the editors may 6 1993 1991 mathematics subject classification primary 55q 10 1994 american mathematical society 0002 9947 94 1 00 25 per page

generalized tensor idempotents and the telescope conjecture we transpose rickard s construction of idempotent representations to the general context of tensor triangulated categories we connect our construction to the telescope conjecture and prove that the latter is of local nature as an illustration we extend the affine telescope theorem of neeman to arbitrary noetherian schemes we also develop supports for non compact objects

on the structure of the hopf algebroid e n e n this paper is devoted to giving a description of e n e n ravenel and hopkins have studied a certain completion f n of e n and have showed that f n f n contains a subalgebra isomorphic to c s n a the algebra of continuous functions on the n th morava stabilizer group taking values in the ring of integers in an unramified degree n extension of the p adic numbers they also show that this completion is split with respect to this subalgebra i e f n f n c s n a a f n we display an injection of e n e n into c s n s n a which extends to the hopkins ravenel completion and identify the image of their subalgebra as the subalgebra of functions which are invariant under translation by elements of s n using this we give a formula for the coaction of c s n a on e n and another proof of the hopkins ravenel splitting theorem

stratifying modular representations of finite groups we classify localising subcategories of the stable module category of a finite group that are closed under tensor product with simple or equiv alently all modules one application is a proof of the telescope conjecture in this context others include new proofs of the tensor product theorem and of the classification of thick subcategories of the finitely generated modules which avoid the use of cyclic shifted subgroups along the way we establish similar classifications for differential graded modules over graded polynomial rings and over graded exterior algebras

ideals in triangulated categories phantoms ghosts and skeleta we begin by showing that in a triangulated category specifying a projective class is equivalent to specifying an ideal i of morphisms with certain properties and that if i has these properties then so does each of its powers we show how a projective class leads to an adams spectral sequence and give some results on the convergence and collapsing of this spectral sequence we use this to study various ideals in the stable homotopy category we examine phantom maps skeletal phantom maps superphantom maps and ghosts a ghost is a map which induces the zero map of homotopy groups we show that ghosts lead to a stable analogue of the lusternik schnirelmann category of a space and we calculate this stable analogue for low dimensional real projective spaces we also give a relation between ghosts and the hopf and kervaire invariant problems in the case ofa modules over ana ring spectrum the ghost spectral sequence is a universal coefficient spectral sequence from the phantom projective class we derive a generalized milnor sequence for filtered diagrams of finite spectra and from this it follows that the group of phantom maps fromxtoycan always be described as alim 1group the last two sections focus on algebraic examples in the derived category of an abelian category we study the ideal of maps inducing the zero map of homology groups and find a natural setting for a result of kelly on the vanishing of composites of such maps we also explain how pure exact sequences relate to phantom maps in the derived category of a ring and give an example showing that phantoms can compose non trivially

an e splitting of spin bordism the work determines the e structure of the k 1 local spin bordism spectrum by giving a multiplicative splitting into e cells the splitting leads to a cellular decomposition of the a map and creates new isomorphisms of conner floyd type introduction and statement of results one of the most important results on spin bordism goes back to anderson brown and peterson abp they showed that two spin manifolds are spin bordant if and only if all stiefel whitney and ko characteristic numbers coincide moreover the spin bordism groups can be computed from the additive 2 local splitting mspin

k 1 local topological modular forms we construct the witten orientation of the topological modular forms spectrum tmf in the k 1 local setting by attaching e cells to the bordism theory mo we also identify the ko homology of tmf with the ring of divided congruences

homotopical localizations of module spectra we prove that stable f localizations where f is any map of spectra preserve ring spectrum structures and module spectrum structures under suitable hypotheses and we use this fact to describe all possible localizations of the integral eilenberg mac lane spectrum hz as a consequence of this study we infer that localizations of stable gems are stable gems and it also follows that there is a proper class of nonequivalent stable localizations

second order numerical methods for first order hamilton jacobi equations we present practical numerical methods which produce provably second order approximations for a class of stationary first order hamilton jacobi partial differential equations using probabilistic methods we derive high order asymptotic expansions for a first order method and then use those results to design second order methods we prove second order convergence for the solution and for its gradient on a subset of the domain where the solution is smooth although we limit our attention to second order schemes in principle the techniques in this paper can be extended to arbitrarily high order methods examples illustrate the rate of convergence as well as global sharp resolution of discontinuities the hamilton jacobi equations we consider correspond to deterministic optimal control problems and our rate of convergence results are valid for the value functions and for the optimal feedback controls

local duality in algebra and topology abstract the first goal of this paper is to provide an abstract framework in which to formulate and study local duality in various algebraic and topological contexts for any stable category c together with a collection of compact objects k c we construct local cohomology and local homology functors satisfying an abstract version of local duality when specialized to the derived category of a commutative ring a and a suitable ideal in a we recover the classical local duality due to grothendieck as well as generalizations by greenlees and may more generally applying our result to the derived category of quasi coherent sheaves on a quasi compact and separated scheme x implies the local duality theorem of alonso tarrio jeremias lopez and lipman as a second objective we establish local duality for quasi coherent sheaves over many algebraic stacks in particular those arising naturally in stable homotopy theory after constructing an appropriate model of the derived category in terms of comodules over a hopf algebroid we show that in familiar cases the resulting local cohomology and local homology theories coincide with functors previously studied by hovey and strickland furthermore our framework applies to global and local stable homotopy theory in a way which is compatible with the algebraic avatars of these theories in order to aid computability we provide spectral sequences relating the algebraic and topological local duality contexts

on the topological hochschild homology of bu i the purpose of this paper and its sequel is to determine the homotopy groups of the spectrum thh here p is an odd prime is the adams summand of p local connective k theory see for example 25 and thh is the topological hochschild homology construction introduced by bokstedt in 3 in the present paper we will determine the mod p homotopy groups of thh and also the integral homotopy groups of thh l where l denotes the periodic adams summand in the sequel we will investigate the integral homotopy groups of thh using our present results as a starting point the thh construction appears to be of basic importance in algebraic k theory because it combines two useful properties it can be used to construct good approximations to the algebraic k theory functor and it is very accessible to calculation we shall review what is known about the first property in a moment the second property was demonstrated by bokstedt s calculation in his paper 4 of the homotopy groups of thh hz p and thh hz here hz p and hz denote the evident eilenberg mac lane spectra it is natural to ask about thh r for other popular ring spectra r and our work is a first step in this direction we pay special attention to the connective case because this is the case which is likely to be relevant in applications see subsection 1 4 below the calculation which we present in this paper is a homotopy theoretic one which uses the adams spectral sequence hereafter abbreviated ass this calculation has several interesting features in particular it is a pleasing example of an ass calculation in which although there are infinitely many differentials it is still possible to get the complete answer here is a summary of the contents of the paper in section 2 we review the facts we need to know about ordinary hochschild homology in section 3 we do the same

convergence of the optimal feedback policies in a numerical method for a class of deterministic optimal control problems we consider a markov chain based numerical approximation method for a class of deterministic nonlinear optimal control problems it is known that methods of this type yield convergent approximations to the value function on the entire domain these results do not easily extend to the optimal control which need not be uniquely defined on the entire domain there are however regions of strong regularity on which the optimal control is well defined and smooth typically the union of these regions is open and dense in the domain using probabilistic methods we prove that on the regions of strong regularity the markov chain method yields a convergent sequence of approximations to the optimal feedback control the result is illustrated with several examples

max plus eigenvector representations for solution of nonlinear h sub spl infin problems basic concepts the h sub spl infin problem for a nonlinear system is considered the corresponding dynamic programming equation is a fully nonlinear first order steady state partial differential equation pde possessing a term which is quadratic in the gradient the solutions are typically nonsmooth and further there is nonuniqueness among the class of viscosity solutions in the case where one tests a feedback control to see if it yields an h sub spl infin controller the pde is a hamilton jacobi bellman equation in the case where the optimal feedback control is being determined as well the problem takes the form of a differential game and the pde is in general an isaacs equation the computation of the solution of a nonlinear steady state first order pde is typically quite difficult in this paper we develop an entirely new class of methods for obtaining the correct solution of such pdes these methods are based on the linearity of the associated semigroup over the max plus or in some cases min plus algebra in particular solution of the pde is reduced to solution of a max plus or min plus eigenvector problem for known unique eigenvalue 0 the max plus multiplicative identity it is demonstrated that the eigenvector is unique and that the power method converges to it an example is included

markov chain approximations for deterministic control problems with affine dynamics and quadratic cost in the control we consider the construction of markov chain approximations for an important class of deterministic control problems the emphasis is on the construction of schemes that can be easily implemented and which possess a number of highly desirable qualitative properties the class of problems covered is that for which the control is affine in the dynamics and with quadratic running cost this class covers a number of interesting areas of application including problems that arise in large deviations risk sensitive and robust control robust filtering and certain problems in computer vision examples are given as well as a proof of convergence

harmonic localization of algebraic k theory spectra the lichtenbaum quillen conjectures hold for the harmonic localization of the k theory spectrum of a nice scheme various consequences of this fact are explored for example the harmonic localization of the k theory of the integers at a regular prime is explicitly identified

a local global principle for the telescope conjecture abstract we prove an etale local global principle for the telescope conjecture and use it to show that the telescope conjecture holds for derived categories of azumaya algebras on noetherian schemes as well as for many classifying stacks and gerbes this specializes to give another proof of the fact that the telescope conjecture holds for noetherian schemes

geometric criteria for landweber exactness the purpose of this paper is to give a new presentation of some of the main results concerning landweber exactness in the context of the homotopy theory of stacks we present two new criteria for landweber exactness over a flat hopf algebroid the first criterion is used to classify stacks arising from landweber exact maps of rings using as extra input only lazard s theorem and cartier s classification of p typical formal group laws this result is then applied to deduce many of the main results concerning landweber exactness in stable homotopy theory and to compute the bousfield classes of certain bp algebra spectra the second criterion can be regarded as a generalization of the landweber exact functor theorem and we use it to give a proof of the original theorem

max plus summation of fenchel transformed semigroups for solution of nonlinear bellman equations abstract max plus methods have been explored for solution of first order nonlinear hamilton jacobi bellman partial differential equations hjb pdes and corresponding nonlinear control problems these methods exploit the max plus linearity of the associated semigroups in particular although the problems are nonlinear the semigroups are linear in the max plus sense these methods have been used successfully to compute solutions although they provide certain advantages they still generally suffer from the curse of dimensionality the natural analog to the laplace transform in ordinary spaces is the fenchel transform over max plus spaces the range space being referred to as the dual space one can transform the semigroup operators into operators on the dual space there are natural operations on the transformed operators which may be used to construct solutions of nonlinear control problems natural building blocks correspond to transforms of operators for linear quadratic problems in this paper a method for exploiting operations in the fenchel transform space is used to develop a method for certain problems such that the computational growth in the most time consuming portion of the computations can be hugely reduced although the curse of dimensionality is not entirely avoided the computational cost reductions are very high for some classes of problems

regularity results for solutions of a class of hamilton jacobi equations the regularity of the gradient of viscosity solutions of first order hamilton jacobi equations is studied under a strict convexity assumption on h t x estimates on the discontinuity set of du are derived such estimates imply that solutions of the above problem are smooth in the complement of a closed n rectifiable set in particular it follows that du belongs to the classsbv i e d2u is a measure with no cantor part

fourier mukai transform in the quantized setting we prove that a coherent dq kernel induces an equivalence between the derived categories of dq modules with coherent cohomology if and only if the graded commutative kernel associated to it induces an equivalence between the derived categories of coherent sheaves