the units of a ring spectrum and a logarithmic cohomology operation recall that if r is a commutative ring then the set r r of invertible elements of r is naturally an abelian group under multiplication this construction is a functor from commutative rings to abelian groups in general there is no obvious relation between the additive group of a ring r and the multiplicative group of units r however under certain circumstances one can define a homomorphism from a subgroup of r to a suitable completion of r e g the natural logarithm q 0 r or the p adic logarithm 1 pzp zp the logarithmic cohomology operation is a homotopy theoretic analogue of the above where r is a commutative s algebra and completion is bousfield localization with respect to a morava k theory the purpose of this paper is to give a formula for the logarithmic operation in certain contexts in terms of power operations before giving our results we briefly explain some of the concepts involved

qualitative behaviour of solutions of signorini problem with perturbing the unknown boundary computational method for the signorini problem in 2d is presented using qualitative properties of the solution on a perturbed boundary the iteration method for finding of the unknown boundary c c is constructed in the first part of the method the boundary a c c c is localized within two mesh points of a fixed fe mesh in the second part introducing the new multigrid algorithm the value a c is accurated convergence of the method is illustrated on the concrete examples

adams filtration and generalized hurewicz maps for infinite loopspaces we study the hurewicz map begin aligned h pi x rightarrow r omega infty x end aligned where omega infty x is the 0th space of a spectrum x and r is the generalized homology theory associated to a connective commutative s algebra r we prove that the decreasing filtration of the domain associated to an r based adams resolution is compatible with a filtration of the range associated to the augmentation ideal filtration of the augmented commutative s algebra sigma infty omega infty x the proof of our main theorem makes much use of composition properties of this filtration and its interaction with topological andre quillen homology an application is a connectivity theorem localize away from p 1 and suppose x is c 1 connected with c 0 if alpha in pi x has adams filtration s and alpha cp s then h alpha 0 in r omega infty x when specialized to mod p homology this implies a finiteness theorem if h x mathbb z p is finitely presented as a module over the steenrod algebra then the image of the hurewicz map in h omega infty x mathbb z p is finite we illustrate these theorems with calculations of the mod 2 hurewicz image of bo its connected covers and omega infty tmf and the mod p hurewicz image of all the spaces in the bp and bp langle n rangle spectra en route we get new proofs of theorems of milnor and wilson in the special case when x is the suspension spectrum of a space z and r h mathbb z 2 we recover results announced by lannes and zarati in the 1980s relating the adams filtration of pi s z to dyer lashof length in h qz mathbb z 2 and generalize them to all primes p for any x we also get parallel results for the hurewicz map for morava e theory where pi x is now given the adams novikov filtration

hkr characters p divisible groups and the generalized chern character in this paper we describe the generalized chern character of classifying spaces of finite groups in terms of hopkins kuhn ravenel generalized group characters for this purpose we study the p divisible group and its level structures associated with the k n localization of the n 1 st morava e theory

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

a cellular nerve for higher categories abstract we realise joyal cell category as a dense subcategory of the category of categories the associated cellular nerve of an category extends the well known simplicial nerve of a small category cellular sets like simplicial sets carry a closed model structure in quillen s sense with weak equivalences induced by a geometric realisation functor more generally there exists a dense subcategory a of the category of a algebras for each operad a in batanin s sense whenever a is contractible the resulting homotopy category of a algebras i e weak categories is equivalent to the homotopy category of compactly generated spaces

minors of the wronskian of the differential equation lny p x y 0 ii dominance of solutions on etudie les equations l n y p x y 0 ou l n est un operateur differentiel disconjugue et p x est de signe fixe on definit une base de l espace solution et on ordonne ses elements selon leurs grandeurs relatives a l infini

on the rigidity theorems of witten in this paper we prove the rigidity theorems predicted by witten in 1986 about the index of certain elliptic operators on manifolds with an s1 action w witten s insight was the culmination of an interesting interchange of ideas between him and hopkins landweber ochanine and stong for the detailed history we refer the reader to la the present account is essentially a reinterpretation of the second author s taubes t original proof of the theorem the senior author s contribution was solely to notice that the rather densely written arguments of the original manuscript could be formulated in terms of the well known fixed point formulae of equivariant index theory and equivariant cohomology in this context the final proof then appears in direct lineage of the atiyah hirzebruch theorem concerning the vanishing of the a genus of spin manifolds admitting a circle action a h and of the even older idea of lusztig concerning circle actions in the complex case there remains however the beautiful physics inspired novelty of connecting these techniques with elliptic function theory

a uniqueness theorem for stable homotopy theory roughly speaking the stable homotopy category of algebraic topology is obtained from the homotopy category of topological spaces by inverting the suspension functor yielding a linear approximation to the homotopy category of spaces the isomorphism classes of objects in the stable homotopy category represent the generalized cohomology theories defined by the eilenberg steenrod axioms es45 without the dimension axiom which distinguishes ordinary from generalized cohomology theories the first construction of the full stable homotopy category was given by boardman b69 nowadays manymodels for the stable homotopy category are known most of which have the additional structure of a closed model category in the sense of quillen q67 in m83 h r margolis introduced a short list of axioms and conjectured that they characterize the stable homotopy category up to an equivalence of categories in theorem 3 2 we prove that these axioms do uniquely specify the stable homotopy category whenever there is some underlying quillen model category we also prove amore structured version theuniqueness theorembelow which states that the model category of spectra itself is uniquely determined by certain equivalent conditions up to so called quillen equivalence of

projective elliptic genera and elliptic pseudodifferential genera abstract in this paper we construct for the first time the projective elliptic genera for a compact oriented manifold equipped with a projective complex vector bundle such projective elliptic genera are rational q series that have topological definition and also have analytic interpretation via the fractional index theorem in 25 without requiring spin condition we prove the modularity properties of these projective elliptic genera as an application we construct elliptic pseudodifferential genera for any elliptic pseudodifferential operator this suggests the existence of putative s 1 equivariant elliptic pseudodifferential operators on loop space whose equivariant indices are elliptic pseudodifferential genera

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

topological hochschild homology of number rings we calculate an explicit formula for the topological hochschild homology of a discrete valuation ring of characteristic zero with finite residue field from this we deduce the topological hochschild homology of global number rings ams subject classification 19d55 16e40 19d50 55t99

the goodwillie tower of the identity functor and the unstable periodic homotopy of spheres we investigate goodwillie s taylor tower of the identity functor from spaces to spaces more specifically we reformulate johnson s description of the goodwillie derivatives of the identity and prove that in the case of an odd dimensional sphere the only layers in the tower that are not contractible are those indexed by powers of a prime moreover in the case of a sphere the tower is finite in v k periodic homotopy

quadratic functions in geometry topology and m theory we describe an interpretation of the kervaire invariant of a riemannian manifold of dimension 4k 2 in terms of a holomorphic line bundle on the abelian variety h 2k 1 m otimes r z our results are inspired by work of witten on the fivebrane partition function in m theory hep th 9610234 hep th 9609122 our construction requires a refinement of the algebraic topology of smooth manifolds better suited to the needs of mathematical physics and is based on our theory of differential functions these differential functions generalize the differential characters of cheeger simons and the bulk of this paper is devoted to their study

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

inner horns for 2 quasi categories abstract dimitri ara s 2 quasi categories which are certain presheaves over andre joyal s 2 cell category 2 are an example of a concrete model that realises the abstract notion of 2 category in this paper we prove that the 2 quasi categories and the fibrations into them can be characterised using the inner horn inclusions and the equivalence extensions introduced by david oury these maps are more tractable than the maps that ara originally used and therefore our result can serve as a combinatorial foundation for the study of 2 quasi categories

lifting homotopy t algebra maps to strict maps abstract the settings for homotopical algebra categories such as simplicial groups simplicial rings a spaces e ring spectra etc are often equivalent to categories of algebras over some monad or triple t in such cases t is acting on a nice simplicial model category in such a way that t descends to a monad on the homotopy category and defines a category of homotopy t algebras in this setting there is a forgetful functor from the homotopy category of t algebras to the category of homotopy t algebras under suitable hypotheses we provide an obstruction theory in the form of a bousfield kan spectral sequence for lifting a homotopy t algebra map to a strict map of t algebras once we have a map of t algebras to serve as a basepoint the spectral sequence computes the homotopy groups of the space of t algebra maps and the edge homomorphism on 0 is the aforementioned forgetful functor we discuss a variety of settings in which the required hypotheses are satisfied including monads arising from algebraic theories and operads we also give sufficient conditions for the e 2 term to be calculable in terms of quillen cohomology groups we provide worked examples in g spaces g spectra rational e algebras and a algebras explicit calculations connected to rational unstable homotopy theory show that the forgetful functor from the homotopy category of e ring spectra to the category of h ring spectra is generally neither full nor faithful we also apply a result of the second named author and nick kuhn to compute the homotopy type of the space e coker j l k 2 r

poincare koszul duality we prove a duality for factorization homology which generalizes both usual poincare duality for manifolds and koszul duality for mathcal e n algebras the duality has application to the hochschild homology of associative algebras and enveloping algebras of lie algebras we interpret our result at the level of topological quantum field theory

moduli of suspension spectra for a 1 connected spectrum e we study the moduli space of suspension spectra which come equipped with a weak equivalence to e we construct a spectral sequence converging to the homotopy of the moduli space in positive degrees in the metastable range we get a complete homotopical classification of the path components of the moduli space our main tool is goodwillie s calculus of homotopy functors

a homotopy coherent cellular nerve for bicategories abstract the subject of this paper is a nerve construction for bicategories introduced by leinster which defines a fully faithful functor from the category of bicategories and normal pseudofunctors to the category of presheaves over joyal s category 2 we prove that the nerve of a bicategory is a 2 quasi category a model for 2 categories due to ara and moreover that the nerve functor restricts to the right part of a quillen equivalence between lack s model structure for bicategories and a bousfield localisation of ara s model structure for 2 quasi categories we deduce that lack s model structure for bicategories is quillen equivalent to rezk s model structure for 2 2 spaces on the category of simplicial presheaves over 2 to this end we construct the homotopy bicategory of a 2 quasi category and prove that a morphism of 2 quasi categories is an equivalence if and only if it is essentially surjective on objects and fully faithful we also prove a quillen equivalence between ara s model structure for 2 quasi categories and the hirschowitz simpson pellissier model structure for quasi category enriched segal categories from which we deduce a few more results about 2 quasi categories including a conjecture of ara concerning weak equivalences of 2 categories

convexity argument for monotone potential operators and its application modelling of many applied problems leads to the nonlinear equations elliptic type with monotone potential operators 6 11 some methods have been developed for di erent classes of nonlinear operators to study such qualitative questions as existence uniqueness linearization of the nonlinear operator and convergence of an approximate solution 2 7 11 in some cases it is possible to construct an abstract iteration technique for linearization of the nonlinear operator and to prove the convergence of an approximate solution in this paper we prove an abstract iterative scheme for monotone potential operators by using the convexity argument note that in the literature di erent convexity arguments or inequalities have been applied to problems of mathematical physics 14 in particular in 9 10 the logarithmic convexity argument was used to establish continuous dependence on the initial data for the cauchy problem the convexity argument introduced here is used as a su cient condition for the convergence of the abstract iteration scheme for monotone potential operators then using the result the convergence of the iteration scheme is proved for concrete nonlinear problems the paper is organized as follows in section 2 we introduce the functional inequality which is called the convexity argument then we consider an abstract operator equation and by using the argument we prove the strong convergence of the iteration

products on mu modules in 2 chapter v elmendorf kriz mandell and may hereafter referred to as ekmm use their new technology of modules over highly structured ring spectra to give new constructions of mu modules such as bp k n and so on which makes it much easier to analyse product structures on these spectra unfortunately their construction only works in its simplest form for modules over mu 12 that are concentrated in degrees divisible by 4 this guarantees that various obstruction groups are trivial in the present paper we extend the ekmm results to the cases where p 2 or the homotopy groups are allowed to be nonzero in all even degrees in this context the obstruction groups are nontrivial we shall show that there are never any obstructions to associativity we prove in section 7 that the obstructions to commutativity are given by a certain power operation this was inspired by a parallel result of mironov in baas sullivan theory 6 in section 8 we shall use formal group theory to derive various formulae for this power operation in section 9 we deduce a number of results about realising 2 local mu modules as mu modules

qualitative behaviour of solutions of unilateral elliptic problems with perturbing the unknown boundary i the theory qualitative properties of solutions of the unilateral elliptic problem and of the signorini problem for the lame system of equations are considered a preliminary analysis reduces each of the nonlinear problems to a mixed linear problem and to additional conditions in the form of inequalities on the part of the boundary in each case it is shown that if a perturbed unknown boundary involves the true one i e c c c c then the penetration condition is failed on c c c c conversely if c c c c then the second additional condition in the form of inequality is failed on c c c c the obtained results for the signorini type problems have a precise physical interpretation in the case of contact problems

the morava k theory of spaces related to bo abstract we calculate the p 2 morava k theory of all of the spaces in the connective omega spectra for z bo bo bso and bspin this leads to a description of the p 2 brown peterson cohomology of many of these spaces of particular interest is the space bo 8 and its relationship to bspin

reedy categories and the vartheta construction we use the notion of multi reedy category to prove that if mathcal c is a reedy category then vartheta mathcal c is also a reedy category this result gives a new proof that the categories vartheta n are reedy categories we then define elegant reedy categories for which we prove that the reedy and injective model structures coincide

homotopical localizations of spaces for a map f of spaces dror farjoun and the author have constructed an f localization functor where a space y is called f local when map f y is an equivalence this very general construction gives all known idempotent homotopy functors of spaces the main theorem of this paper shows that f localization functors always preserve fiber sequences of connected h spaces up to small error terms for instance the localization with respect to the n th morava k theory preserves such fiber sequences up to error terms with at most three nontrivial homotopy groups this implies for example that a k 1 homology equivalence of h spaces must induce an isomorphism of v 1 periodic homotopy groups results are also obtained on the a nullification or a periodization functors which are just the f localization functors for the maps f from spaces a to points two spaces are said to have the same nullity when they give the same nullification functors and it is shown that arbitrary sets of nullity classes have both least upper bounds and greatest lower bounds the a nullifications of nilpotent postnikov spaces are completely determined

on an extension of the notion of reedy category we extend the classical notion of a reedy category so as to allow non trivial automorphisms our extension includes many important examples occurring in topology such as segal s category or the total category of a crossed simplicial group such as connes cyclic category for any generalized reedy category mathbb r and any cofibrantly generated model category mathcal e the functor category mathcal e mathbb r is shown to carry a canonical model structure of reedy type

james maps and en ring spaces we parametrize by operad actions the multiplicative analysis of the total james map given by caruso and ourselves the target of the total james map q c r n x it qdq r n x q o is an en ring space and is a en map where en is the little n cubes operad this implies that has an n fold delooping with domain n x it also implies an algorithm for the calculation of and thus of each q on mod p homology when n 00 and p 2 this algorithm is the essential starting point for kuhn s proof of the whitehead conjecture in 2 4 we analyzed the behavior with respect to pairings of the lames maps iq ex qdq e x used in 4 to obtain the stable splittings of spaces ex in particular we proved that if the given coefficient system e has a sum then the product of the targets is a ring space and the product of the iq s is an exponential h map in fact this is only a fragment of the full structure present in the most interesting examples in the first two sections we give a parametrized elaboration of this analysis we assume the given coefficient system e is a module over an operad and we deduce essentially that

generalized group characters and complex oriented cohomology theories let bg be the classifying space of a finite group g given a multiplicative cohomology theory e the assignment g 7i e bg is a functor from groups to rings endowed with induction transfer maps in this paper we investigate these functors for complex oriented cohomology theories e using the theory of complex representations of finite groups as a model for what one would like to know an analogue of artin s theorem is proved for all complex oriented e the abelian subgroups of g serve as a detecting family for e bg modulo torsion dividing the order of g when e is a complete local ring with residue field of characteristic p and associated formal group of height n we construct a character ring of class functions that computes 1 e bg the domain of the characters is gn p the set of n tuples of elements in g each of which has order a power of p a formula for induction is also found the ideas we use are related to the lubin tate theory of formal groups the construction applies to many cohomology theories of current interest completed versions of elliptic cohomology e n theory etc the nth morava k theory euler characteristic for bg is computed to be the number of g orbits in gn p for various groups g including all symmetric groups we prove that k n bg concentrated in even degrees our results about e bg extend to theorems about e eg gx where x is a finite g cw complex

quasi elliptic cohomology i abstract quasi elliptic cohomology is a variant of elliptic cohomology theories it is the orbifold k theory of a space of constant loops for global quotient orbifolds it can be expressed in terms of equivariant k theories thus the constructions on it can be made in a neat way this theory reflects the geometric nature of the tate curve in this paper we provide a systematic introduction of its construction and definition

derived koszul duality and tq homology completion of structured ring spectra abstract working in the context of symmetric spectra we consider algebraic structures that can be described as algebras over an operad o topological quillen homology or tq homology is the spectral algebra analog of quillen homology and stabilization working in a framework for homotopical descent the associated tq homology completion map appears as the unit map of an adjunction comparing the homotopy categories of o algebra spectra with coalgebra spectra over the associated comonad via tq homology we prove that this adjunction between homotopy categories can be turned into an equivalence by replacing o algebras with the full subcategory of 0 connected o algebras we also construct the spectral algebra analog of the unstable adams spectral sequence that starts from the tq homology groups tq x of an o algebra x and prove that it converges strongly to x when x is 0 connected

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

topological modular forms with level structure the cohomology theory known as mathrm tmf for topological modular forms is a universal object mapping out to elliptic cohomology theories and its coefficient ring is closely connected to the classical ring of modular forms we extend this to a functorial family of objects corresponding to elliptic curves with level structure and modular forms on them along the way we produce a natural way to restrict to the cusps providing multiplicative maps from mathrm tmf with level structure to forms of k theory in particular this allows us to construct a connective spectrum mathrm tmf 0 3 consistent with properties suggested by mahowald and rezk this is accomplished using the machinery of logarithmic structures we construct a presheaf of locally even periodic elliptic cohomology theories equipped with highly structured multiplication on the log etale site of the moduli of elliptic curves evaluating this presheaf on modular curves produces mathrm tmf with level structure

strictly commutative complex orientation theory for a multiplicative cohomology theory e complex orientations are in bijective correspondence with multiplicative natural transformations to e from complex bordism cohomology mu if e is represented by a spectrum with a highly structured multiplication we give an iterative process for lifting an orientation mu rightarrow e to a map respecting this extra structure based on work of arone lesh the space of strictly commutative orientations is the limit of an inverse tower of spaces parametrizing partial lifts stage 1 corresponds to ordinary complex orientations and lifting from stage m 1 to stage m is governed by the existence of an orientation for a family of e modules over a fixed base space f m when e is p local we can say more we find that this tower only changes when m is a power of p and if e is e n local the tower is constant after stage p n moreover if the coefficient ring e is p torsion free the ability to lift from stage 1 to stage p is equivalent to a condition on the associated formal group law that was shown necessary by ando

subgroups of p divisible groups and centralizers in symmetric groups we give a formula relating the transfer maps for the cohomology theories en and ct to the transchromatic generalized character maps of 7 we then apply this to understand the effect of the transchromatic generalized character maps on strickland s isomorphism between the morava e theory of the symmetric group pk modulo a transfer ideal and the global sections of the scheme that classifies subgroups of order p in the formal group associated to en this provides an algebrogeometric interpretation to the ct cohomology of the class of groups arising as centralizers of finite sets of commuting elements in symmetric groups

analytic index for a family of dirac ramond operators we derive a cohomological formula for the analytic index of a family of dirac ramond operators and exhibit its modular properties

elliptic genera and quantum field theory it is shown that in elliptic cohomology as recently formulated in the mathematical literature the supercharge of the supersymmetric nonlinear signa model plays a role similar to the role of the dirac operator ink theory this leads to several insights concerning both elliptic cohomology and string theory some of the relevant calculations have been done previously by schellekens and warner in a different context

the bousfield kuhn functor and topological andre quillen cohomology we construct a natural transformation from the bousfield kuhn functor evaluated on a space to the topological andre quillen cohomology of the k n local spanier whitehead dual of the space and show that the map is an equivalence in the case where the space is a sphere this results in a method for computing unstable v n periodic homotopy groups of spheres from their morava e cohomology as modules over the dyer lashof algebra of morava e theory we relate the resulting algebraic computations to the algebraic geometry of isogenies between lubin tate formal groups

on existence uniqueness and convergence of approximate solution of boundary value problems related to the nonlinear operator au k u 2 u g u we study the problems of solvability and linearization for the nonlinear boundary value problems with nonlinear operator au k u 2 u g u solvability in h 1 a b of the problems is obtained by using monotone potential operator theory and browder minty theorem sufficient conditions for the solvability are obtained in explicit form for the linearization of the considered nonlinear problems monotone iterative scheme is developed the scheme permits use of the variational finite difference scheme for the numerical solution the considered nonlinear problems sufficient conditions for the convergence of the iteration method are presented computational experiments illustrate high accuracy of the presented method

a classification of the solutions of a difference equation according to their behavior at infinity on considere l equation aux differences lineaires d ordre n ly t p t y t 0 ou p t est de signe defini sur a a a 1 a est un entier

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

representation and character theory in 2 categories we develop a 2 categorical generalization of the theory of group representations and characters we categorify the concept of the trace of a linear transformation associating to any endofunctor of any small category a set called its categorical trace in a linear situation the categorical trace is a vector space and we associate to any two commuting self equivalences a number called their joint trace for a group acting on a linear category we define an analog of the character which is the function on commuting pairs of group elements given by the joint traces of the corresponding functors we call this function the 2 character of such functions of commuting pairs and more generally n tuples appear in the work of hopkins kuhn and ravenel michael j hopkins nicholas j kuhn douglas c ravenel generalized group characters and complex oriented cohomology theories j amer math soc 13 3 2000 553 594 electronic on equivariant morava e theories we define the concept of induced categorical representation and show that the corresponding 2 character is given by the same formula as was obtained in michael j hopkins nicholas j kuhn douglas c ravenel generalized group characters and complex oriented cohomology theories j amer math soc 13 3 2000 553 594 electronic for the transfer map in the second equivariant morava e theory

minors of the wronskian of the differential equation lny p x y 0 etude de certains mineurs du wronskien satisfaisant a une equation differentielle similaire proprietes de l equation initiale essentielles pour la theorie des oscillations des solutions de la seconde equation

circle equivariant classifying spaces and the rational equivariant sigma genus the circle equivariant spectrum mstring mathbb c is the equivariant analogue of the cobordism spectrum mu langle 6 rangle of stably almost complex manifolds with c1 c2 0 given a rational elliptic curve c greenlees topology 44 1213 1227 2005 constructs a ring mathbb t spectrum ec representing the associated mathbb t equivariant elliptic cohomology the core of the present paper is the construction when c is a complex elliptic curve of a map of ring mathbb t spectra mstring mathbb c to ec which is the rational equivariant analogue of the sigma orientation of ando et al invent math 146 595 687 2001 we support this by a theory of characteristic classes for calculation and a conceptual description in terms of algebraic geometry in particular we prove a conjecture the first author made in ando geom topol 7 91 153 2003

orbifold genera product formulas and power operations we generalize the definition of orbifold elliptic genus and introduce orbifold genera of chromatic level h using h tuples rather than pairs of commuting elements we show that our genera are in fact orbifold invariants and we prove integrality results for them if the genus arises from an h8 map into the morava lubin tate theory eh then we give a formula expressing the orbifold genus of the symmetric powers of a stably almost complex manifold m in terms of the genus of m itself our formula is the p typical analogue of the dijkgraaf moore verlinde verlinde formula for the orbifold elliptic genus r dijkgraaf et al elliptic genera of symmetric products and second quantized strings comm math phys 185 1 1997 197 209 it depends only on h and not on the genus

localization of andre quillen goodwillie towers and the periodic homology of infinite loopspaces abstract let k n be the n th morava k theory at a prime p and let t n be the telescope of a v n self map of a finite complex of type n in this paper we study the k n homology of x the 0th space of a spectrum x and many related matters we give a sampling of our results let p x be the free commutative s algebra generated by x it is weakly equivalent to the wedge of all the extended powers of x we construct a natural map s n x l t n p x l t n x of commutative algebras over the localized sphere spectrum l t n s the induced map of commutative cocommutative k n hopf algebras s n x k n p x k n x satisfies the following properties it is always monic it is an isomorphism if x is n connected n 1 x is torsion and t i x 0 for 1 i n 1 it is an isomorphism only if k i x 0 for 1 i n 1 it is universal the domain of s n x preserves k n isomorphisms and if f is any functor preserving k n isomorphisms then any natural transformation f x k n x factors uniquely through s n x the construction of our natural transformation uses the telescopic functors constructed and studied previously by bousfield and the author and thus depends heavily on the nilpotence theorem of devanitz hopkins and smith our proof that s n x is always monic uses topological andre quillen homology and goodwillie calculus in nonconnective settings

the sigma orientation is an h map in an earlier paper the authors constructed a natural map called the sigma orientation from the thom spectrum mu 6 to any elliptic spectrum mu 6 is an h ring spectrum and in this paper we show that if e c t is the elliptic spectrum associated to the universal deformation of a supersingular elliptic curve over a perfect field of characteristic p 0 then the sigma orientation is a map of h ring spectra

power operations in elliptic cohomology and representations of loop groups the first part describes power operations in elliptic cohomology in terms of isogenies of the underlying elliptic curve the second part discusses a relationship between equivariant elliptic cohomology and representations of loop groups the third part investigates the representation theoretic considerations which give rise to the power operations discussed in the first part

an inverse problem for an elastoplastic medium in this paper within the range of the so called deformation theory of plasticity the determination of unknown coefficients in a nonlinear system of equilibrium equations from over specified data measured at the boundary is considered this inverse problem is reformulated as a minimization problem for a certain functional it is shown that this functional has at least one solution quasisolution in an admissible class of coefficients then the existence of an exact solution of inverse problem is proved a numerical algorithm and examples related to direct and inverse problems are presented

computational modeling of elastoplastic behaviour of homogeneous materials under indentation monotone operator theory and finite element analysis with remeshing algorithm is developed for study of elastoplastic behaviour of homogeneous materials under indentation convexity and continuity properties of the nonlinear functional related to plasticity operator is analyzed in the considered formulation the problem is nonlinear even for physically linear pure elastic materials since the unknown contact domain on each step of indentation is assumed to be unknown numerical algorithm presented is based on a locally adaptable mesh in order to find the unknown contact domain with high accuracy computational results show high predictive capability of the proposed model for indentation of inelastic homogeneous materials