restriction to symmetric subgroups of unitary representations of rank one semisimple lie groups we consider the spherical complementary series of rank one lie groups h n so0 n 1 f for f r c h we prove that there exist finitely many discrete components in its restriction under the subgroup hn 1 so0 n 1 1 f this is proved by imbedding the complementary series into analytic continuation of holomorphic discrete series of g n su n 1 su n 1 x su n 1 and su 2n 2 and by the branching of holomorphic representations under the corresponding subgroup g n 1

integral deligne cohomology for real varieties we develop an integral version of deligne cohomology for smooth proper real varieties for this purpose the role played by singular cohomology in the complex case has to be replaced by the ordinary bigraded gal mathbb c mathbb r equivariant cohomology of lewis et al bull am math soc n s 4 2 208 212 1981 the equivariant counterpart of singular cohomology the theory is aimed at giving more precise information about the 2 primary components of regulators we establish basic properties and give a geometric interpretation for the groups in dimension 2 in weights 1 and 2

hz algebra spectra are differential graded algebras we show that the homotopy theory of differential graded algebras coincides with the homotopy theory of h algebra spectra namely we construct quillen equivalences between the quillen model categories of unbounded differential graded algebras and h algebra spectra we also construct quillen equivalences between the differential graded modules and module spectra over these algebras we use these equivalences in turn to produce algebraic models for rational stable model categories we show that basically any rational stable model category is quillen equivalent to modules over a differential graded algebra with many objects

uniqueness of the multiplicative cyclotomic trace abstract making use of the theory of noncommutative motives we characterize the topological dennis trace map as the unique multiplicative natural transformation from algebraic k theory to topological hochschild homology thh and the cyclotomic trace map as the unique multiplicative lift through topological cyclic homology tc moreover we prove that the space of all multiplicative structures on algebraic k theory is contractible we also show that the algebraic k theory functor from small stable categories to spectra is lax symmetric monoidal which in particular implies that e n ring spectra give rise to e n 1 ring algebraic k theory spectra along the way we develop a multiplicative morita theory establishing a symmetric monoidal equivalence between the category of small idempotent complete stable categories and the morita localization of the category of spectral categories

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

on the homotopy type of higher orbifolds and haefliger classifying spaces abstract we describe various equivalent ways of associating to an orbifold or more generally a higher etale differentiable stack a weak homotopy type some of these ways extend to arbitrary higher stacks on the site of smooth manifolds and we show that for a differentiable stack x arising from a lie groupoid g the weak homotopy type of x agrees with that of b g using this machinery we are able to find new presentations for the weak homotopy type of certain classifying spaces in particular we give a new presentation for the borel construction m g e g of an almost free action of a lie group g on a smooth manifold m as the classifying space of a category whose objects consist of smooth maps r n m which are transverse to all the g orbits where n dim m dim g we also prove a generalization of segal s theorem which presents the weak homotopy type of haefliger s groupoid q as the classifying space of the monoid of self embeddings of r q b emb r q and our generalization gives analogous presentations for the weak homotopy type of the lie groupoids 2 q s p and r q which are related to the classification of foliations with transverse symplectic forms and transverse metrics respectively we also give a short and simple proof of segal s original theorem using our machinery

replacing model categories with simplicial ones in this paper we show that model categories of a very broad class can be replaced up to quillen equivalence by simplicial model categories

simplicial structures on model categories and functors we produce a highly structured way of associating a simplicial category to a model category which improves on work of dwyer and kan and answers a question of hovey we show that model categories satisfying a certain axiom are quillen equivalent to simplicial model categories a simplicial model category provides higher order structure such as composable mapping spaces and homotopy colimits we also show that certain homotopy invariant functors can be replaced by weakly equivalent simplicial or continuous functors this is used to show that if a simplicial model category structure exists on a model category then it is unique up to simplicial quillen equivalence

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

the global derived period map abstract we develop the global period map in the context of derived geometry generalising griffiths classical period map as well as the infinitesimal derived period map we begin by constructing the derived period domain which classifies hodge filtrations and enhances the classical period domain we analyze the monodromy action then we associate to any polarized smooth projective map of derived stacks a canonical morphism of derived analytic stacks from the base into the quotient of the derived period domain by monodromy we conclude the paper by discussing a few examples and a derived torelli problem in the appendix we describe how to present derived analytic artin stacks as hypergroupoids which may be of independent interest

discrete approximations for complex kac moody groups abstract we construct a map from the classifying space of a discrete kac moody group over the algebraic closure of the field with p elements to the classifying space of a complex topological kac moody group and prove that it is a homology equivalence at primes q different from p this generalizes a classical result of quillen friedlander and mislin for lie groups as an application we construct unstable adams operations for general kac moody groups compatible with the frobenius homomorphism our results rely on new integral homology decompositions for certain infinite dimensional unipotent subgroups of discrete kac moody groups

kloosterman sums for clifford algebras and a lower bound for the positive eigenvalues of the laplacian for congruence subgroups acting on hyperbolic spaces

local structures on stratified spaces abstract we develop a theory of conically smooth stratified spaces and their smooth moduli including a notion of classifying maps for tangential structures we characterize continuous space valued sheaves on these conically smooth stratified spaces in terms of tangential data and we similarly characterize 1 excisive invariants of stratified spaces these results are based on the existence of open handlebody decompositions for conically smooth stratified spaces an inverse function theorem a tubular neighborhood theorem an isotopy extension theorem and functorial resolutions of singularities to smooth manifolds with corners

topological hypercovers and 1 realizations we show that if u is a hypercover of a topological space x then the natural map hocolim u x is a weak equivalence this fact is used to construct topological realization functors for the 1 homotopy theory of schemes over real and complex fields in an appendix we also prove a theorem about computing homotopy colimits of spaces that are not cofibrant

iterated wreath product of the simplex category and iterated loop spaces generalising segal s approach to 1 fold loop spaces the homotopy theory of n fold loop spaces is shown to be equivalent to the homotopy theory of reduced tn spaces where tn is an iterated wreath product of the simplex category a sequence of functors from tn to g allows for an alternative description of the segal spectrum associated to a g space this yields a canonical reduced tn set model for each eilenberg maclane space

the role of the ramanujan conjecture in analytic number theory we discuss progress towards the ramanujan conjecture for the group gln and its relation to various other topics in analytic number theory

some irreducible unitary representations of g k for a simple algebraic group g over an algebraic number field k let k be an algebraic number field and let gk be the group of k rational points of a simply connected simple linear algebraic group g defined over k we construct a new family of irreducible unitary representations of gk as follows it is well known that gk embeds diagonally as a lattice in ga where a is the ring of adeles of k let be an irreducible unitary representation of ga we show that the restriction of to gk is irreducible and that is determined by up to unitary equivalence many of these restrictions are not in the support of the regular representation of gk

rigidity and exotic models for v1 local g equivariant stable homotopy we prove that the v 1 local g equivariant stable homotopy category for g a finite group has a unique g equivariant model at p 2 this means that at the prime 2 the homotopy theory of g spectra up to fixed point equivalences on k theory is uniquely determined by its triangulated homotopy category and basic mackey structure the result combines the rigidity result for k local spectra of the second author with the equivariant rigidity result for g spectra of the first author further when the prime p is at least 5 and does not divide the order of g we provide an algebraic exotic model as well as a g equivariant exotic model for the v 1 local g equivariant stable homotopy category showing that for primes p ge 5 equivariant rigidity fails in general

vanishing theorems for real algebraic cycles we establish the analogue of the friedlander mazur conjecture for teh s reduced lawson homology groups of real varieties which says that the reduced lawson homology of a real quasi projective variety x vanishes in homological degrees larger than the dimension of x in all weights as an application we obtain a vanishing of homotopy groups of the mod 2 topological groups of averaged cycles and a characterization in a range of indices of dos santos real lawson homology as the homotopy groups of the topological group of averaged cycles we also establish an equivariant poincare duality between equivariant friedlander walker real morphic cohomology and dos santos real lawson homology we use this together with an equivariant extension of the mod 2 beilinson lichtenbaum conjecture to compute some real lawson homology groups in terms of bredon cohomology

sobolev w1p spaces on closed subsets of rn abstract for each p n we use local oscillations and doubling measures to give intrinsic characterizations of the restriction of the sobolev space w p 1 r n to an arbitrary closed subset of r n

counting lattice points for a locally compact second countable group g and a lattice subgroup we give an explicit quantitative solution of the lattice point counting problem in general domains in g provided that i g has finite upper local dimension and the domains satisfy a basic regularity condition ii the mean ergodic theorem for the action of g on g holds with a rate of convergence the error term we establish matches the best current result for balls in symmetric spaces of simple higher rank lie groups but holds in much greater generality a significant advantage of the ergodic theoretic approach we use is that the solution to the lattice point counting problem is uniform over families of lattice subgroups provided they admit a uniform spectral gap in particular the uniformity property holds for families of finite index subgroups satisfying a quantitative variant of property we discuss a number of applications including counting lattice points in general domains in semisimple s algebraic groups counting rational points on group varieties with respect to a height function and quantitative angular or conical equidistribution of lattice points in symmetric spaces and in affine symmetric varieties we note that the mean ergodic theorems which we establish are based on spectral methods including the spectral transfer principle and the kunze stein phenomenon we formulate and prove appropriate analogues of both of these results in the set up of adele groups and they constitute a necessary step in our proof of quantitative results in counting rational points

etale realization on the a1 homotopy theory of schemes abstract we compare friedlander s definition of the etale topological type for simplicial schemes to another definition involving realizations of pro simplicial sets this can be expressed as a notion of hypercover descent for etale homotopy we use this result to construct a homotopy invariant functor from the category of simplicial presheaves on the etale site of schemes over s to the category of pro spaces after completing away from the characteristics of the residue fields of s we get a functor from the morel voevodsky a 1 homotopy category of schemes to the homotopy category of pro spaces

universal homotopy theories abstract begin with a small category c the goal of this short note is to point out that there is such a thing as a universal model category built from c we describe applications of this to the study of homotopy colimits the dwyer kan theory of framings sheaf theory and the homotopy theory of schemes

homotopical and operator algebraic twisted k theory using the framework for multiplicative parametrized homotopy theory introduced in joint work with c schlichtkrull we produce a multiplicative comparison between the homotopical and operator algebraic constructions of twisted k theory both in the real and complex case we also improve several comparison results about twisted k theory of c algebras to include multiplicative structures our results can also be interpreted in the infty categorical setup for parametrized spectra

trace operator and the dirichlet problem for elliptic equations on arbitrary bounded open sets abstract we consider the dirichlet problem and the weak dirichlet problem on a general possibly nonregular bounded domain for elliptic linear equation with uniformly elliptic divergence form operator we investigate carefully the relationship between weak integral probabilistic soft and the perron wiener brelot solutions of the problem we also study the relationship between solutions of the dirichlet problem and weak dirichlet problem to this end we extend the usual notion of the trace operator and consider the dirichlet problem with martin boundary

modules over motivic cohomology abstract for fields of characteristic zero we show that the homotopy category of modules over the motivic ring spectrum representing motivic cohomology is equivalent to voevodsky s big category of motives the proof makes use of some highly structured models for motivic stable homotopy theory motivic spanier whitehead duality the homotopy theories of motivic functors and of motivic spaces with transfers as introduced from ground up in this paper working with rational coefficients we extend the equivalence for fields of characteristic zero to all perfect fields by employing the techniques of alterations and homotopy purity in motivic homotopy theory

combinatorial model categories have presentations abstract we show that every combinatorial model category is quillen equivalent to a localization of a diagram category where diagram category means diagrams of simplicial sets this says that every combinatorial model category can be built from a category of generators and a set of relations

best possible rates of distribution of dense lattice orbits in homogeneous spaces this paper establishes upper and lower bounds on the speed of approximation in a wide range of natural diophantine approximation problems the upper and lower bounds coincide in many cases giving rise to optimal results in diophantine approximation which were inaccessible previously our approach proceeds by establishing more generally upper and lower bounds for the rate of distribution of dense orbits of a lattice subgroup in a connected lie or algebraic group g acting on suitable homogeneous spaces g h the upper bound is derived using a quantitative duality principle for homogeneous spaces reducing it to a rate of convergence in the mean ergodic theorem for a family of averaging operators supported on h and acting on g in particular the quality of the upper bound on the rate of distribution we obtain is determined explicitly by the spectrum of h in the automorphic representation on l 2 g we show that the rate is best possible when the representation in question is tempered and show that the latter condition holds in a wide range of examples

discrete components in restriction of unitary representations of rank one semisimple lie groups we consider spherical principal series representations of the semisimple lie group of rank one g so n 1 k g so n 1 k k r c hk r c h there is a family of unitarizable representations of g for in an interval on rr the so called complementary series and subquotients or subrepresentations of g for being negative integers we consider the restriction of g g under the subgroup h so n 1 1 k h so n 1 1 k we prove the appearing of discrete components the corresponding results for the exceptional lie group f4 20 f4 20 and its subgroup spin0 8 1 spin0 8 1 are also obtained

on planar sobolev lpm extension domains abstract for each m 1 and p 2 we characterize bounded simply connected sobolev l p m extension domains r 2 our criterion is expressed in terms of certain intrinsic subhyperbolic metrics in its proof is based on a series of results related to the existence of special chains of squares joining given points x and y in an important geometrical ingredient for obtaining these results is a new square separation theorem it states that under certain natural assumptions on the relative positions of a point x and a square s there exists a similar square q which touches s and has the property that x and s belong to distinct connected components of q

complete segal spaces arising from simplicial categories in this paper we compare several functors which take simplicial categories or model categories to complete segal spaces which are particularly nice simplicial spaces which like simplicial categories can be considered to be models for homotopy theories we then give a characterization up to weak equivalence of complete segal spaces arising from these functors

motivic spheres and the image of the suslin hurewicz map we show that an old conjecture of a a suslin characterizing the image of a hurewicz map from quillen k theory in degree n to milnor k theory in degree n admits an interpretation in terms of unstable mathbb a 1 homotopy sheaves of the general linear group using this identification we establish suslin s conjecture in degree 5 for infinite fields having characteristic unequal to 2 or 3 we do this by linking the relevant unstable mathbb a 1 homotopy sheaf of the general linear group to the stable mathbb a 1 homotopy of motivic spheres

on homotopy varieties given an algebraic theory t a homotopy t algebra is a simplicial set where all equations from t hold up to homotopy all homotopy t algebras form a homotopy variety we will give a characterization of homotopy varieties analogous to the characterization of varieties

galois equivariance and stable motivic homotopy theory for a finite galois extension of fields l k with galois group g we study a functor from the g equivariant stable homotopy category to the stable motivic homotopy category over k induced by the classical galois correspondence we show that after completing at a prime and the motivic hopf map this results in a full and faithful embedding whenever k is real closed and l k i it is a full and faithful embedding after completion if a motivic version of serre s finiteness theorem is valid we produce strong necessary conditions on the field extension l k for this functor to be full and faithful along the way we produce several results on the stable c2 equivariant betti realization functor and prove convergence theorems for the p primary c2 equivariant adams spectral sequence

bigraded equivariant cohomology of real quadrics we give a complete description of the bigraded bredon co homology ring of smooth projective real quadrics with coefficients in the constant mackey functor z these invariants are closely related to the integral motivic cohomology ring which is not known for these va rieties some of the results and techniques introduced can be applied to other geometrically cellular real varieties

on the boundary values of sobolev wp1 functions abstract for each p n we use local oscillations to give intrinsic characterizations of the trace of the sobolev space w p 1 to the boundary of an arbitrary domain r n

on sobolev extension domains in r n abstract we describe a class of sobolev w p k extension domains r n determined by a certain inner subhyperbolic metric in this enables us to characterize finitely connected sobolev w p 1 extension domains in r 2 for each p 2

equivariant factorization homology of global quotient orbifolds abstract we introduce equivariant factorization homology extending the axiomatic framework of ayala francis to encompass multiplicative invariants of manifolds equipped with finite group actions examples of such equivariant factorization homology theories include bredon equivariant homology and twisted versions of hochschild homology our main result is that equivariant factorization homology satisfies an equivariant version of excision and is uniquely characterised by this property we also discuss applications to representation theory such as constructions of categorical braid group actions

the unitary spectrum for real rank one groups let g be a connected real semi simple lie group with finite center one of the main problems in harmonic analysis is to determine the unitary spectrum of g in this paper we treat this question in the case when real rank of g is i although the answer was known for g of classical type previously 2 5 7 15 we have redone this work sometimes giving simpler arguments with the arbitrary rank case in mind we have tried to deal with the problem of classifying unitary representations in as systematic a way as possible pro ceeding by induction on the dimension of g we have concentrated on the case when g is linear and has a compact cartan subgroup the other cases being known already as an application we also give a list of the unitary representations that contribute to the l2 index formula for the dirac operator with coefficients as is apparent from the calculations in 19 this is intimately connected with one of our main techniques for determining the unitary spectrum the dirac in equality we will deal with some related problems in a future paper

diagram spaces and symmetric spectra abstract we present a general homotopical analysis of structured diagram spaces and discuss the relation to symmetric spectra the main motivating examples are the i spaces which are diagrams indexed by finite sets and injections and j spaces which are diagrams indexed by quillen s localization construction 1 on the category of finite sets and bijections we show that the category of i spaces provides a convenient model for the homotopy category of spaces in which every e space can be rectified to a strictly commutative monoid similarly the commutative monoids in the category of j spaces model graded e spaces using the theory of j spaces we introduce the graded units of a symmetric ring spectrum the graded units detect periodicity phenomena in stable homotopy and we show how this can be applied to the theory of topological logarithmic structures

monoidal uniqueness of stable homotopy theory we show that the monoidal product on the stable homotopy category of spectra is essentially unique this strengthens work of this author with schwede on the uniqueness of models of the stable homotopy theory of spectra also the equivalences constructed here give a unified construction of the known equivalences of the various symmetric monoidal categories of spectra s modules w spaces orthogonal spectra simplicial functors with symmetric spectra as an application we show that with an added assumption about underlying model structures margolis axioms uniquely determine the stable homotopy category of spectra up to monoidal equivalence

orbispaces orthogonal spaces and the universal compact lie group this paper identifies the homotopy theories of topological stacks and orbispaces with unstable global homotopy theory at the same time we provide a new perspective by interpreting it as the homotopy theory of spaces with an action of the universal compact lie group the upshot is a novel way to construct and study genuine cohomology theories on stacks orbifolds and orbispaces defined from stable global homotopy types represented by orthogonal spectra the universal compact lie group which is neither compact nor a lie group is a well known object namely the topological monoid mathcal l of linear isometric self embeddings of mathbb r infty the underlying space of mathcal l is contractible and the homotopy theory of mathcal l spaces with respect to underlying weak equivalences is just another model for the homotopy theory of spaces however the monoid mathcal l contains copies of all compact lie groups in a specific way and we define global equivalences of mathcal l spaces by testing on corresponding fixed points we establish a global model structure on the category of mathcal l spaces and prove it to be quillen equivalent to the global model category of orthogonal spaces and to the category of orbispaces i e presheaves of spaces on the global orbit category

singular chains on topological stacks i abstract we extend the functor sing of singular chains to the category of topological stacks and establish its main properties we prove that sing respects weak equivalences and takes a morphism of topological stacks that is both a serre and a reedy fibration to a kan fibration of simplicial sets when restricted to the category of topological spaces sing coincides with the usual singular functor

left fibrations and homotopy colimits for a small category mathbf a we prove that the homotopy colimit functor from the category of simplicial diagrams on mathbf a to the category of simplicial sets over the nerve of mathbf a establishes a left quillen equivalence between the projective or reedy model structure on the former category and the covariant model structure on the latter we compare this equivalence to a quillen equivalence in the opposite direction previously established by lurie from our results we deduce that a categorical equivalence of simplicial sets induces a quillen equivalence on the corresponding over categories equipped with the covariant model structures also we show that a version of quillen s theorem a for infty categories easily follows

derived azumaya algebras and twisted k theory abstract we construct a relative version of topological k theory in the sense of 11 over an arbitrary quasi compact quasi separated c scheme x this has as input a perf x linear stable category and output a sheaf of spectra on x c the space of complex points of x we then characterize the values of this functor on inputs of the form perf x a for a a derived azumaya algebra over x in such cases we show that this coincides with the twisted topological k theory of x c for some appropriately defined twist of k theory we use this to provide a topological analogue of a classical result of quillen s on the algebraic k theory of severi brauer varieties

homotopical algebraic geometry i topos theory this is the rst of a series of papers devoted to lay the foundations of algebraic geometry in homotopical and higher categorical contexts in this rst part we investigate a notion of higher topos for this we use s categories i e simplicially enriched categories as models for certain kind of 1 categories and we develop the notions of s topologies s sites and stacks over them we prove in particular that for an s category t endowed with an s topology there exists a model category of stacks over t generalizing the model category structure on simplicial presheaves over a grothendieck site of a joyal and r jardine we also prove some analogs of the relations between topologies and localizing subcategories of the categories of presheaves by proving that there exists a one to one correspodence between s topologies on an s category t and certain left exact bouseld localizations of the model category of pre stacks on t based on the above results we study the notion of model topos introduced by c rezk and we relate it to our model categories of stacks over s sites in the second part of the paper we present a parallel theory where s categories s topologies and s sites are replaced by model categories model topologies and model sites we prove that a canonical way to pass from the theory of stacks over model sites to the theory of stacks over s sites is provided by the simplicial localization construction of dwyer and kan as an example of application we propose a denition of etale k theory of ring spectra extending the etale k theory of commutative rings