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

algebraic k theory of group rings and the cyclotomic trace map we prove that the farrell jones assembly map for connective alge braic k theory is rationally injective under mild homological niteness condi tions on the group and assuming that a weak version of the leopoldt schneider conjecture holds for cyclotomic elds this generalizes a result of bokstedt hsiang and madsen and leads to a concrete description of a large direct summand of kn z g z q in terms of group homology in many cases the number theoretic conjectures are true so we obtain rational injectivity results about assembly maps in particular for whitehead groups under niteness assumptions on the group only the proof uses the cyclotomic trace map to topological cyclic homology bokstedt hsiang madsen s functor c and new general injectivity results about the assembly maps for topological hochschild homology and c

projective manifolds whose tangent bundles are numerically effective in 1979 mori mo proved the so called hartshorne frankel conjecture every projective n dimensional manifold with ample tangent bundle is isomorphic to the complex projective space p a differential geometric analogon assuming the existence of a k ihler metric on x with positive holomorphic bisectional curvature is independently due to siu yau sy thus it seems natural to classify projective manifolds x whose tangent bundle tx satisfy a degenerate condition of ampleness numerical effectivity abbreviated by nef this means that the tautological quotient line bundle d 1 on f tx is numerically effective i e c 0

mountain pass solutions for singularly perturbed nonlinear dirichlet problems abstract we consider the following singularly perturbed nonlinear elliptic problem 2 u u f u 0 u 0 in u 0 on where the nonlinearity f is of subcritical growth and is a bounded domain in r n under certain conditions of f there exists a mountain pass solution u for above problem and the solution u exhibits a spike layer as 0 to see the location of the spike layer for small 0 some additional assumptions certain nondegeneracy for a limiting problem or monotonicity of f t t have been assumed in this paper we characterize the location of the spike layer without such additional assumptions for n 3

the homotopy theory of cyclic sets the aim of this note is to show that the homotopy theory of the cyclic sets of connes 3 is equivalent to that of s0 2 spaces i e spaces with a circle action and hence to that of spaces over k z 2

cyclic homology and equivariant homology the purpose of this paper is to explore the relationship between the cyclic homology and cohomology theories of connes 9 11 see also loday and quillen 20 and if equivariant homology and cohomology theories here ii is the circle group the most general results involve the definitions of the cyclic homology of cyclic chain complexes and the notions of cyclic and cocyclic spaces so precise statements will be postponed until w 3 in this introduction we explain some of the formal similarities between the cyclic theory and the equivariant theory and give two examples where the general results apply let a be an associative algebra over a commutative ring k then one can form the cyclic homology hc a and cohomology hc a of a these groups have periodicity operators

sign changing and constant sign solutions for p laplacian problems with jumping nonlinearities abstract by variational methods we prove the existence of a sign changing solution for the p laplacian equation under dirichlet boundary condition with jumping nonlinearity having relation to the fucik spectrum of p laplacian we also provide the multiple existence results for the p laplacian problems

cyclic symmetries of an quiver representations abstract this note contains a combinatorial construction of symmetries arising in symplectic geometry partially wrapped or infinitesimal fukaya categories algebraic geometry derived categories of singularities and k theory waldhausen s s construction our specific motivation in the spirit of expectations of kontsevich is a combinatorial construction of categorical quantizations of lagrangian skeleta of symplectic manifolds the main result of this paper gives an immediate solution in the one dimensional case of ribbon graphs

existence and multiplicity of solutions for elliptic equations with jumping nonlinearities abstract in this paper we are concerned with a semilinear elliptic dirichlet problem with jumping nonlinearity and using a completely variational method we show that the number of solutions may be arbitrarily large provided the number of jumped eigenvalues is large enough in order to prove this fact we show that for every positive integer k when a suitable parameter is large enough there exists a solution which presents k peaks under the assumptions we consider in this paper new unexpected phenomena are observed in the study of this problem and new methods are required to construct the k peaks solutions and describe their asymptotic behavior weak limits of the rescaled solutions localization of the concentration points of the peaks asymptotic profile of the rescaled peaks etc

sign changing semi classical solutions of the brezis nirenberg problems with jump nonlinearities in high dimensions abstract in this paper we consider the following elliptic equation e 2 u u u u p 2 u u 2 2 u in u 0 on where r n n 4 is a bounded domain u max u 0 0 are constants e 0 is a small parameter and 2 p 2 2 n n 2 we obtain a least energy sign changing solution to this problem for e 0 that is sufficiently small by means of the variational method furthermore by combining the elliptic and local energy estimates we study the concentration behavior of this least energy sign changing solution and the location of the positive and negative spikes as e 0 our results partially complete the studies in 6 2011 in the sense that due to sobolev embedding only subcritical cases are considered in that paper

singularly perturbed nonlinear dirichlet problems with a general nonlinearity let be a bounded domain in r n n 3 with a boundary c 2 we consider the following singularly perturbed nonlinear elliptic problem on e 2 u u f u 0 u 0 on u 0 on where the nonlinearity f is of subcritical growth under rather strong conditions on f it has been known that for small e 0 there exists a mountain pass solution u e of above problem which exhibits a spike layer near a maximum point of the distance function d from as e 0 in this paper we construct a solution u e of above problem which exhibits a spike layer near a maximum point of the distance function under certain conditions on f which we believe to be almost optimal

positive least energy solutions and phase separation for coupled schrodinger equations with critical exponent in this paper we study the following coupled schrodinger system which can be seen as a critically coupled perturbed brezis nirenberg problem left begin array ll delta u lambda 1 u mu 1 u 3 beta uv 2 quad x in omega delta v lambda 2 v mu 2 v 3 beta vu 2 quad x in omega u geqq 0 v geqq 0 rm in omega quad u v 0 quad rm on partial omega end array right here omega subset mathbb r 4 is a smooth bounded domain lambda 1 omega 0 and beta neq 0 where lambda 1 omega is the first eigenvalue of with the dirichlet boundary condition note that the nonlinearity and the coupling terms are both critical in dimension 4 that is frac 2n n 2 4 when n 4 we show that this critical system has a positive least energy solution for negative positive small and positive large for the case in which lambda 1 lambda 2 we obtain the uniqueness of positive least energy solutions we also study the limit behavior of the least energy solutions in the repulsive case beta to infty and phase separation is expected these seem to be the first results for this schrodinger system in the critical case

the cyclic homology of crossed product algebras in their article 9 on cyclic homology feigin and tsygan have given a spectral sequence for the cyclic homology of a crossed product algebra generalizing burghelea s calculation 4 of the cyclic homology of a group algebra for an analogous spectral sequence for the hochschild homology of a crossed product algebra see brylinski 2 3 in this article we give a new derivation of this spectral sequence and generalize it to negative and periodic cyclic homology hc a and hp a the method of proof is itself of interest since it involves a natural generalization of the notion of a cyclic module in which the condition that the morphism n n is cyclic of order n 1 is relaxed to the condition that it be invertible we call this category the paracyclic category given a paracyclic module p we can define a chain complex c p with differentials b and b which respectively lower and raise degree the condition that the module p is paracyclic translates to the condition on c p that 1 bb bb is invertible our main result is to show that there is an analogue of the eilenberg zilber theorem for bi paracyclic modules it is then easy to obtain a new expression for the cyclic homology of a crossed product algebra which leads immediately to the spectral sequence of feigin and tsygan if m is a module over a commutative ring k we will denote by m k the iterated tensor product defined by m 0 k and m k 1 m k m if m and n are graded modules we will denote by m n their graded tensor product we would like to thank j block j l brylinski and c kassel for a number of interesting discussions on the results presented here

higher direct images of dualizing sheaves of lagrangian fibrations let f x s be a lagrangian fibration between projective varieties we prove that inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i i s if s is smooth suppose that x is an irreducible symplectic manifold or a certain moduli space of semistable torsion free sheaves on a k 3 surface the hodge numbers satisfy h p q s h p q inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i where n dim s if s inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 03i and x is an irreducible symplectic manifold there exists a hypersurface m f of the kuranishi space of x such that every member of the kuranishi family over m f admits a lagrangian fibration over inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 04i