k theory of log schemes ii log syntomic k theory abstract we prove that suitably truncated topological log syntomic etale homotopy k theory of proper semistable schemes in mixed characteristic surjects onto the kummer log etale p adic k theory of the generic fiber as a corollary we get that the log syntomic etale cohomology is a direct factor of the log syntomic etale cohomology of homotopy k theory sheaves the proofs use p adic hodge theory computations of p adic nearby cycles

generalization of selberg s frac 3 16 theorem and affine sieve an analogue of the well known frac 3 16 lower bound for the first eigenvalue of the laplacian for a congruence hyperbolic surface is proven for a congruence tower associated with any non elementary subgroup l of sl 2 z the proof in the case that the hausdorff of the limit set of l is bigger than frac 1 2 is based on a general result which allows one to transfer such bounds from a combinatorial version to this archimedian setting in the case that delta is less than frac 1 2 we formulate and prove a somewhat weaker version of this phenomenon in terms of poles of the corresponding dynamical zeta function these spectral gaps are then applied to sieving problems on orbits of such groups

dynamical spectral rigidity among mathbb z _2 symmetric strictly convex domains close to a circle appendix b coauthored with h hezari

compact isospectral sets of surfaces on etudie des ensembles de surfaces qui sont isospectrales par rapport a l operateur de laplace beltrami

algebraic cycles and higher k theory here g x is the grothendieck group of coherent sheaves on x 13 gr refers to the graded group defined by the y filtration on g x cf kratzer 14 soult 20 and ch x is the chow group of codimension i algebraic cycles defined by fulton 9 the left hand isomorphism is a formal consequence of the existence of a i structure on g x while the existence of r is the central theme of the b f m rr theorem the main purpose of this paper is to define a theory of higher chow groups ch x n n 2 0 so as to obtain isomorphisms

artin tate motivic sheaves with finite coefficients over an algebraic variety we propose a construction of a tensor exact category fxm of artin tate motivic sheaves with finite coefficients z m over an algebraic variety x over a field k of characteristic prime to m in terms of etale sheaves of z m modules over x among the objects of fxm in addition to the tate motives z m j there are the cohomological relative motives with compact support mccm y x of varieties y quasi finite over x exact functors of inverse image with respect to morphisms of algebraic varieties and direct image with compact supports with respect to quasi finite morphisms of varieties y x act on the exact categories fxm assuming the existence of triangulated categories of motivic sheaves dm x z m over algebraic varieties x over k and a weak version of the six operations in these categories we identify fxm with the exact subcategory in dm x z m consisting of all the iterated extensions of the tate twists mccm y x j of the motives mccm y x an isomorphism of the z m modules ext between the tate motives z m j in the exact category fxm with the motivic cohomology modules predicted by the beilinson lichtenbaum etale descent conjecture recently proved by voevodsky rost et al holds for smooth varieties x over k if and only if the similar isomorphism holds for artin tate motives over fields containing k when k contains a primitive m root of unity the latter condition is equivalent to a certain koszulity hypothesis as shown in our previous paper positselski mixed artin tate motives with finite coefficients mosc math j 11 2011 317 402

the group sk 2 of a biquaternion algebra m rost has proved the existence of an exact sequence relating the group sk1 kernel of the reduced norm for k1 of a biquaternion algebra d whose center is a field f with galois cohomology groups of f in this paper we relate the group sk2 kernel of the reduced norm for k2 of d with galois cohomology of f through an exact sequence

rigidity for pseudo pretheories assume f is a homotopy invariant pseudo pretheory with torsion values and x is a smooth scheme of finite type over a field k we show for certain field extensions k k the map f x f xk is an isomorphism

an analogue of the prime number theorem for closed orbits of axiom a flows for an axiom a flow restricted to a basic set we extend the zeta function to an open set containing w s h where h is the topological entropy this enables us to give an asymptotic formula for the number of closed orbits by adapting the wiener ikehara proof of the prime number theorem

hermitian k theory of the integers rognes and weibel used voevodsky s work on the milnor conjecture to deduce the strong dwyer friedlander form of the lichtenbaum quillen conjecture at the prime 2 in consequence the 2 completion of the classifying space for algebraic k theory of the integers z 1 2 can be expressed as a fiber product of well understood spaces bo and bgl f3 over bu similar results are now obtained for hermitian k theory and the classifying spaces of the integral symplectic and orthogonal groups for the integers z 1 2 this leads to computations of the 2 primary hermitian k groups and affirmation of the lichtenbaum quillen conjecture in the framework of hermitian k theory

discretization of riemannian manifolds applied to the hodge laplacian for kappa geq 0 and r 0 0 let bbb m n kappa r 0 be the set of all connected compact n dimensional riemannian manifolds such that k g leq kappa and inj m g geq r 0 we study the relation between the k rm th positive eigenvalue of the hodge laplacian on differential forms and the k rm th positive eigenvalue of the combinatorial laplacian associated to an open cover acting on v c ech cochains we show that for a fixed sufficiently small varepsilon 0 there exist positive constants c 1 and c 2 depending only on n kappa and varepsilon such that for any m in bbb m n kappa r 0 and for any varepsilon discretization x of m we have c 1 lambda k p x leq lambda k p m leq c 2 lambda k p x for any k leq k k depends on x moreover we find a lower bound for the spectrum of the combinatorial laplacian and a lower bound for the spectrum of the hodge laplacian

on the trend to global equilibrium in spatially inhomogeneous entropy dissipating systems the linear fokker planck equation we study the long time behavior of kinetic equations in which transport and spatial confinement in an exterior potential or in a box are associated with a degenerate collision operator acting only in the velocity variable we expose a general method based on logarithmic sobolev inequalities and the entropy to overcome the well known problem due to the degeneracy in the position variable of the existence of infinitely many local equilibria this method requires that the solution be somewhat smooth in this paper we apply it to the linear fokker planck equation and prove decay to equilibrium faster than o t 1 for all 0 2001 john wiley sons inc

the homotopy fixed point theorem and the quillen lichtenbaum conjecture in hermitian k theory abstract we settle two conjectures for computing higher grothendieck witt groups also known as hermitian k groups of noetherian schemes x under some mild conditions it is shown that the comparison map from the hermitian k theory of x to the homotopy fixed points of k theory under the natural z 2 action is a 2 adic equivalence we also prove that the mod 2 comparison map between the hermitian k theory of x and its etale version is an isomorphism on homotopy groups in the same range as for the quillen lichtenbaum conjecture in k theory applications compute higher grothendieck witt groups of complex algebraic varieties and rings of 2 integers in number fields and hence values of dedekind zeta functions

algebraic geometry of topological spaces i we use techniques from both real and complex algebraic geometry to study k theoretic and related invariants of the algebra c x of continuous complex valued functions on a compact hausdorff topological space x for example we prove a parameterized version of a theorem by joseph gubeladze we show that if m is a countable abelian cancellative torsion free semi normal monoid and x is contractible then every finitely generated projective module over c x m is free the particular case m mathbb n 0 n gives a parameterized version of the celebrated theorem proved independently by daniel quillen and andrei suslin that finitely generated projective modules over a polynomial ring over a field are free the conjecture of jonathan rosenberg which predicts the homotopy invariance of the negative algebraic k theory of c x follows from the particular case m mathbb z n we also give algebraic conditions for a functor from commutative algebras to abelian groups to be homotopy invariant on c algebras and for a homology theory of commutative algebras to vanish on c algebras these criteria have numerous applications for example the vanishing criterion applied to nil k theory implies that commutative c algebras are k regular as another application we show that the familiar formulas of hochschild kostant rosenberg and loday quillen for the algebraic hochschild and cyclic homology of the coordinate ring of a smooth algebraic variety remain valid for the algebraic hochschild and cyclic homology of c x applications to the conjectures of be linson soule and farrell jones are also given

coarse non amenability and covers with small eigenvalues given a closed riemannian manifold m and a virtual epimorphism pi 1 m twoheadrightarrow mathbb f 2 of the fundamental group onto a free group of rank 2 we construct a tower of finite sheeted regular covers left m n right n 0 infty of m such that 1 mn 0 as n this is the first example of such a tower which is not obtainable up to uniform quasi isometry or even up to uniform coarse equivalence by the previously known methods where 1 m is supposed to surject onto an amenable group

random construction of riemann surfaces we develop a new approach for the study of typical riemann surfaces with high genus the method that we use is the construction of random riemann surfaces from oriented cubic graphs this construction enables us to get a control over the global geometry properties of compact riemann surfaces we use the theory of random regular graphs to show that almost all such surfaces have large first eigenvalues and large cheeger constants moreover a closer analysis of the probability space of oriented cubic graphs shows that on a typical surface there is a large embedded hyperbolic ball

a cohomological classification of vector bundles on smooth affine threefolds we give a cohomological classification of vector bundles of rank 2 on a smooth affine threefold over an algebraically closed field having characteristic unequal to 2 as a consequence we deduce that cancellation holds for rank 2 vector bundles on such varieties the proofs of these results involve three main ingredients first we give a description of the first nonstable a1 homotopy sheaf of the symplectic group second these computations can be used in concert with f morel s a1 homotopy classification of vector bundles on smooth affine schemes and obstruction theoretic techniques stemming from a version of the postnikov tower in a1 homotopy theory to reduce the classification results to cohomology vanishing statements third we prove the required vanishing statements

counting isospectral manifolds abstract given a simple lie group h of real rank at least 2 we show that the maximum cardinality of a set of isospectral non isometric h locally symmetric spaces of volume at most x grows at least as fast as x c log x log log x 2 where c c h is a positive constant in contrast with the real rank 1 case this bound comes surprisingly close to the total number of such spaces as estimated in a previous work of belolipetsky and lubotzky 2 our proof uses sunada s method results of 2 and some deep results from number theory we also discuss an open number theoretical problem which would imply an even faster growth estimate

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

rigidity for equivariant pseudo pretheories abstract we prove versions of the suslin and gabber rigidity theorems in the setting of equivariant pseudo pretheories of smooth schemes over a field with an action of a finite group examples include equivariant algebraic k theory presheaves with equivariant transfers equivariant suslin homology and bredon motivic cohomology

affine representability results in mathbb a 1 homotopy theory i vector bundles we establish a general affine representability result in a1 homotopy theory over a general base we apply this result to obtain representability results for vector bundles in a1 homotopy theory our results simplify and significantly generalize morel s a1 representability theorem for vector bundles

isoperimetric control of the spectrum of a compact hypersurface upper bounds for the eigenvalues of the laplace beltrami operator on a hypersurface bounding a domain in some ambient riemannian manifold are given in terms of the isoperimetric ratio of the domain these results are applied to the extrinsic geometry of isometric embeddings

on the local global conjecture for integral apollonian gaskets we prove that a set of density one satisfies the local global conjecture for integral apollonian gaskets that is for a fixed integral primitive apollonian gasket almost every in the sense of density admissible passing local obstructions integer is the curvature of some circle in the gasket

optimal estimate of the spectral gap for the degenerate goldstein taylor model in this paper we study the decay to the equilibrium state for the solution of a generalized version of the goldstein taylor system posed in the one dimensional torus mathbb t mathbb r mathbb z by allowing that the nonnegative cross section can vanish in a subregion x x in mathbb t vert sigma x 0 of the domain with meas x 0 with respect to the lebesgue measure

homology of gln injectivity conjecture for gl4 the homology of gln r and sln r is studied where r is a commutative ring with many units our main theorem states that the natural map h4 gl3 r k h4 gl4 r k is injective where k is a field with char k 2 3 for an algebraically closed field f we prove a better result namely h 4 rm gl 3 f mathbb z to h 4 rm gl 4 f mathbb z is injective we will prove a similar result replacing gl by sl this is used to investigate the indecomposable part of the k group k4 r

on l1 exponential trend to equilibrium for conservative linear kinetic equations on the torus we compute the growth bound of weighted advection semigroups on the n dimensional flat torus with general not necessarily bounded measurable nonnegative collision frequencies and give a compactness theorem in l1 spaces of interest for the stability of essential type of kinetic semigroups under perturbation by scattering operators this provides us with a characterisation of the existence of a spectral gap for general conservative neutron transport like equations various related questions are also dealt with

on the exponential decay to equilibrium of the degenerate linear boltzmann equation abstract in this paper we study the decay to the equilibrium state for the solution of the linear boltzmann equation in the torus t d r d z d d n by allowing that the non negative cross section can vanish in a subregion x x t d x 0 of the domain with meas x 0 with respect to the lebesgue measure we show that the geometrical characterization of x is the key property to produce exponential decay to equilibrium

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

heegaard splittings the virtually haken conjecture and property τ the behaviour of finite covers of 3 manifolds is a very important but poorly understood topic there are three increasingly strong conjectures in the field that have remained open for over twenty years the virtually haken conjecture the positive virtual b1 conjecture and the virtually fibred conjecture any of these would have profound ramifications for 3 manifold theory in this paper we explore the interaction of these conjectures with the following seemingly unrelated areas eigenvalues of the laplacian and heegaard splittings

rigidity for linear framed presheaves and generalized motivic cohomology theories abstract a rigidity property for the homotopy invariant stable linear framed presheaves is established as a consequence a variant of gabber rigidity theorem is obtained for a cohomology theory representable in the motivic stable homotopy category by a torsion spectrum with gw k of rank coprime to the exponential characteristic of the base field k it is shown that the values of such cohomology theories at an essentially smooth henselian ring and its residue field coincide the result is applicable to cohomology theories representable by n torsion spectra as well as to the ones representable by periodic spectra and spectra related to witt groups

rigidity for henselian local rings and mathbb a 1 representable theories we prove that for a large class of mathbb a 1 representable theories including all orientable theories it is possible to construct transfer maps and to prove rigidity theorems similar to those of gabber for algebraic k theory this extends rigidity results of panin and yagunov from algebraically closed fields to arbitrary infinite ones

the construction of weight two arithmetic cohomology let x be a regular noetherian scheme in a letter to soul6 be beilinson conjectured for each non negative integer n the existence of complexes f n x of zariski sheaves on x whose hypercohomology should be in the same relation to algebraic k theory as classical cohomology is to topological ktheory and proposed certain properties that these complexes should satisfy this investigation was continued in l where the conjectured complexes were moved to the a at other points we work only up to 2torsion or up to torsion involving primes dividing the residue field characteristics there seems good reason to expect our complex f 2 x to in fact exactly satisfy all the axioms we now sketch the construction of f 2 x and recall the axioms let a be a regular noetherian ring let w s p e c a t z s p e c a t t t 1 let b bl b 2 b be a finite sequence of exceptional units of a i e b i and

third homology of general linear groups over rings with many units for a commutative ring r with many units we describe the kernel of h3 inc h3 gl2 r z h3 gl3 r z moreover we show that the elements of this kernel are of order at most two as an application we study the indecomposable part of k3 r

ends of locally symmetric spaces with maximal bottom spectrum let x be a symmetric space of non compact type and gnx a locally sym metric space then the bottom spectrum l1 gnx satisfies the inequality l1 gnx e l1 x we show that if equality l1 gnx l1 x holds then gnx has either one end which is necessarily of infinite volume or two ends one of infinite volume and another of finite vol ume in the latter case gnx is isometric to r 1 n endowed with a multi warped metric where n is compact

euler class groups and the homology of elementary and special linear groups abstract we improve homology stability ranges for elementary and special linear groups over rings with many units our result implies stability for unstable quillen k groups and proves a conjecture of bass for commutative local rings with infinite residue fields we show that the obstruction to further stability is given by milnor witt k theory as an application we construct euler classes of projective modules with values in the cohomology of the milnor witt k theory sheaf for d dimensional commutative noetherian rings with infinite residue fields we show that the vanishing of the euler class is necessary and sufficient for an oriented projective module p of rank d to split off a rank 1 free direct summand along the way we obtain a new presentation of milnor witt k theory and of symplectic k 2 simplifying the classical matsumoto moore presentation

on the convergence to equilibrium for degenerate transport problems we give a counterexample which shows that the asymptotic rate of convergence to the equilibrium state for the transport equation with a degenerate cross section and in the periodic setting cannot be better than t 1 2 in the general case we suggest moreover that the geometrical properties of the cross section are the key feature of the problem and impose through the distribution of the forward exit time the speed of convergence to the stationary state

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 the k theory of local fields in this paper we establish a connection between the quillen k theory of certain local fields and the de rham witt complex of their rings of integers with logarithmic poles at the maximal ideal the fieldsk we consider are complete discrete valuation fields of characteristic zero with perfect residue field k of characteristic p 2 when k contains the pth roots of unity the relationship between the k theory with z p coefficients and the de rham witt complex can be described by a sequence

a bloch wigner exact sequence over local rings abstract in this article we extend the bloch wigner exact sequence over local rings where their residue fields have more than nine elements moreover we prove van der kallen s theorem on the presentation of the second k group of local rings such that their residue fields have more than four elements note that van der kallen proved this result when the residue fields have more than five elements although we prove our results over local rings all our proofs also work over semilocal rings where all their residue fields have similar properties as the residue field of local rings

rigidity in motivic homotopy theory we show that extensions of algebraically closed fields induce full and faithful functors between the respective motivic stable homotopy categories with finite coefficients

on the linear boltzmann equation in evolutionary domains with an absorbing boundary we consider the linear boltzmann equation in under the effect of an absorbing moving barrier we prove the existence and uniqueness of the solution and consider the problem of the time asymptotic convergence to equilibrium we then propose a numerical strategy to study the problem and provide quantitative results concerning some relevant test cases

gersten conjecture for equivariant k theory and applications for a reductive group scheme g over a regular semi local ring a we prove the gersten conjecture for the equivariant k theory as a consequence we show that if f is the field of fractions of a then k g 0 a cong k g 0 f generalizing the analogous result for a dvr by serre inst hautes etudes sci publ math 34 37 52 1968 we also show the rigidity for the k theory with finite coefficients of a henselian local ring in the equivariant setting we use this rigidity theorem to compute the equivariant k theory of algebraically closed fields

the gersten conjecture for milnor k theory we prove that the n th milnor k group of an essentially smooth local ring over an infinite field coincides with the n n motivic cohomology of the ring this implies levine s generalized bloch kato conjecture

reduced k theory of azumaya algebras abstract in the theory of central simple algebras often we are dealing with abelian groups which arise from the kernel or co kernel of functors which respect transfer maps for example k functors since a central simple algebra splits and the functors above are trivial in the split case one can prove certain calculus on these functors the common examples are kernel or co kernel of the maps k i f k i d where k i are quillen k groups d is a division algebra and f its center or the homotopy fiber arising from the long exact sequence of above map or the reduced whitehead group sk 1 in this note we introduce an abstract functor over the category of azumaya algebras which covers all the functors mentioned above and prove the usual calculus for it this for example immediately shows that k theory of an azumaya algebra over a local ring is almost the same as k theory of the base ring the main result is to prove that reduced k theory of an azumaya algebra over a henselian ring coincides with reduced k theory of its residue central simple algebra the note ends with some calculation trying to determine the homotopy fibers mentioned above

homology of classical lie groups made discrete ii h2 h3 and relations with scissors congruences the present work extends a number of earlier results see s 7 15 181 for example the following fundamental exact sequence essentially due to bloch and wigner in somewhat different form but not published by them can be found in dupont and sah 7 o o z th3 sl 2 n a a o 2 12 the group y c is defined in 1 4 1 8 and is isomorphic to the group pc considered in dupont and sah 7 also the study of the scissors congruence problem had led to two exact sequences in dupont s theorems 1 3 and 1 43 that roughly correspond to the f eigenspaces of 2 12 under the action of complex conjugation o a h su 2 s3 l wo w z h su 2 to 0 1

bloch wigner theorem over rings with many units the bloch wigner theorem over an algebraically closed field f determines a connection between the third homology of sl2 f and the group describing the nontrivial relations on the tensors of the form x otimes 1 x x in f times backslash 1 the purpose of this article is to provide a version of this theorem over rings with large number of unites for examples semi local rings with infinite residue fields

geometric models for higher grothendieck witt groups in mathbb a 1 homotopy theory we show that the higher grothendieck witt groups a k a algebraic hermitian k groups are represented by an infinite orthogonal grassmannian in the mathbb a 1 homotopy category of smooth schemes over a regular base for which 2 is a unit in the ring of regular functions we also give geometric models for various mathbb p 1 and s 1 loop spaces of hermitian k theory

on the distribution of free path lengths for the periodic lorentz gas consider the domain and let the free path length be defined as the distribution of values of is studied in the limit as for all it is shown that the value is critical for this problem in other words the limiting behavior of depends only on whether is larger or smaller than

third homology of general linear groups the third homology group of gln r is studied where r is a ring with many units with center z r the main theorem states that if k1 z r q k1 r q e g r a commutative ring or a central simple algebra then h3 gl2 r q h3 gl3 r q is injective if r is commutative q can be replaced by a field k such that 1 2 k for an infinite field r resp an infinite field r such that r r 2 we get the better result that h3 gl2 r z 12 h3 gl3 r z 12 resp h3 gl2 r z h3 gl3 r z is injective as an application we study the third homology group of sl2 r and the indecomposable part of k3 r