desingularization of quasi excellent schemes in characteristic zero grothendieck proved in ega iv that if any integral scheme of finite type over a locally noetherian scheme x admits a desingularization then x is quasi excellent and conjectured that the converse is probably true we prove this conjecture for noetherian schemes of characteristic zero namely starting with the resolution of singularities for algebraic varieties of characteristic zero we prove the resolution of singularities for noetherian quasi excellent q schemes

lie theory for nilpotent l algebras the deligne groupoid is a functor from nilpotent differential graded lie algebras concentrated in positive degrees to groupoids in the special case of lie algebras over a field of characteristic zero it gives the associated simply connected lie group we generalize the deligne groupoid to a functor from l algebras concentrated in degree n to n groupoids we actually construct the nerve of the n groupoid which is an enriched kan complex the construction of gamma is quite explicit it is based on dupont s proof of the de rham theorem and yields higher dimensional analogues of holonomy and of the campbell hausdorff formula in the case of abelian l algebras i e chain complexes the functor is the dold kan simplicial set

wave chaos in acoustics and elasticity interpreting wave phenomena in terms of an underlying ray dynamics adds a new dimension to the analysis of linear wave equations forming explicit connections between spectra and wavefunctions on the one hand and the properties of a related ray dynamics on the other hand is a comparatively new research area especially in elasticity and acoustics the theory has indeed been developed primarily in a quantum context it is increasingly becoming clear however that important applications lie in the field of mechanical vibrations and acoustics we provide an overview over basic concepts in this emerging field of wave chaos this ranges from ray approximations of the green function to periodic orbit trace formulae and random matrix theory and summarizes the state of the art in applying these ideas in acoustics both experimentally and from a theoretical numerical point of view

computing motivic zeta functions on log smooth models we give an explicit formula for the motivic zeta function in terms of a log smooth model it generalizes the classical formulas for snc models but it gives rise to much fewer candidate poles in general this formula plays an essential role in recent work on motivic zeta functions of degenerating calabi yau varieties by the second named author and his collaborators as a further illustration we explain how the formula for newton non degenerate polynomials can be viewed as a special case of our results

combinatorial formulas for vassiliev invariants from chern simons gauge theory we analyze the perturbative series expansion of the vacuum expectation value of a wilson loop in chern simons gauge theory in the temporal gauge from the analysis emerges the notion of the kernel of a vassiliev invariant the kernel of a vassiliev invariant of order n is not a knot invariant since it depends on the regular knot projection chosen but it differs from a vassiliev invariant by terms that vanish on knots with n singular crossings we conjecture that vassiliev invariants can be reconstructed from their kernels we present the general form of the kernel of a vassiliev invariant and we describe the reconstruction of the full primitive vassiliev invariants at orders two three and four at orders two and three we recover known combinatorial expressions for these invariants at order four we present new combinatorial expressions for the two primitive vassiliev invariants present at this order

iterated vanishing cycles convolution and a motivic analogue of a conjecture of steenbrink we prove a motivic analogue of steenbrink s conjecture 25 conjecture 2 2 on the hodge spectrum proved by m saito in 21 to achieve this we construct and compute motivic iterated vanishing cycles associated with two functions we are also led to introduce a more general version of the convolution operator appearing in the motivic thom sebastiani formula throughout the article we use the framework of relative equivariant grothendieck rings of varieties endowed with an algebraic torus action

automatic classification of protein structure by using gauss integrals we introduce a method of looking at analyzing and comparing protein structures the topology of a protein is captured by 30 numbers inspired by vassiliev knot invariants to illustrate the simplicity and power of this topological approach we construct a measure scaled gauss metric sgm of similarity of protein shapes under this metric protein chains naturally separate into fold clusters we use sgm to construct an automatic classification procedure for the cath2 4 database the method is very fast because it requires neither alignment of the chains nor any chain chain comparison it also has only one adjustable parameter we assign 95 51 of the chains into the proper c class a architecture t topology and h homologous superfamily fold find all new folds and detect no false geometric positives using the sgm we display a map of the space of folds projected onto two dimensions show the relative locations of the major structural classes and zoom into the space of proteins to show architecture topology and fold clusters the existence of a simple measure of a protein fold computed from the chain path will have a major impact on automatic fold classification

semiclassical theory of flexural vibrations of plates we study the biharmonic equation of flexural vibrations of elastic plates by a semiclassical method that can easily be generalized for other models of wave propagation three terms of the asymptotic number of levels for plates with smooth boundaries are derived and the trace formula for the density of states is obtained the main difference between this formula and the gutzwiller trace formula for billiards is the existence of a specific phase factor obtained while reflecting from the boundary six hundred eigenvalues of a clamped stadium plate are obtained by a specially developed numerical algorithm and the trace formula is assessed looking at its fourier transform an extra contribution occurs for a free plate due to the existence of boundary modes

curve counting via stable pairs in the derived category for a nonsingular projective 3 fold x we define integer invariants virtually enumerating pairs c d where c x is an embedded curve and d c is a divisor a virtual class is constructed on the associated moduli space by viewing a pair as an object in the derived category of x the resulting invariants are conjecturally equivalent after universal transformations to both the gromov witten and dt theories of x for calabi yau 3 folds the latter equivalence should be viewed as a wall crossing formula in the derived category

vanishing cycles for formal schemes let k be a non archimedean field and let x be a formal scheme locally finitely presented over the ring of integers k see 1 in this work we construct and study the vanishing cycles functor from the category of etale sheaves on the generic fibre x of x which is a k analytic space to the category of etale sheaves on the closed fibre xs of x which is a scheme over the residue field of k we prove that if x is the formal completion x of a scheme x finitely presented over k along the closed fibre then the vanishing cycles sheaves of x are canonically isomorphic to those of x as defined in sga7 exp xiii in particular the vanishing cycles sheaves of x depend only on x and any morphism x induces a homomorphism from the pullback of the vanishing cycles sheaves of x under s ys xs to those of y furthermore we prove that for each x one can find a nontrivial ideal of k such that if two morphisms x coincide modulo this ideal then the homomorphisms between the vanishing cycles sheaves induced by and coincide these facts were conjectured by p deligne in 1 we associate with a formal scheme x locally finitely presented over k a k analytic space x in the sense of ber1 and ber2 in 2 we find that the morphism y x which is induced by an etale morphism of formal schemes y x possesses a certain property morphisms of k analytic spaces with this property are called quasi etale and they give rise to a quasi etale site xqet of a k analytic space x there is a canonical morphism of sites xqet xet where xet is the etale site introduced in ber2 we show that the inverse image functor identifies the category of etale sheaves x et with a full subcategory of

knot probabilities in random diagrams we consider a natural model of random knotting choose a knot diagram at random from the finite set of diagrams with n crossings we tabulate diagrams with 10 and fewer crossings and classify the diagrams by knot type allowing us to compute exact probabilities for knots in this model as expected most diagrams with 10 and fewer crossings are unknots about 78 of the roughly 1 6 billion 10 crossing diagrams for these crossing numbers the unknot fraction is mostly explained by the prevalence of tree like diagrams which are unknots for any assignment of over under information at crossings the data shows a roughly linear relationship between the log of knot type probability and the log of the frequency rank of the knot type analogous to zipf s law for word frequency the complete tabulation and all knot frequencies are included as supplementary data

the euler lagrange equation and heat flow for the mobius energy we prove the following results 1 a unique smooth solution exists for a short time for the heat equation associated with the mobius energy of loops in a euclidean space starting with any simple smooth loop 2 a critical loop of the energy is smooth if it has cube integrable curvature combining this with an earlier result of m freedman z wang and the author we show that any local minimizer of the energy must be smooth 3 circles are the only two dimensional critical loops with cube integrable curvature the technique also applies to a family of other knot energies similar problems are open for energies of surfaces or more generally for embedded submanifolds in a fixed riemannian manifold 2000 john wiley sons inc

a trace formula for varieties over a discretely valued field we study the motivic serre invariant of a smoothly bounded alge braic or rigid variety x over a complete discretely valued field k with perfect residue field k if k has characteristic zero we extend the definition to arbi trary k varieties using bittner s presentation of the grothendieck ring and a process of neron smoothening of pairs of varieties the motivic serre invariant can be considered as a measure for the set of unramified points on x under certain tameness conditions it admits a cohomological interpretation by means of a trace formula in the curve case we use t saito s geometric criterion for cohomological tameness to obtain more detailed results we discuss some applications to weil chatelet groups chow motives and the structure of the grothendieck ring of varieties msc 2000 11s15 14g05 14g22

integral geometry of plane curves and knot invariants we study the integral expression of a knot invariant obtained as the second coefficient in the perturbative expansion of witten s chern simons path integral associated with a knot one of the integrals involved turns out to be a generalization of the classical crofton integral on convex plane curves and it is related with the invariants of generic plane curves recently defined by arnold with deep motivations in symplectic and contact geometry quadratic bounds on these plane curve invariants are derived using their relationship with the knot invariant

motivic serre invariants and weil restriction we study the interactions between weil restriction for formal schemes and rigid varieties greenberg schemes and motivic serre invariants and their behavior with respect to finite extensions of the base ring r which is a complete discrete valuation ring with perfect residue field

motivic zeta functions of degenerating calabi yau varieties we study motivic zeta functions of degenerating families of calabi yau varieties our main result says that they satisfy an analog of igusa s monodromy conjecture if the family has a so called galois equivariant kulikov model we provide several classes of examples where this condition is verified we also establish a close relation between the zeta function and the skeleton that appeared in kontsevich and soibelman s non archimedean interpretation of the syz conjecture in mirror symmetry

on steenbrink s conjecture let f x c be a holomorphic function on a complex manifold using the monodromy and mixed hodge structure on the cohomology of milnor fiber we can associate the spectrum sp f x of f at x e xo f 1 0 it is a fractional laurent polynomial n t with n z this notion is first introduced by steenbrink 15 cteq in the isolated singularity case cf also 19 and 16 in general if f has isolated singularity n are nonnegative and 0 n 0 counted with multiplicity are called the exponents or spectra 19 of f assume f has one dimensional singular locus and consider a deformation of the form f g where g is a linear form i e a function such that g x 0 and dg 0 at x then f g has isolated singularity if g is generic let zk be a local irreducible component of singf at x and mk its multiplicity along z zk x f is viewed as constant deformation of isolated singularity let k j be its exponents which are constant on z 20 we can also define ilk j e 0 1 in a compatible way with k j so that exp 2ni 3k j are the eigenvalues of the monodromy along z then steenbrink conjectured

germs of arcs on singular algebraic varieties and motivic integration we study the scheme of formal arcs on a singular algebraic variety and its images under truncations we prove a rationality result for the poincare series of these images which is an analogue of the rationality of the poincare series associated to p adic points on a p adic variety the main tools which are used are semi algebraic geometry in spaces of power series and motivic integration a notion introduced by m kontsevich in particular we develop the theory of motivic integration for semi algebraic sets of formal arcs on singular algebraic varieties we prove a change of variable formula for birational morphisms and we prove a geometric analogue of a result of oesterle

equivariant motivic integration and proof of the integral identity conjecture for regular functions we develop denef loeser s motivic integration to an equivariant version and use it to prove the full integral identity conjecture for regular functions in comparison with hartmann s work the equivariant grothendieck ring defined in this article is more elementary and it yields the application to the conjecture

on the mixed hodge structure on the cohomology of the milnor fibre in 18 the second author introduced a mixed hodge structure on the cohomology of the milnor fibre of an isolated hypersurface singularity for the definition of a mixed hodge structure el 17 sect 3 4 the weight filtration is essentially the monodromy weight filtration 18 sect 4 and thus simplifies the jordan normal form of the unipotent part of the monodromy the significance of the hodge filtration however is not so dear and its description seems difficult to use this paper gives another description of the hodge filtration which we hope is easier to understand and apply its definition does not use resolution of singularities instead it relies on the theory of holonomic modules in one variable with regular singularities this paper arose from conversations at the 1980 arbeitstagung in bonn where varchenko s conjectured hodge filtration 22 had been discussed in brieskorn s seminar we have since learned that varchenko obtained similar results to those in this paper in the summer and autumn of 1980 23 25 in sect 1 the milnor fibration is embedded in a family of smooth projective hypersurfaces the hodge theory of a smooth projective hypersurface is explained in the fashion of brylinski 2 in sect 2 in sect 3 we recall the description of the gauss manin system from 8 our new formula for the hodge filtration is explained in sects 4 5 in sect 6 we prove that it gives the same result as 18 we prove a thom sebastiani formula for the hodge filtration in sect 7 which leads to a proof of conjecture 5 4 of 18 in sect 8 we show how to prove varchenko s result about the jordan normal forms of multiplication by f in the jacobian ring of f and the logarithm of the unipotent monodormy of f with our method in the last chapter the mixed hodge structure is calculated for two examples

experiments on elastomechanical wave functions in chaotic plates and their statistical features we measure the amplitude of the elastomechanical displacement at a fine grid of points on a free plate having the shape of a sinai stadium the obtained displacement field formally corresponds to a wave function in a quantum system while the distribution of the squared amplitudes agrees with the prediction of random matrix theory rmt there is a strong deviation of the spatial correlator from the standard prediction for quantum chaotic systems we show that this is due to the presence of two modes leading to a beating phenomenon we construct a proper extension of the spatial correlator within the framework of rmt

motivic integration on smooth rigid varieties and invariants of degenerations we develop a theory of motivic integration for smooth rigid varieties as an application we obtain a motivic analogue for rigid varieties of serre s invariant for p adic varieties our construction provides new geometric birational invariants of degenerations of algebraic varieties for degenerations of calabi yau varieties our results take a stronger form

spectral hirzebruch milnor classes of singular hypersurfaces we introduce spectral hirzebruch milnor classes for singular hypersurfaces these can be identified with steenbrink spectra in the isolated singularity case and may be viewed as their global analogues in general their definition uses vanishing cycles of mixed hodge modules and the todd class transformation these are compatible with the pushforward by proper morphisms and the classes can be calculated by using resolutions of singularities formulas for hirzebruch milnor classes of projective hypersurfaces in terms of these classes are given in the case where the multiplicity of a generic hyperplane section is not 1 these formulas using hyperplane sections instead of hypersurface ones are easier to calculate in certain cases here we use the thom sebastiani theorem for the underlying filtered d modules of vanishing cycles from which we can deduce the thom sebastiani type theorem for spectral hirzebruch milnor classes for the chern classes after specializing to y 1 we can give a relatively simple formula for the localized milnor classes which implies a new formula for the euler numbers of projective hypersurfaces using iterated hyperplane sections applications to log canonical thresholds and du bois singularities are also explained for instance the latter can be detected by using hirzebruch milnor classes in the projective hypersurface case

vanishing cycles for non archimedean analytic spaces in this work we develop a formalism of vanishing cycles for non archimedean analytic spaces which is an analog of that for complex analytic spaces from sga7 exp xiv as an application we prove that in the equicharacteristic case the stalks of the vanishing cycles sheaves of a scheme x at a closed point x e x depend only on the formal completion spf ox of x at x in particular any continuous homomorphism ox x y y induces a homorphism from the stalks of the vanishing cycles sheaves of x at x to those of y at y furthermore we prove that given ox x and o y there exists n 1 such that for any pair of continuous homomorphisms ox x oy y that coincide modulo the n th power of the maximal ideal of oy y the induced homomorphisms between the stalks of the vanishing cycles sheaves coincide these facts generalize a result of g laumon from lau see remark 7 6 throughout the paper we fix a non archimedean field k whose valuation is not assumed to be nontrivial in 1 we study etale galois sheaves on k analytic spaces to define the vanishing cycles functor and to work with it we use the language of pro analytic spaces i e pro objects of the category of analytic spaces sga4 exp i examples of such objects are the germs of analytic spaces as in ber2 3 4 another example is considered in 3 in 4 we define the vanishing cycles functor and establish its basic properties in 5 we show that the vanishing cycles sheaves are trivial for smooth morphisms in 6 we prove a comparison theorem for vanishing cycles this theorem is more general than its analog over c from sga7 exp xiv and its proof does not use hironaka s theorem on resolution of singularities in 7 we apply the comparison theorem to prove the properties of the vanishing cycles sheaves of schemes formulated above it is worthwhile to note that this application is obtained by considering non archimedean analytic geometry over fields with trivial valuation like ber3 this work arose from a suggestion of p deligne to apply the etale cohomology theory from ber2 to the study of the vanishing cycles sheaves of schemes i am very grateful to him for useful discussions on the subject i also

a trace formula for rigid varieties and motivic weil generating series for formal schemes we establish a trace formula for rigid varieties x over a complete discretely valued field which relates the set of unramified points on x to the galois action on its etale cohomology next we show that the analytic milnor fiber of a morphism f at a point x completely determines the formal germ of f at x we develop a theory of motivic integration for formal schemes of pseudo finite type over a complete discrete valuation ring r and we introduce the weil generating series of a regular formal r scheme mathfrak x of pseudo finite type via the construction of a gelfand leray form on its generic fiber when mathfrak x is the formal completion of a morphism f from a smooth irreducible variety to the affine line then its weil generating series coincides modulo normalization with the motivic zeta function of f when mathfrak x is the formal completion of f at a closed point x of the special fiber f 1 0 we obtain the local motivic zeta function of f at x

motivic zeta functions of abelian varieties and the monodromy conjecture we prove for abelian varieties a global form of denef and loeser s motivic monodromy conjecture in arbitrary characteristic more precisely we prove that for every tamely ramified abelian variety a over a complete discretely valued field with algebraically closed residue field its motivic zeta function has a unique pole at chai s base change conductor c a of a and that the order of this pole equals one plus the potential toric rank of a moreover we show that for every embedding of ql in c the value exp 2 ic a is an l adic tame monodromy eigenvalue of a the main tool in the paper is edixhoven s filtration on the special fiber of the neron model of a which measures the behavior of the neron model under tame base change

counting function for a sphere of anisotropic quartz we calculate the leading weyl term of the counting function for a monocrystalline quartz sphere in contrast to other studies of counting functions the anisotropy of quartz is a crucial element in our investigation hence we do not obtain a simple analytical form but we carry out a numerical evaluation to this end we employ the radon transform representation of the green s function we compare our result to a previously measured unique data set of several tens of thousands of resonances

definable sets motives and p adic integrals 0 1 let x be a scheme reduced and separated of finite type over z for p a prime number one may consider the set x zp of its zp rational points for every n in n there is a natural map n x zp x z p assigning to a zp rational point its class modulo p the image yn p of x zp by n is exactly the set of z p rational points which can be lifted to zp rational points denote by nn p the cardinality of the finite set yn p by a result of the first author 7 the poincare series

proofs of the integral identity conjecture over algebraically closed fields recently it has become well known that the conjectural integral identity is of crucial importance in the motivic donaldson thomas invariants theory for noncommutative calabi yau threefolds the purpose of this article is to consider different versions of the identity for regular functions and formal functions and to give them the positive answer for the algebraically closed ground fields technically the result on the motivic milnor fiber by hrushovski loeser using hrushovski kazhdan s motivic integration and nicaise s computations on motivic integrals on special formal schemes are main tools

simple hironaka resolution in characteristic zero building upon work of villamayor and bierstone milman we give a proof of the canonical hironaka principalization and desingularization the idea of homogenized ideals introduced in the paper gives a priori the canonicity of algorithm and radically simplifies the proof

the thom sebastiani theorem for the euler characteristic of cyclic l algebras abstract let l be a cyclic l algebra of dimension 3 with finite dimensional cohomology only in dimension one and two by transfer theorem there exists a cyclic l algebra structure on the cohomology h l the inner product plus the higher products of the cyclic l algebra defines a superpotential function f on h 1 l we associate with an analytic milnor fiber for the formal function f and define the euler characteristic of l is to be the euler characteristic of the etale cohomology of the analytic milnor fiber in this paper we prove a thom sebastiani type formula for the euler characteristic of cyclic l algebras as applications we prove the joyce song formulas about the behrend function identities for semi schur objects in the derived category of coherent sheaves over calabi yau threefolds a motivic thom sebastiani type formula and a conjectural motivic joyce song formula for the motivic milnor fiber of cyclic l algebras are also discussed

on grothendieck rings and algebraically constructible functions we investigate grothendieck rings appearing in real geometry notably for arc symmetric sets and focus on the relative case in analogy with the properties of the ring of algebraically constructible functions defined by mccrory and parusinski we study in particular the duality and link operators including their behaviour with respect to motivic milnor fibres with signs

donaldson thomas type invariants via microlocal geometry we prove that donaldson thomas type invariants are equal to weighted euler characteristics of their moduli spaces in particular such invariants depend only on the scheme structure of the moduli space not the symmetric obstruction theory used to define them we also introduce new invariants generalizing donaldson thomas type invariants to moduli problems with open moduli space these are useful for computing donaldson thomas type invariants over stratifications

topology of nonarchimedean analytic spaces and relations to complex algebraic geometry this note surveys basic topological properties of nonarchimedean analytic spaces in the sense of berkovich including the recent tameness results of hrushovski and loeser we also discuss interactions between the topology of nonarchimedean analytic spaces and classical algebraic geometry

geometric criteria for tame ramification we prove an a campo type formula for the tame monodromy zeta function of a smooth and proper variety over a discretely valued field k as a first application we relate the orders of the tame monodromy eigenvalues on the l adic cohomology of a k curve to the geometry of a relatively minimal sncd model and we show that the semi stable reduction theorem and saito s criterion for cohomological tameness are immediate consequences of this result as a second application we compute the error term in the trace formula for smooth and proper k varieties we see that the validity of the trace formula would imply a partial generalization of saito s criterion to arbitrary dimension

motivic degree zero donaldson thomas invariants given a smooth complex threefold x we define the virtual motive operatorname hilb n x operatorname vir of the hilbert scheme of n points on x in the case when x is calabi yau operatorname hilb n x operatorname vir gives a motivic refinement of the n point degree zero donaldson thomas invariant of x the key example is x 3 where the hilbert scheme can be expressed as the critical locus of a regular function on a smooth variety and its virtual motive is defined in terms of the denef loeser motivic nearby fiber a crucial technical result asserts that if a function is equivariant with respect to a suitable torus action its motivic nearby fiber is simply given by the motivic class of a general fiber this allows us to compute the generating function of the virtual motives operatorname hilb n mathbb c 3 operatorname vir via a direct computation involving the motivic class of the commuting variety we then give a formula for the generating function for arbitrary x as a motivic exponential generalizing known results in lower dimensions the weight polynomial specialization leads to a product formula in terms of deformed macmahon functions analogous to gottsche s formula for the poincare polynomials of the hilbert schemes of points on surfaces

on the self linking of knots this note describes a subcomplex f of the de rham complex of parametrized knot space which is combinatorial over a number of universal anomaly integrals the self linking integrals of guadaguini martellini and mintchev perturbative aspects of chern simons field theory phys lett b 227 111 1989 and bar natan perturbative aspects of the chern simons topological quantum field theory ph d thesis princeton university 1991 also on the vassiliev knot invariants to appear in topology are seen to represent the first nontrivial element in h0 f occurring at level 4 and are anomaly free however already at the next level an anomalous term is possible