uniformization of analytic spaces in this article we give an explicit elementary proof of a local version of resolution of singularities in characteristic zero local means that the centres of blowing up are chosen locally so that a finite number of finite sequences of local blowings up may be required to cover a neighbourhood of a given point hence we use the term uniformization rather than local desingularization although uniformization in our sense is considerably stronger than the original idea of zariski 16 in the last two decades several mathematicians notably abhyankar 1 hironaka and spivakovsky 12 and youssin 14 15 have proposed simpler or more explicit versions of the inductive procedure in hironaka s proof of his great theorem 8 these approaches would seem to lead to some form of uniformization though full details of none of them have yet appeared the idea of seeking an explicit local procedure to determine the centres of blowing up is that a sufficiently good local choice should globalize automatically our method for example gives global resolution of singularities of surfaces in any codimension throughout this paper k denotes either the field of real numbers r or the field of complex numbers c we will work in the category of analytic spaces over k although our techniques apply as well to algebraic spaces over a field

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

equivariant poincare polynomials and counting points over finite fields abstract suppose a finite group acts as a group of automorphisms of a smooth complex algebraic variety which is defined over a number field we show how in certain circumstances an equivariant comparison theorem in l adic cohomology may be used to convert the computation of the graded character of the induced action on cohomology into questions about numbers of rational points of varieties over finite fields this is carried through in three applications first for the symmetric group acting on the moduli space of n points on a genus zero curve second for a unitary reflection group acting on the complement of its reflecting hyperplanes and third for the symmetric group action on the space of configurations of points in any smooth variety which satisfies certain strong purity conditions

a principalization algorithm for locally monomial ideal sheaves on 3 folds with an application to toroidalization abstract suppose that i is an ideal sheaf on a nonsingular variety x a principalization of i is a proper birational morphism x x such that x is nonsingular and i o x is locally principal we provide a fast and simple algorithm to construct a principalization of a locally monomial ideal sheaf on a 3 fold as an application we prove the existence of toroidalization of locally toroidal morphisms of 3 folds over an algebraically closed field of characteristic zero

on the distribution of points of bounded height on equivariant compactifications of vector groups we prove asymptotic formulas for the number of rational points of bounded height on smooth equivariant compactifications of the affine space

hypersurface singularities in positive characteristic we present results on multiplicity theory differential operators on smooth schemes play a central role in the study of the multiplicity of an embedded hypersurface at a point this follows from the fact that the multiplicity is defined by the taylor development of the defining equation at such point and the taylor development involves higher order differentials on the other hand the multiplicity of a hypersurface can be expressed in terms of general projections defined at etale neighborhoods of such point given a local embedding of a d 1 dimensional hypersurface in a smooth d dimensional scheme a general projection on a smooth d 1 dimensional scheme is a finite map on the hypersurface and the multiplicity at the point is the degree of this finite map we relate both approaches in fact we study invariants of embedded hypersurfaces defined in terms of differential operators which express properties of the ramification of the morphism of particular interest is the case of hypersurfaces over fields of positive characteristic a central result in multiplicity theory of hypersurfaces is a form of elimination of one variable in the description of highest multiplicity locus this form of elimination known over fields of characteristic zero is achieved with the notion of tschirnhausen polynomial introduced by abhyankar this is a key point in the proof of embedded desingularization in this paper we provide a characteristic free approach to this form of elimination our alternative approach is based on projections on smooth d 1 dimensional schemes the properties of this new form of elimination remain weaker in positive characteristic than it does in characteristic zero when it comes to compatibility of elimination with permissible monoidal transformation this opens the way to new questions on resolution problems we also discuss the behavior of invariants attached to a singularity at a point with this form of elimination

good points and constructive resolution of singluarities in this paper we present a new algorithm of resolution of singularities over fields of characteristic zero making use of invariants that come from abhyankar s good point theory abl we also prove new properties on constructive or algorithmic desingularization let us explain what we mean by an algorithm of resolution consider a pair x w where w is a regular variety over a base field k of characteristic zero not necessarily irreducible i e w smooth over k and xc w is a closed non empty subscheme call c the class of all such pairs over different base fields k natural maps i x w1 x w arise within this class for instance if w1 w is a smooth map over a fixed field k or if w1 w arises from an arbitrary change of base field fix now a totally ordered set i and suppose assigned for each pair 7 x w of u a function p x i which is upper semi continuous and takes only finitely many values say 1 c r c i let max bp be the biggest ai and set

a new perspective on single and multi variate differential equations we elaborate upon a new method of solving linear differential equations of arbitrary order which is applicable to a wide class of single and multi variate equations our procedure separates the operator part of the equation under study in to a part containing a function of the euler operator and constants and another one retaining the rest the solution of the equation is then obtained from the monomials or the monomial symmetric functions for the multi variate case which are the eigenfunctions of the euler operator novel exponential forms of the solutions of the differential equations enable one to analyze the underlying symmetries of the equations and explore the algebraic structures of the solution spaces in a straightforward manner the procedure allows one to derive various properties of the orthogonal polynomials and functions in a unified manner after showing how the generating functions and rodriguez formulae emerge naturally in this method we briefly outline the generalization of the present approach to the multi variate case

quantum k theory of grassmannians we show that equivariant k theoretic 3 point gromov witten invariants of genus zero on a grassmann variety are equal to triple intersec tions computed in the equivariant k theory of a two step flag manifold thus generalizing an earlier result of buch kresch and tamvakis in the process we show that the gromov witten variety of curves passing through 3 general points is irreducible and rational our applications include pieri and giambelli formulas for the quantum k theory ring of a grassmannian which determine the multiplication in this ring we also compute the dual schubert basis for this ring and show that its structure constants satisfy s3 symmetry our for mula for gromov witten invariants can be partially generalized to cominuscule homogeneous spaces by using a construction of chaput manivel and perrin

formulae relating the bernstein and iwahori matsumoto presentations of an affine hecke algebra abstract we consider the anti dominant variants of the elements occurring in the bernstein presentation of an affine hecke algebra h we find explicit formulae for in terms of the iwahori matsumoto generators tw w ranging over the extended affine weyl group of the root system r in the case i r is arbitrary and is a minuscule coweight or ii r is attached to gln and mek where ek is a standard basis vector and m 1 in the above cases certain minimal expressions for play a crucial role such minimal expressions exist in fact for any coweight for gln we give a sheaf theoretic interpretation of the existence of a minimal expression for the corresponding perverse sheaf on the affine schubert variety x t is the push forward of an explicit perverse sheaf on the demazure resolution m x t x t this approach yields for a minuscule coweight of any r or for an arbitrary coweight of gln a conceptual albeit less explicit expression for the coefficient w of the basis element tw in terms of the cohomology of a fiber of the demazure resolution

algebraic treatment of second poschl teller morse rosen and eckart equations the method of an earlier paper see ibid vol 20 p 4075 1987 is applied to the non compact case to solve a family of second poschl teller morse rosen and eckart equations with quantised coupling constants both discrete and continuous spectra bound state and scattering wavefunctions transmission coefficients are found from the matrix elements of group representations

equimultiplicity algebraic elimination and blowing up abstract given a variety x over a perfect field we study the partition defined on x by the multiplicity into equimultiple points and the effect of blowing up at smooth equimultiple centers over fields of characteristic zero we prove resolution of singularities by using the multiplicity as an invariant instead of the hilbert samuel function

a proof of kurdyka s conjecture on arc analytic functions we prove a conjecture of kurdyka stating that every arc symmetric semialgebraic set is precisely the zero locus of an arc analytic semialgebraic function this implies in particular that arc symmetric semialgebraic sets are in one to one correspondence with radical ideals of the ring of arc analytic semialgebraic functions

pro p iwahori hecke algebra and supersingular representations the motivation of this paper is the search for a langlands correspondence modulo p we show that the pro p iwahori hecke ring h 1 of a split reductive p adic group g over a local field f of finite residue field fq with q elements admits an iwahori matsumoto presentation and a bernstein z basis and we determine its centre we prove that the ring h 1 is finitely generated as a module over its centre these results are proved in 12 only for the iwahori hecke ring let p be the prime number dividing q and let k be an algebraically closed field of characteristic p a character from the centre of h 1 to k which is as null as possible will be called null the simple h k modules with a null central character are called supersingular when g gl n we show that each simple h k module of dimension n containing a character of the affine subring h aff is supersingular the proof uses the minimal expressions of haines generalized to h 1 and that the number of such modules is equal to the number of irreducible k representations of the weil group wf of dimension n when the action of an uniformizer pf in the hecke algebra side and of the determinant of a frobenius frf in the galois side are fixed i e the number nn q of unitary irreducible polynomials in fq x of degree n one knows that the converse is true by explicit computations when n 2 11 and when n 3 rachel ollivier

an elementary coordinate dependent local resolution of singularities and applications abstract an elementary classical analysis resolution of singularities method is developed extensively using explicit coordinate systems the algorithm is designed to be applicable to subjects such as oscillatory integrals and critical integrability exponents as one might expect the trade off for such an elementary method is a weaker theorem than hironaka s work h hironaka resolution of singularities of an algebraic variety over a field of characteristic zero i ann of math 2 79 1964 109 203 h hironaka resolution of singularities of an algebraic variety over a field of characteristic zero ii ann of math 2 79 1964 205 326 or its subsequent simplications and extensions such as e bierstone p milman canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant invent math 128 2 1997 207 302 s encinas o villamayor good points and constructive resolution of singularities acta math 181 1 1998 109 158 j kollar resolution of singularities seattle lectures preprint a n varchenko newton polyhedra and estimates of oscillatory integrals funct anal appl 18 3 1976 175 196 nonetheless the methods of this paper can be used to prove a variety of theorems of interest in analysis as illustration two consequences are given first and most notably a general theorem regarding the existence of critical integrability exponents are established secondly a quick proof of a well known inequality of lojasiewicz s lojasiewicz ensembles semi analytiques inst hautes etudes sci bures sur yvette 1964 is given the arguments here are substantially different from the general algorithms such as h hironaka resolution of singularities of an algebraic variety over a field of characteristic zero i ann of math 2 79 1964 109 203 h hironaka resolution of singularities of an algebraic variety over a field of characteristic zero ii ann of math 2 79 1964 205 326 or the elementary arguments of e bierstone p milman semianalytic and subanalytic sets inst hautes etudes sci publ math 67 1988 5 42 and h sussman real analytic desingularization and subanalytic sets an elementary approach trans amer math soc 317 2 1990 417 461 the methods here have as antecedents the earlier work of the author m greenblatt a direct resolution of singularities for functions of two variables with applications to analysis j anal math 92 2004 233 257 phong and stein d h phong e m stein the newton polyhedron and oscillatory integral operators acta math 179 1997 107 152 and varchenko a n varchenko newton polyhedra and estimates of oscillatory integrals funct anal appl 18 3 1976 175 196

functorial desingularization of quasi excellent schemes in characteristic zero the nonembedded case we prove that any reduced noetherian quasi excellent scheme of characteristic zero admits a strong desingularization which is functorial with respect to all regular morphisms as a main application we deduce that any reduced formal variety of characteristic zero admits a strong functorial desin gularization also we show that as an easy formal consequence of our main result one obtains strong functorial desingularization for many other spaces of characteristic zero including quasi excellent stacks and formal schemes and complex or non archimedean analytic spaces moreover these functors easily generalize to non compact setting by use of generalized convergent blow up sequences with regular centers

canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant this article contains an elementary constructive proof of resolution of singularities in characteristic zero our proof applies in particular to schemes of finite type and to analytic spaces so we recover the great theorems of hironaka we introduce a discrete local invariant rm inv x a whose maximum locus determines a smooth centre of blowing up leading to desingularization to define rm inv x we need only to work with a category of local ringed spaces x x cal o x satisfying certain natural conditions if a in x then rm inv x a depends only on widehat cal o x a more generally rm inv x is defined inductively after any sequence of blowings up whose centres have only normal crossings with respect to the exceptional divisors and lie in the constant loci of rm inv x cdot the paper is self contained and includes detailed examples one of our goals is that the reader understand the desingularization theorem rather than simply know it is true

techniques for the study of singularities with applications to resolution of 2 dimensional schemes we give an overview of invariants of algebraic singularities over perfect fields we then show how they lead to a synthetic proof of embedded resolution of singularities of 2 dimensional schemes

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

liftable derivations for generically separably algebraic morphisms of schemes we consider dominant generically algebraic e g generically fi nite and tamely ramified if the characteristic is positive morphisms n x s y s of s schemes where y 5 are ncetherian and integral and x is a krull scheme e g normal ncetherian and study the sheaf of tangent vector fields on y that lift to tangent vector fields on x we give an easily com putable description of these vector fields using valuations along the critical locus we apply this to answer the question when the liftable derivations can be defined by a tangency condition along the discriminant in particular if it is a blow up of a coherent ideal we show that tangent vector fields that preserve the ratliff rush ideal equals in 1 n for high n associated to are liftable and that all liftable tangent vector fields preserve the integral closure of we also generalise in positive characteristic seidenberg s theorem that all tangent vector fields can be lifted to the normalisation assuming tame ramification

a game for the resolution of singularities we propose a combinatorial game on finite graphs called stratify t hat is played by two protagonists a and b the game captures the logical structure of a proof of the resolution of singularities in each round the graph of the game is modified by the moves of the players when it assumes a final configuration a has won otherwise the game goes on forever and nobody wins in particular b cannot win the game but can only prevent a from winning we show that a always possesses a winning strategy regardless of the initial shape of the graph and of the moves of b this implies translating the game back to algebraic geometry that there is a choice of centres for the blowup of singular varieties in characteristic zero which eventually leads to their resolution the algebra needed for this implication is elementary the transcription from varieties to graphs and from blowups to modifications of the graph thus axiomatizes the proof of the resolution of singularities in principle the same logic could also work in positive characteristic once an appropriate descent in dimension is settled

a new realisation of dynamical groups and factorisation method a new method of algebraisation of quantum mechanical eigenvalue equations is presented in this method the dynamical algebra is represented on the space of group matrix elements the ladder operators of the dynamical algebra are obtained from infeld hull miller factorisations the method is used to study the first poschl teller equation even in the non symmetric case the energy spectrum and the exact normalised solutions are obtained in agreement with the results of non algebraic methods

motivic integration over deligne mumford stacks abstract the aim of this article is to develop the theory of motivic integration over deligne mumford stacks and to apply it to the birational geometry of deligne mumford stacks

a canonical bundle formula a higher dimensional analogue of kodaira s canonical bundle formula is obtained as applications we prove that the log canonical ring of a klt pair with 3is finitely generated and that there exists an effectively computable natural number m such that mk x induces the iitaka fibering for every algebraic threefold x with kodaira dimension 1

on special representations of p adic reductive groups let f be a non archimedean locally compact field let g be a split connected reductive group over f for a parabolic subgroup q g and a ring l we consider the g representation on the l module c g q l

some examples of square integrable representations of semisimple p adic groups we construct irreducible representations of the hecke algebra of an affine weyl group analogous to kilmoyer s reflection representation corresponding to finite weyl groups and we show that in many cases they correspond to a square integrable representation of a simple p adic group

on polynomial solutions of the heun equation by making use of a recently developed method to solve linear differential equations of arbitrary order we find a wide class of polynomial solutions to the heun equation we construct the series solution to the heun equation before identifying the polynomial solutions the heun equation extended by the addition of a term x is also amenable for polynomial solutions

pro p iwahori hecke ring and supersingular representations the motivation of this paper is the search for a langlands correspondence modulo p we show that the pro p iwahori hecke ring open image in new window of a split reductive p adic group g over a local field f of finite residue field fq with q elements admits an iwahori matsumoto presentation and a bernstein z basis and we determine its centre we prove that the ring open image in new window is finitely generated as a module over its centre these results are proved in 11 only for the iwahori hecke ring let p be the prime number dividing q and let k be an algebraically closed field of characteristic p a character from the centre of open image in new window to k which is as null as possible will be called null the simple open image in new window modules with a null central character are called supersingular when g gl n we show that each simple open image in new window module of dimension n containing a character of the affine subring open image in new window is supersingular using the minimal expressions of haines generalized to open image in new window and that the number of such modules is equal to the number of irreducible k representations of the weil group wf of dimension n when the action of an uniformizer pf in the hecke algebra side and of the determinant of a frobenius frf in the galois side are fixed i e the number nn q of unitary irreducible polynomials in fq x of degree n one knows that the converse is true by explicit computations when n 2 10 and when n 3 rachel ollivier

the weierstrass approximation theorem for maps between real algebraic varieties p n r for some n endowed with the zariski topology and the sheaf of rvalued regular functions morphisms between real algebraic varieties will be called regular maps every real algebraic variety carries also the euclidean topology that is the topology induced by the usual metric topology on r unless explicitly stated otherwise all topological notions related to real algebraic varieties will refer to the euclidean topology given two real algebraic varieties x and y we regard the set r x y of all regular maps from x into y as a subset of the space c x y of all continuous maps fromx into y endowed with the compact open topology if x and y are nonsingular then we also regard r x y as a subset of the space c x y of all c maps from x into y endowed with the c compact open topology the weak c topology in the terminology used in 6