on arithmetic fundamental lemmas we present a relative trace formula approach to the gross zagier formula and its generalization to higher dimensional unitary shimura varieties as a crucial ingredient we formulate a conjectural arithmetic fundamental lemma for unitary rapoport zink spaces we prove the conjecture when the rapoport zink space is associated to a unitary group in two or three variables

regular formal moduli spaces and arithmetic transfer conjectures we define various formal moduli spaces of p divisible groups which are regular and morphisms between them we formulate arithmetic transfer conjectures which are variants of the arithmetic fundamental lemma conjecture of the third author in the presence of ramification these conjectures include the at conjecture of our previous joint work we prove these conjectures in low dimensional cases

the fundamental lemma of jacquet and rallis we prove both the group version and the lie algebra version of the fundamental lemma appearing in a relative trace formula of jacquet and rallis in the function field case when the characteristic is greater than the rank of the relevant groups in the appendix by gordon our results are transferred to the p adic field case for sufficiently large p

constructible exponential functions motivic fourier transform and transfer principle we introduce spaces of exponential constructible functions in the motivic setting for which we construct direct image functors in the absolute and relative settings this allows us to define a motivic fourier transformation for which we get various inversion statements we also define spaces of motivic schwartz bruhat functions on which motivic fourier transformation induces isomorphisms our motivic integrals specialize to nonarchimedean integrals we give a general transfer principle comparing identities between functions defined by exponential integrals over local fields of characteristic zero resp of positive characteristic having the same residue field we also prove new results about p adic integrals of exponential functions and stability of this class of functions under p adic integration

integrability of oscillatory functions on local fields transfer principles for oscillatory functions on local fields coming from motivic exponential functions we show that integrability over q n p implies integrability over f p t n for large p and vice versa more generally the integrability only depends on the isomorphism class of the residue field of the local field once the characteristic of the residue field is large enough this principle yields general local integrability results for harish chandra characters in positive characteristic as we show in other work transfer principles for related conditions such as boundedness and local integrability are also obtained the proofs rely on a thorough study of loci of integrability to which we give a geometric meaning by relating them to zero loci of functions of a specific kind

fibration de hitchin et endoscopie we propose a geometric interpretation of the theory of elliptic endoscopy due to langlands and kottwitz in terms of the hitchin fibration as applications we prove a global analog of a purity conjecture due to goresky kottwitz and macpherson for unitary groups this global purity statement has been used in a joint work with g laumon to prove the fundamental lemma over a local fields of equal characteristics

directional homogenization of elliptic equations in non divergence form abstract the paper concerns regularities of solutions of elliptic equations in non divergence form with directional homogenization the main interest is to investigate the differences of regularities of datums that are needed and of solutions that are obtained between directions with homogenization and without homogenization we establish uniform interior l p 1 p estimates for d 2 u e and uniform interior c 0 1 estimates for d d x u e where x are the non homogenization directions our results are optimal

w1 p estimates for elliptic problems with neumann boundary conditions in lipschitz domains abstract we study w 1 p estimates in lipschitz domains for second order elliptic equations and systems of divergence form with real valued bounded measurable coefficients for any fixed p 2 we prove that a weak reverse holder inequality implies the w 1 p estimates for solutions with neumann boundary conditions as a result we are able to show that if the coefficient matrix of elliptic equation is symmetric and in vmo r n the w 1 p estimate holds for 3 2 e p 3 e if n 3 and for 4 3 e p 4 e if n 2 and for elliptic systems the w 1 p estimate is true for 2 n n 1 e p 2 n n 1 e

the gan gross prasad conjecture for u n u n abstract we prove the gan gross prasad conjecture for u n u n under some local conditions using a relative trace formula we deduce some new cases of the gan gross prasad conjecture for u n 1 u n from the case of u n u n

convergence rates in l 2 for elliptic homogenization problems we study rates of convergence of solutions in l2 and h1 2 for a family of elliptic systems mathcal l varepsilon with rapidly oscillating coefficients in lipschitz domains with dirichlet or neumann boundary conditions as a consequence we obtain convergence rates for dirichlet neumann and steklov eigenvalues of mathcal l varepsilon most of our results which rely on the recently established uniform estimates for the l2 dirichlet and neumann problems in kenig and shen math ann 350 867 917 2011 commun pure appl math 64 1 44 2011 are new even for smooth domains

uniform w 1 p estimates for systems of linear elasticity in a periodic medium abstract let l e be a family of elliptic systems of linear elasticity with rapidly oscillating periodic coefficients we obtain the uniform w 1 p estimate u e p c f p in a lipschitz domain in r n for solutions to the dirichlet problem l e u e div f in and u e 0 on where 1 p 1 2 1 2 n and c 0 are constants independent of e 0 the ranges are sharp for n 2 or 3

gradient estimates for parabolic and elliptic systems from linear laminates we establish several gradient estimates for second order divergence type parabolic and elliptic systems the coefficients and data are assumed to be holder or dini continuous in the time variable and all but one spatial variable this type of system arises from the problems of linearly elastic laminates and composite materials for the proof we use campanato s approach in a novel way non divergence type equations under similar conditions are also discussed

homogenization of elliptic equations with principal part not in divergence form and hamiltonian with quadratic growth in this paper we consider the following problem here the coefficients aij and bi are smooth periodic with respect to the second variable and the matrix aij ij is uniformly elliptic the hamiltonian h is locally lipschitz continuous with respect to u and du and has quadratic growth with respect to du the hamilton jacobi beliman equations of some stochastic control problems are of this type our aim is to pass to the limit in 0 as tends to zero we assume the coefficients bi to be centered with respect to the invariant measure of the problem see the main assumption 3 13 then we derive l h and w p0 2 estimates for the solutions of 0 we also prove the following corrector s result this allows us to pass to the limit in 0 and to obtain this problem is of the same type as the initial one when 0 is the hamilton jacobi bellman equation of a stochastic control problem then 00 is also a hamilton jacobi bellman equation but one corresponding to a modified set of controls

le lemma fondamental pour les groupes unitaires let g be an unramified reductive group over a nonarchimedian local field f the so called langlands fundamental lemma is a family of conjectural identities between orbital integrals for g f and orbital integrals for endoscopic groups of g in this paper we prove the langlands fundamental lemma in the particular case where f is a finite extension of fp t g is a unitary group and p rank g waldspurger has shown that this particular case implies the langlands fundamental lemma for unitary groups of rank p when f is any finite extension of qp we follow in part a strategy initiated by goresky kottwitz and macpherson our main new tool is a deformation of orbital integrals which is constructed with the help of the hitchin fibration for unitary groups over projective curves

homology of affine springer fibers in the unramified case assuming a certain purity conjecture we derive a formula for the complex cohomology groups of the affine springer fiber corresponding to any unramified regular semisimple element we use this calculation to present a complex analog of the fundamental lemma for function fields we show that the kappa orbital integral that arises in the fundamental lemma is equal to the lefschetz trace of the frobenius acting on the etale cohomology of a related variety

the lp boundary value problems on lipschitz domains abstract let be a bounded lipschitz domain in r n we develop a new approach to the invertibility on l p of the layer potentials associated with elliptic equations and systems in as a consequence for n 4 and 2 n 1 n 1 e p 2 where e 0 depends on we obtain the solvability of the l p neumann type boundary value problems for second order elliptic systems the analogous results for the biharmonic equation are also established

on w1 p estimates for elliptic equations in divergence form first we will apply the method to study w 1 p regularity for a nonlinear elliptic operator in divergence form we would like to point out that in the particular case of a linear elliptic equation this method gives an alternative proof to the classical one which uses the general theory of singular integrals the utility of the method that we describe below is that hypotheses a and b are obtained directly by studying the deviation of the coefficients of a from the coefficients of a0 and this is usually not a difficult task a more interesting application of this kind of approximation method is to elliptic homogenization problems for which we obtain results that give w 1 p estimates with a weak hypothesis of regularity in the coefficients for instance we are able to study the following cases

lipschitz estimates in almost periodic homogenization we establish uniform lipschitz estimates for second order elliptic systems in divergence form with rapidly oscillating almost periodic coefficients we give interior estimates as well as estimates up to the boundary in bounded c1 domains with either dirichlet or neumann data the main results extend those in the periodic setting due to avellaneda and lin for interior and dirichlet boundary estimates and later kenig lin and shen for the neumann boundary conditions in contrast to these papers our arguments are constructive and thus the constants are in principle computable and the results for the neumann conditions are new even in the periodic setting since we can treat nonsymmetric coefficients we also obtain uniform w1 p estimates 2015 wiley periodicals inc

constructible motivic functions and motivic integration 1 1 in this paper intended to be the first in a series we lay new general foundations for motivic integration and give answers to some important issues in the subject since its creation by maxim kontsevich 23 motivic integration developed quickly and has spread out in many directions in a nutshell in motivic integration numbers are replaced by geometric objects like virtual varieties or motives but classicaly not only numbers are defined using integrals but also interesting classes of functions the previous constructions of motivic integration were all quite geometric and it was quite unclear how they could be generalized to handle integrals depending on parameters the new approach we present here based on cell decomposition allows us to develop a very general theory of motivic integration taking parameters into account more precisely we define a natural class of functions constructible motivic functions which is stable under integration the basic idea underlying our approach is to construct more generally push forward morphisms f which are functorial they satisfy f g f g so that performing motivic integration corresponds to taking the push forward to the point this strategy has many technical advantages in essence it allows to reduce the construction of f to the case of closed immersions and projections and in the latter case we can perform induction on the relative dimension the basic case being that of relative dimension 1

orientability of moduli spaces of spin 7 instantons and coherent sheaves on calabi yau 4 folds abstract suppose x g is a compact spin 7 manifold e g a riemannian 8 manifold with holonomy spin 7 or a calabi yau 4 fold let g be u m or su m and p x be a principal g bundle we show that the infinite dimensional moduli space b p of all connections on p modulo gauge is orientable in a certain sense we deduce that the moduli space m p spin 7 b p of irreducible spin 7 instanton connections on p modulo gauge as a manifold or derived manifold is orientable this improves theorems of cao and leung 9 and munoz and shahbazi 42 if x is a calabi yau 4 fold the derived moduli stack m of complexes of coherent sheaves on x is a 2 shifted symplectic derived stack m by pantev toen vaquie vezzosi 46 and so has a notion of orientation by borisov joyce 7 we prove that m is orientable by relating algebro geometric orientations on m to differential geometric orientations on b p for u m bundles p x and using orientability of b p this has applications to defining donaldson thomas type invariants counting semistable coherent sheaves on a calabi yau 4 fold as in donaldson and thomas 15 cao and leung 8 and borisov and joyce 7

intersection homology ii in 19 20 we introduced topological invariants ih x called intersection homology groups for the study of singular spaces x these groups depend on the choice of a perversity p a perversity is a function from 2 3 to the non negative integers such that both c and c 2 c are positive and increasing functions of c 2 1 the group ihr is defined for spaces x called pseudomanifolds a pseudomanifold of dimension n is a space that admits a stratification x x n x n z x n 3 x 1 x 0

orientability for gauge theories on calabi yau manifolds abstract we study orientability issues of moduli spaces from gauge theories on calabi yau manifolds our results generalize and strengthen those for donaldson thomas theory on calabi yau manifolds of dimensions 3 and 4 we also prove a corresponding result in the relative situation which is relevant to the gluing formula in dt theory

the orientation of yang mills moduli spaces and 4 manifold topology si x est une 4 variete lisse orientee fermee dont la forme intersection q h 2 x z torsion z est definie negative alors la forme est equivalente sur les entiers a la forme standard 1 1 1 soit x une 4 variete lisse orientee fermee avec les trois proprietes suivantes 1 h 1 x z n a pas de 2 torsion 2 la forme intersection q sur h 2 x torsion a une partie positive de rang 1 ou 2 3 la forme intersection est paire alors q est equivalente sur les entiers a une des formes 01 10 01 10 01 10

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

fourier transform and the global gan gross prasad conjecture for unitary groups by the relative trace formula approach of jacquet rallis we prove the global gan gross prasad conjecture for unitary groups under some local restrictions for the automorphic representations

compactness methods in the theory of homogenization ii equations in non divergence form we prove c0 c1 and c1 1 a priori estimates for solutions of boundary value problems for elliptic operators with periodic coefficients of the form j 1ai j x 2 xi xj the constants in these estimates are independent of the small parameter and hence our results imply strengthened versions of the classical averaging theorems these results generalize to a wide class of linear operators in non divergence form including equations with lower order terms and nonuniformly oscillating coefficients as well as to certain nonlinear problems which we discuss in the last section

the su 3 casson invariant for integral homology 3 spheres we derive a gauge theoretic invariant of integral homology 3 spheres which counts gauge orbits of irreducible perturbed flat su 3 connections with sign given by spectral flow to compensate for the dependence of this sum on perturbations the invariant includes contributions from the reducible perturbed flat orbits our formula for the correction term generalizes that given by walker in his extension of casson s su 2 invariant to rational homology 3 spheres

optimal quantitative estimates in stochastic homogenization for elliptic equations in nondivergence form we prove quantitative estimates for the stochastic homogenization of linear uniformly elliptic equations in nondivergence form under strong independence assumptions on the coefficients we obtain optimal estimates on the subquadratic growth of the correctors with stretched exponential type bounds in probability like the theory of gloria and otto ann probab 39 3 779 856 2011 ann appl probab 22 1 1 28 2012 for divergence form equations the arguments rely on nonlinear concentration inequalities combined with certain estimates on the green s functions and derivative bounds on the correctors we obtain these analytic estimates by developing a c1 1 regularity theory down to microscopic scale which is of independent interest and is inspired by the c0 1 theory introduced in the divergence form case by the first author and smart ann sci ec norm super 4 49 2 423 481 2016

smoothness of linear laminates linearly elastic laminates are examples of materials whose equilibrium equations may have only bounded measurable coefficients yet whose solutions may be fairly smooth this is quite different from general experience where regularity of solutions is determined by the closeness of the system to a diagonal one a particular situation where a laminate may appear is a highly twinned elastic or ferroelectric crystal and there are questions related to these materials which make it useful to know some properties of these special systems of equations

an application of gauge theory to four dimensional topology soit x une u variete compacte reguliere simplement connexe orientee avec la propriete que la forme associee q est definie positive alors cette forme est equivalente sur les entiers a la forme diagonale standard soit dans une base q u 1 u 2 u 2 u 2 1 u 2 2 u 2 2