langlands program trace formulas and their geometrization the langlands program relates galois representations and auto morphic representations of reductive algebraic groups the trace formula is a powerful tool in the study of this connection and the langlands functorial ity conjecture after giving an introduction to the langlands program and its geometric version which applies to curves over finite fields and over the complex field i give a survey of my recent joint work with robert langlands and ngo bao chau fln fn on a new approach to proving the functoriality conjecture using the trace formulas and on the geometrization of the trace formulas in particular i discuss the connection of the latter to the categorifi cation of the langlands correspondence

conservative descent for semi orthogonal decompositions abstract motivated by the local flavor of several well known semi orthogonal decompositions in algebraic geometry we introduce a technique called conservative descent which shows that it is enough to establish these decompositions locally the decompositions we have in mind are those for projectivized vector bundles and blow ups due to orlov and root stacks due to ishii and ueda our technique simplifies the proofs of these decompositions and establishes them in greater generality for arbitrary algebraic stacks

on a vanishing conjecture appearing in the geometric langlands correspondence 0 1 this paper should be regarded as a sequel to 7 there it was shown that the geometric langlands conjecture for gln follows from a certain vanishing conjecture the goal of the present paper is to prove this vanishing conjecture let x be a smooth projective curve over a ground field k let e be an m dimensional local system on x and let bunm be the moduli stack of rank m vector bundles on x the geometric langlands conjecture says that to e we can associate a perverse sheaf fe on bunm which is a hecke eigensheaf with respect to e the vanishing conjecture of 7 says that for all integers n m a certain functor ave depending on e and a parameter d z which maps the category d bunn to itself vanishes identically when d is large enough the fact that the vanishing conjecture implies the geometric langlands conjecture may be regarded as a geometric version of the converse theorem moreover as will be explained in the sequel the vanishing of the functor ave is analogous to the condition that the rankin selberg convolution of e viewed as an m dimensional galois representation and an automorphic form on gln with n m is well behaved both the geometric langlands conjecture and the vanishing conjecture can be formulated in any of the sheaf theoretic situations e g q adic sheaves when char k d modules when char k 0 and sheaves with coefficients in a finite field f again when char k when the ground field is the finite field fq and we are working with adic coefficients it was shown in 7 that the vanishing conjecture can be deduced from lafforgue s theorem that establishes the full langlands correspondence for global fields of positive characteristic cf 9

borcherds products and arithmetic intersection theory on hilbert modular surfaces we prove an arithmetic version of a theorem of hirzebruch and zagier saying that hirzebruch zagier divisors on a hilbert modular surface are the coecients of an elliptic modular form of weight two moreover we determine the arithmetic selfintersection number of the line bundle of modular forms equipped with its petersson metric on a regular model of a hilbert modular surface and study faltings heights of arithmetic hirzebruch zagier divisors

general constraint qualifications in nondifferentiable programming we show that a familiar constraint qualification of differentiable programming has nonsmooth counterparts as a result necessary optimality conditions of kuhn tucker type can be established for inequality constrained mathematical programs involving functions not assumed to be differentiable convex or locally lipschitzian these optimality conditions reduce to the usual karush kuhn tucker conditions in the differentiable case and sharpen previous results in the locally lipschitzian case

modular forms and special cubic fourfolds abstract we study the degrees of special cubic divisors on moduli space of cubic fourfolds with at worst ade singularities in this paper we show that the generating series of the degrees of such divisors is a level three modular form

cores of tangent cones and clarke s tangent cone it is known that clarke s tangent cone at any point of any subset of rn is always both unique and convex by contrast nearly all other notions of convex tangent cone in the literature are monotone in the sense that if a convex cone k is a tangent cone at a point x0 of a set c rn then k k c c automatically implies that k is a tangent cone of the same type for c at x0 this carries the rider that such tangent cones are not unique and in general even maximal convex tangent cones are not unique in this paper it is shown that for most monotone types one can identify preferred tangent cones for c at x0 called core tangent cones of that type such that maximal core tangent cones are unique and convex

on the geometric langlands conjecture let x be a smooth projective curve and g be a reductive group over a finite field fq the field kx of rational functions on x is what number theorists call a global field and we can do the theory of automorphic functions over it namely we consider the quotient g kx g a here a is the ring of adeles corresponding to kx and study the space of functions on it as a representation of the adele group g a it was essentially an observation of a weil that the quotient g kx g a is closely related to the set of isomorphism classes of principal g bundles on our curve x however g bundles on x possess a richer structure one can view the above set as a set of fq points of an algebraic variety or rather a stack denoted bung moreover instead of studying the vector space of functions on the discrete set of points of bung we will consider the category of adic sheaves on it basically what people call the geometric langlands program is an attempt to understand a spectral decomposition of the above category under the action of the so called hecke functors the latter will be defined in the talk the answer predicted by the geometric langlands conjecture links this spectral decomposition to the moduli stack of local systems on x with respect to the langlands dual group

heights of kudla rapoport divisors and derivatives of l functions we study special cycles on integral models of shimura varieties associated with unitary similitude groups of signature n 1 1 we construct an arithmetic theta lift from harmonic maass forms of weight 2 n to the arithmetic chow group of the integral model of a unitary shimura variety by associating to a harmonic maass form f a linear combination of kudla rapoport divisors equipped with the green function given by the regularized theta lift of f our main result is an equality of two complex numbers 1 the height pairing of the arithmetic theta lift of f against a cm cycle and 2 the central derivative of the convolution l function of a weight n cusp form depending on f and the theta function of a positive definite hermitian lattice of rank n 1 when specialized to the case n 2 this result can be viewed as a variant of the gross zagier formula for shimura curves associated to unitary groups of signature 1 1 the proof relies on among other things a new method for computing improper arithmetic intersections

spans of special cycles of codimension less than 5 we show that the span of special cycles in the r th chow group of a shimura variety of orthogonal type is finite dimensional if r 5 as our main tool we develop the theory of jacobi forms with rational index m matn q

superconnections theta series and period domains abstract we use superconnections to define and study some natural differential forms on the period domain d that parametrizes polarized hodge structures of given type on a rational quadratic vector space v these forms depend on a choice of vectors v 1 v r v and have a gaussian shape that peaks on the locus where v 1 v r become hodge classes we show that they can be rescaled so that one can form theta series by summing over a lattice l r v r these series define differential forms on arithmetic quotients d we compute their cohomology class explicitly in terms of the cohomology classes of hodge loci in d when the period domain is a hermitian symmetric domain of type iv we show that the components of our forms of appropriate degree recover the forms introduced by kudla and millson in particular our results provide another way to establish the main properties of these forms

convex subcones of the contingent cone in nonsmooth calculus and optimization the tangential approximants most useful in nonsmooth analysis and optimization are those which lie between the clarke tangent cone and the bouligand contigent cone a study of this class of tangent cones is undertaken here it is shown that although no convex subcone of the contingent cone has the isotonicity property of the contingent cone there are such convex subcones which are more accurate approximants than the clarke tangent cone and possess an associated subdifferential calculus that is equally strong in addition a large class of convex subcones of the contingent cone can replace the clarke tangent cone in necessary optimality conditions for a nonsmooth mathematical program however the clarke tangent cone plays an essential role in the hypotheses under which these calculus rules and optimality conditions are proven overall the results obtained here suggest that the most complete theory of nonsmooth analysis combines a number of different tangent cones

lefschetz properties for noncompact arithmetic ball quotients we prove a lefschetz property for the restriction of the cohomology of noncompact arithmetic ball quotients to ball quotients of smaller dimension

endoscopy theta liftings and period integrals for the unitary group in three variables l packets for the quasi split unitary group in three variables u 3 we shall give an explicit parametrization of these l packets using theta liftings and describe some relations between the structure of l packets and certain period integrals we also prove that an endoscopic l packet contains a unique generic representation in local and global cases suppose that g is a reductive algebraic group over a local or global field

recent advances in the langlands program these are the notes for the lecture given by the author at the mathematical current events special session of the ams meeting in baltimore on january 17 2003 topics reviewed include the langlands correspondence for gl n in the function field case and its proof by v drinfeld and l lafforgue the geometric langlands correspondence for gl n and its proof by d gaitsgory k vilonen and the author and the work of a beilinson and v drinfeld on the quantization of the hitchin system and the langlands correspondence for an arbitrary semisimple algebraic group

indecomposable vector bundles and stable higgs bundles over smooth projective curves we prove that the number of geometrically indecomposable vector bundles of xed rank r and degree d over a smooth projective curve x dened over a nite eld is given by a polynomial depending only on r d and the genus g of x in the weil numbers of x we provide a closed formula expressed in terms of generating series for this polynomial we also show that the same polynomial computes the number of points of the moduli space of stable higgs bundles of rank r and degree d over x this entails a closed formula for the poincar e polynomial of the moduli spaces of stable higgs bundles over a compact riemann surface and hence also for the poincar e polynomials of the twisted character varieties for the groups gl r

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

a multiplication in cyclic homology we define a multiplication on the cyclic homology of a commutative cocommutative bialgebra h with superproduct in the case when h is a field of characteristic zero the cyclic homology becomes a polynomial algebra in one generator the loday quillen multiplication is trivial in that case in this paper we define an associative multiplication on the cyclic homology of certain commutative algebras h the definition is based on the cartesian product of sets instead of the shuffle product as in the case of lq and ga when h contains q our product is compatible with the hodge decomposition of hc h so it has the form hc n h hc j m h hc n m h the existence of this multiplication in the case h k is classical since it is just given by the tensor product of complex line bundle however its significance came to our attention in joint work with john klein as a consequence of our study of fiberwise morse theory on circle bundles ik and in other joint works with r penner and r hain by doing morse theory on circles john klein and i realized that there was a rationally nontrivial associative multiplication on the cyclic homology of the integral group ring on any cyclic group and that it had important consequences for pseudoisotopy theory this will be explained in a later paper in this paper we show that this multiplication in cyclic homotopy is defined first in the case when the algebra is h km k m where m is the underlying additive group of a ring then in the case when h is a hopf algebra with superproduct in the sense of gaucher ga and finally in the more general case of any algebra example 12 4 in fact we do not need additive inverses in m nor do we need an antipode for h so we can take m to be a monoid and h to be a bialgebra when h k i e m 1 the cyclic homology becomes a divided power algebra which is of course isomorphic to a polynomial algebra in one generator if k is a field of characteristic zero we assume throughout the paper that k is a commutative ring with 1 and that all tensor products are taken over k this includes those tensor products which appear in the definition of cyclic homology we also use connes original definition that a cyclic object in a category c is a covariant functor c where is connes cyclic category since this is the same as a contravariant functor c an outline of the paper is as follows in the first section we give a topological definition of the multiplication in hc km the construction is so elementary that received by the editors april 14 1994 1991 mathematics subject classification primary 18g60 secondary 16w30 this research was supported by nsf grant no dms 90 02512 c 1999 american mathematical society

on the converse theorem for borcherds products abstract we prove a new converse theorem for borcherds multiplicative theta lift which improves the previously known results to this end we develop a newform theory for vector valued modular forms for the weil representation which might be of independent interest we also derive lower bounds for the ranks of the picard groups and the spaces of holomorphic top degree differential forms of modular varieties associated to orthogonal groups

green forms and the arithmetic siegel weil formula we construct natural green forms for special cycles in orthogonal and unitary shimura varieties in all codimensions and for compact shimura varieties of type mathrm o p 2 and mathrm u p 1 we show that the resulting local archimedean height pairings are related to special values of derivatives of siegel eisentein series a conjecture put forward by kudla relates these derivatives to arithmetic intersections of special cycles and our results settle the part of his conjecture involving local archimedean heights

cm values of regularized theta lifts and harmonic weak maas forms of weight 1 we study special values of regularized theta lifts at complex multiplication cm points in particular we show that cm values of borcherds products can be expressed in terms of finitely many fourier coefficients of certain harmonic weak maas forms of weight 1 as it turns out these coefficients are logarithms of algebraic integers whose prime ideal factorization is determined by special cycles on an arithmetic curve our results imply a conjecture of duke and li and give a new proof of the modularity of a certain arithmetic generating series of weight 1 studied by kudla rapoport and yang

the hodge conjecture and arithmetic quotients of complex balls let s be a closed shimura variety uniformized by the complex n ball associated with a standard unitary group the hodge conjecture predicts that every hodge class in h 2k s mathbb q k 0 dots n is algebraic we show that this holds for all degrees k away from the neighborhood bigl tfrac13n tfrac23n bigr of the middle degree we also prove the tate conjecture for the same degrees as the hodge conjecture and the generalized form of the hodge conjecture in degrees away from an interval depending on the codimension c of the subvariety centered at the middle dimension of s these results are derived from a general theorem that applies to all shimura varieties associated with standard unitary groups of any signature the proofs make use of arthur s endoscopic classification of automorphic representations of classical groups as such our results rely on the stabilization of the trace formula for the disconnected groups gl n rtimes langle theta rangle associated with base change

faltings heights of cm cycles and derivatives of l functions we study the faltings height pairing of arithmetic special divisors and cm cycles on shimura varieties associated to orthogonal groups we compute the archimedean contribution to the height pairing and derive a conjecture relating the total pairing to the central derivative of a rankin l function we prove the conjecture in certain cases where the shimura variety has dimension 0 1 or 2 in particular we obtain a new proof of the gross zagier formula

regularized theta lifts for orthogonal groups over totally real fields we define a regularized theta lift from sl2 to orthogonal groups over totally real fields it takes harmonic whittaker forms to automorphic green functions and weakly holomorphic whittaker forms to meromorphic modular forms on orthogonal groups with zeros and poles supported on special divisors generalizing borcherds work on automorphic products to prove our results we use the spectral expansion of the lift and study its relationship with the cohomological theta lift of kudla and millson

hall algebras of curves commuting varieties and langlands duality we construct an isomorphism between the universal spherical hall algebra of a smooth projective curve of genus g and a convolution algebra in the equivariant k theory of the genus g commuting varieties c mathfrak gl r big x i y i in mathfrak gl r 2g sum i 1 g x i y i 0 big we can view this isomorphism as a version of the geometric langlands duality in the formal neighborhood of the trivial local system for the group glr we extend this to all reductive groups and we compute the image under our correspondence of the skyscraper sheaf supported on the trivial local system

tautological classes on moduli spaces of hyper kahler manifolds we study algebraic cycles on moduli spaces f h of h polarized hyperkahler manifolds following previous work of marian oprea and pandharipande on the tautological conjecture on moduli spaces of k3 surfaces we first define the tautological ring on f h we then study the images of these tautological classes in the cohomology groups of f h and prove that most of them are linear combinations of noether lefschetz cycle classes in particular we prove the cohomological version of the tautological conjecture on moduli space of k3 n type hyperkahler man ifolds with n 2 secondly we prove the cohomological generalized franchetta conjecture on universal family of these hyperkahler mani folds

the noether lefschetz conjecture and generalizations we prove the noether lefschetz conjecture on the moduli space of quasi polarized k3 surfaces this is deduced as a particular case of a general theorem that states that low degree cohomology classes of arithmetic manifolds of orthogonal type are dual to the classes of special cycles i e sub arithmetic manifolds of the same type for compact manifolds this was proved in 3 here we extend the results of 3 to non compact manifolds this allows us to apply our results to the moduli spaces of quasi polarized k3 surfaces

a physical origin for singular support conditions in geometric langlands theory we explain how the nilpotent singular support condition introduced into the geometric langlands conjecture by arinkin and gaitsgory arises naturally from the point of view of 4 dimensional mathcal n 4 supersymmetric gauge theory we define what it means in topological quantum field theory to restrict a category of boundary conditions to the full subcategory of objects compatible with a fixed choice of vacuum both in functorial field theory and in the language of factorization algebras for b twisted mathcal n 4 gauge theory with gauge group g the moduli space of vacua is equivalent to mathfrak h w and the nilpotent singular support condition arises by restricting to the vacuum 0 in mathfrak h w we then investigate the categories obtained by restricting to points in larger strata and conjecture that these categories are equivalent to the geometric langlands categories with gauge symmetry broken to a levi subgroup and furthermore that by assembling such for the groups mathrm gl n with n ge 1 one finds a hidden factorization algebra structure for the geometric langlands theory