virtual classes and virtual motives of quot schemes on threefolds abstract for a simple rigid vector bundle f on a calabi yau 3 fold y we construct a symmetric obstruction theory on the quot scheme quot y f n and we solve the associated enumerative theory we discuss the case of other 3 folds exploiting the critical structure on the local model quot a 3 o r n we construct a virtual motive in the sense of behrend bryan szendr i for quot y f n for an arbitrary vector bundle f on a smooth 3 fold y we compute the associated motivic partition function we obtain new examples of higher rank motivic donaldson thomas invariants

symmetric operations in algebraic cobordism abstract in this article we describe certain new cohomological operations in algebraic cobordisms these operations give the natural obstructions for the cobordism class to be represented by the embedding also they permit to work with algebraic cobordisms and chow groups in a more subtle way than the landweber novikov operations related to 2 torsion effects we describe applications to the computation of the algebraic cobordisms of a pfister quadrics and to the problem of rationality of cycles

applications of atiyah hirzebruch spectral sequences for motivic cobordism in this paper we study applications of atiyah hirzebruch spectral sequences for motivic cobordism recently found by hopkins and morel for example we compute the mod 2 motivic cohomology of a reduced cech complex of a splitting variety according to orlov vishik and voevodsky the cobordism ring mgl 2 spec mathbb r is computed and compared with the results of the real cobordism theory of hu and kriz moreover we study algebraic cobordism of the classifying spaces bg for algebraic groups over mathbb c for example the chow ring ch bg 2 2 for the exceptional lie group g 2 is determined by using motivic cobordism and motivic cohomology

examples of non algebraic classes in the brown peterson tower for every n ge 0 we construct classes in the brown peterson cohomology bp langle n rangle of smooth projective complex algebraic varieties which are not in the image of the cycle map from the corresponding motivic brown peterson cohomology this generalizes the examples of atiyah and hirzebruch to all finite levels in the brown peterson tower

the nil hecke ring and cohomology of g p for a kac moody group g abstract let g be the group with borel subgroup b associated to a kac moody lie algebra unk with weyl group w and cartan subalgebra unk then h g b has among others four distinguished structures i an algebra structure ii a distinguished basis given by the schubert cells iii a module for w and iv a module for hecke type operators aw for w unk w we construct a ring r which we refer to as the nil hecke ring which is very simply and explicitly defined as a functor of w together with the w module unk alone and such that all these four structures on h g b arise naturally from the ring r

the dt pt correspondence for smooth curves we show a version of the dt pt correspondence relating local curve counting invariants encoding the contribution of a fixed smooth curve in a calabi yau threefold we exploit a local study of the hilbert chow morphism about the cycle of a smooth curve we compute via quot schemes the global donaldson thomas theory of a general abel jacobi curve of genus 3

motivic bivariant characteristic classes let k0 v x k0 v x be the relative grothendieck group of varieties over x obj v x obj v with v vk qp resp v vcan the category of quasi projective algebraic resp compact complex analytic varieties over a base field k then we constructed the motivic hirzebruch class transformation ty k0 v x h x q y ty k0 v x h x q y in the algebraic context for k of characteristic zero with h x ch x h x ch x resp in the complex algebraic or analytic context with h x h2 bm x it unifies the well known three characteristic class transformations of singular varieties macpherson s chern class baum fulton macpherson s todd class and the l class of goresky macpherson and cappell shaneson in this paper we construct a bivariant relative grothendieck group k0 v x y k0 v x y for v vk qp resp vcan so that k0 v x pt k0 v x k0 v x pt k0 v x in the algebraic context with k of characteristic zero resp complex analytic context we also construct in the algebraic context in any characteristic two grothendieck transformations mcy ymot k0 vqp x y kalg x y z y and ty k0 vqp x y h x y q y ty k0 vqp x y h x y q y with kalg f kalg f the bivariant algebraic k theory of f perfect complexes and hh the bivariant operational chow groups or the even degree bivariant homology in the case k ck c evaluating at y 0y 0 we get a motivic lift t0t0 of fulton macpherson s bivariant riemann roch transformation kalg h q kalg h q the covariant transformations mcy k0 vqp x pt g0 x z y mcy k0 vqp x pt g0 x z y and ty k0 vqp x pt h x q y ty k0 vqp x pt h x q y agree for k of characteristic zero with our motivic chern and hirzebruch class transformations defined on k0 vqp x k0 vqp x finally evaluating at y 1y 1 for k of characteristic zero we get a motivic lift t 1t 1 of ernstrom yokura s bivariant chern class transformation f ch

schubert calculus for algebraic cobordism we establish a schubert calculus for bott samelson resolutions in the algebraic cobordism ring of a complete flag variety g b extending the results of bressler evens 4 to the algebro geometric setting

frequency domain synthesis of hereditary control systems via anisochronic state space a method of control feedback design for systems with hereditary properties is presented which is based on the concept of anisochronic state space a new kind of state feedback control is proposed using as state variables the system accumulation attributes only while the other delay dynamics are represented by functional terms assigning the control parameters is approached as a placement of characteristic function zeros however owing to the infinite set of these zeros in the hereditary system only the dominant zeros are placed for this purpose a new technique applying the conformal mapping of transcendental characteristic function of the hereditary system has been developed

permutation representations on schubert varieties this paper defines and studies permutation representations on the equivariant cohomology of schubert varieties as representations both over bbb c and over bbb c t 1 t 2 ldots t n we show these group actions are the same as an action of simple transpositions studied geometrically by m brion and give topological meaning to the divided difference operators of berstein gelfand gelfand demazure kostant kumar and others we analyze these representations using the combinatorial approach to equivariant cohomology introduced by goresky kottwitz macpherson we find that each permutation representation on equivariant cohomology produces a representation on ordinary cohomology that is trivial though the equivariant representation is not

lefschetz theorems for torsion algebraic cycles in codimension 2 abstract let y be a smooth projective variety over c and x be a smooth hypersurface in y we prove that the natural restriction map on chow groups of codimension two cycles is an isomorphism when restricted to the torsion subgroups provided dim y 5 we prove an analogous statement for a very general hypersurface x p 4 of degree 5 in the more general setting of a very general hypersurface x of sufficiently high degree in a fixed smooth projective four fold y under some additional hypothesis we prove that the restriction map is an isomorphism on l primary torsion for almost all primes l as a consequence we obtain a weak lefschetz theorem for torsion in the griffiths groups of codimension 2 cycles and prove the injectivity of the abel jacobi map when restricted to torsion in this griffiths group thereby providing a partial answer to a question of nori

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

torsion cohomology classes and algebraic cycles on complex projective manifolds abstract atiyah and hirzebruch gave examples of even degree torsion classes in the singular cohomology of a smooth complex projective manifold which are not poincare dual to an algebraic cycle we notice that the order of these classes must be small compared to the dimension of the manifold however building upon a construction of kollar one can provide such examples with arbitrary high prime order the dimension being fixed this method also provides examples of torsion algebraic cycles which are non trivial in the griffiths group and lie in a arbitrary high level of the h saito filtration on chow groups

some backward uniqueness results soit 7b u un operateur non borne lineaire ou non lineaire sur un espace de hilbert h tel que 7b u 0 0 soit une fonction solution de l equation d evolution d dt 7b u 0 et telle que t 0 pour un t 0 on etudie la propriete d unicite retrograde sur 0 t pour des inegalites d t dt a t t h n t t d a 1 2 t ou a t t 0 est une famille d operateurs lineaires non bornes autoadjoint sur h et u l 2 0 t

drinfeld modules and torsion in the chow groups of certain threefolds let e b be an elliptic surface defined over the algebraic closure of a finite field of characteristic greater than 5 let w be a resolution of singularities of e b e we show that the l adic abel jacobi map from the l power torsion in the second chow group of w to h 3 w zl 2 ql zl is an isomorphism for almost all primes l a main tool in the proof is the assertion that certain cm cycles in fibres of w b are torsion which is proven using results from the theory of drinfeld modular curves

the chow ring of the classifying space bso 2n c we compute the chow ring of the classifying space bso 2n c in the sense of totaro using the fibration gl 2n so 2n bso 2n bgl 2n and a computation of the chow ring of gl 2n so 2n in a previous paper we find this chow ring is generated by chern classes and a characteristic class defined by edidin and graham which maps to 2n 1 times the euler class under the usual class map from the chow ring to ordinary cohomology moreover we show this class represents 1 2n 1 n 1 times the nth chern class of the representation of so 2n whose highest weight vector is twice that of the half spin representation

k theory formulas for orthogonal and symplectic orbit closures abstract the complex orthogonal and symplectic groups both act on the complete flag variety with finitely many orbits we study two families of polynomials introduced by wyser and yong representing the k theory classes of the closures of these orbits our polynomials are analogous to the grothendieck polynomials representing k classes of schubert varieties and we show that like grothendieck polynomials they are uniquely characterized among all polynomials representing the relevant classes by a certain stability property we show that the same polynomials represent the equivariant k classes of symmetric and skew symmetric analogues of knutson and miller s matrix schubert varieties we derive explicit expressions for these polynomials in special cases including a pfaffian formula relying on a more general degeneracy locus formula of anderson finally we show that taking an appropriate limit of our representatives recovers the k theoretic schur q functions of ikeda and naruse

the schubert calculus braid relations and generalized cohomology let x be the flag variety of a compact lie group and let h be a complex oriented generalized cohomology theory we introduce operators on h x which generalize operators introduced by bernstein gel fand and gel fand for rational cohomology and by demazure for k theory using the becker gottlieb transfer we give a formula for these operators which enables us to prove that they satisfy braid relations only for the two classical cases thereby giving a topological interpretation of a theorem proved by the authors and extended by gutkin one of the central issues in lie theory is the geometry of the flag variety associated to a compact lie group g an important problem concerning the flag variety is the schubert calculus which studies the ring structure of the cohomology of the flag variety work initiated by borel bott and kostant which culminated in a paper by bernstein gel fand and gel fand bgg gave a complete solution to the problem demazure studied the same problem for k theory moreover these techniques have been generalized to the kac moody situation by kac peterson kostant kumar and others this work has focussed on algebro geometric properties of the flag variety here on the other hand we study the flag variety from the point of view of algebraic topology as a consequence not only do we recover the classical results described above but we extend these results to a certain class of cohomology theories those which are complex oriented examples of complex oriented theories include ordinary cohomology k theory complex cobordism and elliptic cohomology since the context we have chosen in very general the proofs are universal and are often simpler than the classical arguments in the work of bgg a crucial role is played by operators ai associated to each simple reflection si of the weyl group of g defined by demazure in k theory these operators ai satisfy the braid relations which are the relations between pairs of simple reflections in this paper we generalize the a to give operators d acting on h g t for any complex oriented theory h we prove that braid relations are satisfied only for cohomology theories with the formal group law of rational cohomology or of k theory theorem received by the editors june 21 1988 1980 mathematics subject classification 1985 revision primary 55n20 57t15 the second author was supported by an nsf graduate fellowship 3 1990 american mathematical society 0002 9947 90 1 00 25 perpage

residue construction of hecke algebras abstract hecke algebras are usually defined algebraically via generators and relations we give a new algebra geometric construction of affine and double affine hecke algebras the former is known as the iwahori hecke algebra and the latter was introduced by cherednik in terms of residues more generally to any generalized cartan matrixaand a pointqin a one dimensional complex algebraic group c we associate an associate algebrahq ifais of finite type and c c the algebrahqis theaffinehecke algebra of the corresponding finite root system ifais of affine type and c c thenhqis essentially the cherednik algebra the case c c corresponds to degenerate counterparts of the above objects considered by drinfeld and lusztig finally taking c to be an elliptic curve one gets some new elliptic analogues of the affine hecke algebra

unique continuation for some evolution equations this work is concerned with unique continuation results for some evolution equations essentially of parabolic type to be more precise let l be an evolution operator acting on functions defined on some connected open set b of r r x r l is said to have the unique continuation property if every solution u of lu 0 which vanishes one some nonempty open set u i of fl vanishes in the horizontal component of o in 0 i e in x t e 0 3x x t e co the first general result of this type was established by mizohata 6 in the case where l is a second order parabolic operator with smooth coefficients however mizohata s proof is somewhat delicate it is based in particular on an extended class of calderon zygmund operators and does not lead obviously to a weakening of the smoothness of the coefficients on the other hand it is important for the applications to deal with nonsmooth coefficients since l is often obtained as the linearization of some nonlinear operator at a nonnecessarily smooth solution we refer to lions 4 and kernevez and lions 3 for applications to control theory and to 1 l for applications to transversality techniques let us now describe the content of this work in the first section we prove a unique continuation theorem when l is a second order parabolic equation our proof is simple and based on the derivation of a carleman estimate which is reminiscent of the classical carleman estimates for second order elliptic operators see 121 this carleman inequality allows the weakening of the 118 0022 0396187 3 00

some remarks concerning mod nk theory the spectral sequence predicted by a beilinson relating motivic cohomology to algebraic k theory has been established for smooth quasi projective varieties over a field cf fs l1 among other properties verified this spectral sequence has the expected multiplicative behavior involving cup product in motivic cohomology and product in algebraic k theory and a good multiplicative mod n reduction relating mod n motivic cohomology to mod n algebraic k theory the purpose of this short paper is to draw two simple conclusions concerning mod n algebraick theory of smooth quasi projective varieties over an algebraically closed field by comparing this spectral sequence to its localization which we show converges to mod n etale k theory in theorem 2 1 we verify that the map from mod n algebraic k theory to mod n etale k theory of a smooth quasi projective variety x over an algebraically closed field in which n is invertible is surjective in degrees greater than or equal to twice the dimension of x this refines the surjectivity result of dfst and extends an argument by a suslin in s1 then in example 3 4 we show that certain projective smooth 3 folds x constructed by s bloch and h esnault have non zero elements in k0 x z n which are killed by multiplication by a sufficiently high power of the bott element k2 x z n although the existence of such examples is perhaps not surprising there have been attempts to prove that multiplication by is always injective the authors gratefully acknowledge useful conversations with h esnault and a suslin

transfers of chern classes in bp cohomology and chow rings the bp module structure of bp bg for extraspecial 2 groups is studied using transfer and chern classes these give rise to p torsion elements in the kernel of the cycle map from the chow ring to ordinary cohomology first obtained by totaro

relative gromov witten invariants we define relative gromov witten invariants of a symplectic manifold relative to a codimension two symplectic submanifold these invariants are the key ingredients in the symplectic sum formula of ip4 the main step is the construction of a compact space of v stable maps simple special cases include the hurwitz numbers for algebraic curves and the enumerative invariants of caporaso and harris

equivariant cobordism for torus actions abstract we study the equivariant cobordism groups for the action of a split torus t on varieties over a field k of characteristic zero we show that for t acting on a variety x there is an isomorphism t x b t l x as applications we show that for a connected linear algebraic group g acting on a k variety x the forgetful map g x x is surjective with rational coefficients as a consequence we describe the rational algebraic cobordism ring of algebraic groups and flag varieties we prove a structure theorem for the equivariant cobordism of smooth projective varieties with torus action using this we prove various localization theorems and a form of bott residue formula for such varieties as an application we show that the equivariant cobordism of a smooth variety x with torus action is generated by the invariant cobordism cycles in x as b t module

cohomologie non ramifiee et conjecture de hodge entiere en nous appuyant sur la conjecture de bloch kato en k theorie de milnor nous etablissons un lien general entre le defaut de la conjecture de hodge entiere pour la cohomologie de degre 4 et le troisieme groupe de cohomologie non ramifiee a coefficients q z ceci permet de montrer que sur un solide en anglais threefold uniregle le troisieme groupe de cohomologie non ramifiee a coefficients q z s annule ce que la k theorie algebrique ne permet d obtenir que dans certains cas ceci permet a l inverse de deduire d exemples ayant leur source en k theorie que la conjecture de hodge entiere pour la cohomologie de degre 4 peut etre en defaut pour les varietes rationnellement connexes pour certaines familles a un parametre de surfaces on etablit un lien entre la conjecture de hodge entiere et l indice de la fibre generique

s duality for surfaces with a_n type singularities we show that the generating series of euler characteristics of hilbert schemes of points on any algebraic surface with at worst a n type singularities is described by the theta series determined by integer valued positive definite quadratic forms and the dedekind eta function in particular it is a fourier development of a meromorphic modular form with possibly half integer weight the key ingredient is to apply the flop transformation formula of donaldson thomas type invariants counting two dimensional torsion sheaves on threefolds proved in the author s previous paper

examples for the mod p motivic cohomology of classifying spaces let bg be the classifying space of a compact lie group g some examples of computations of the motivic cohomology h eg z p are given by comparing with h bg z p ch eg and bp bg

torsion algebraic cycles and etale cobordism we prove that the classical integral cycle class map from algebraic cycles to etale cohomology factors through a quotient of l adic etale cobordism over an algebraically closed field of positive characteristic this shows that there is a strong topological obstruction for cohomology classes to be algebraic and that examples of atiyah hirzebruch and totaro also work in positive characteristic

motivic decompositions of twisted flag varieties and representations of hecke type algebras motivated by the motivic galois group and the kostant kumar results on equivariant cohomology of flag varieties we provide a uniform de scription of motivic direct sum decompositions with integer coefficients of versal flag varieties in terms of integer representations of the associated affine nil hecke algebra h more generally we establish an equivalence between the h motivic subcat egory generated by the motive of e b and the category of projective modules of the associated rational algebra d of push pull operators where e is a tor sor for a split semisimple linear algebraic group g over a field k b is a borel subgroup of g h is an algebraic oriented cohomology theory in the sense of levine morel e g chow ring ch or an algebraic cobordism the algebra d can be think of as an integer analogue of the hopf algebra of the h motivic galois group of e b as an application taking h ch and specializing the coefficients to the finite field fp we obtain that p modular projective representations of d h are generated by an irreducible h module corresponding to the generalized rost voevodsky motive for g p

open gromov witten theory on symplectic manifolds and symplectic cutting abstract let x be a symplectic manifold and l be a lagrangian submanifold diffeomorphic to s n r p n or a lens space of a certain type using the symplectic cut and symplectic sum constructions we express the open gromov witten invariants of x l in terms of open gromov witten invariants of a pair x l determined by l and the standard gromov witten invariants of a symplectic manifold x determined by x l we also describe other applications of this approach

comparison of cobordism theories abstract relying on results of hopkins morel we show that for x a quasi projective variety over a field of characteristic zero the canonical map n x mgl 2 n n x is an isomorphism

algebraic cobordism revisited we define a cobordism theory in algebraic geometry based on normal crossing degenerations with double point singularities the main result is the equivalence of double point cobordism to the theory of algebraic cobordism previously defined by levine and morel double point cobordism provides a simple geometric presentation of algebraic cobordism theory as a corollary the lazard ring given by products of projective spaces rationally generates all nonsingular projective varieties modulo double point degenerations

controller design for delay systems via eigenvalue assignment on a new result in the distribution of quasi polynomial roots this paper considers the eigenvalue distribution of a linear time invariant lti system with time delays and its application to some controllers design for a delay plant via eigenvalue assignment first a new result on the root distribution for a class of quasi polynomials is developed based on the extension of the hermite biehler theorem then such result is applied to proportional integral pi controller parameter design for a first order plant with time delay through pole placement the complete region of pi gains can be obtained so that the rightmost eigenvalues in the infinite eigenspectrum of the closed loop system with delay plant are assigned to desired positions in the complex plane furthermore on the basis of the previous result this paper also extended the pi control to the proportional integral derivative pid control it is worth pointing out that this work aims to improve the performance of the closed loop system on the premise of guaranteeing the stability

segre classes and kempf laksov formula in algebraic cobordism in this paper we extend to algebraic cobordism the classical damon kempf laksov formula which expresses the chow ring schubert classes of grassmann bundles as schur determinants in chern classes the basic building block of our closed formula which is written as a sum of determinants is represented by a generalisation of segre classes that we introduce and describe

schubert calculus in complex cobordism we study the structure of the complex cobordism ring of the flag variety of a compact connected lie group an explicit procedure for determining products of basis elements is obtained generalizing the work of bernstein gel fand gel fand on ordinary cohomology and of kostant kumar on k theory bott samelson resolutions are used to replace the classical basis of schubert cells

k theoretic chern class formulas for vexillary degeneracy loci abstract using raising operators and geometric arguments we establish formulas for the k theory classes of degeneracy loci in classical types we also find new determinantal and pfaffian expressions for classical cases considered by giambelli the loci where a generic matrix drops rank and where a generic symmetric or skew symmetric matrix drops rank in an appendix we construct a k theoretic euler class for even rank vector bundles with quadratic form refining the chow theoretic class introduced by edidin and graham we also establish a relation between top chern classes of maximal isotropic subbundles which is used in proving the type d degeneracy locus formulas

the local gromov witten theory of curves the local gromov witten theory of curves is solved by localization and degeneration methods localization is used for the exact evalu ation of basic integrals in the local gromov witten theory of p 1 a tqft formalism is defined via degeneration to capture higher genus curves together the results provide a compete and effective solution the local gromov witten theory of curves is equivalent to the lo cal donaldson thomas theory of curves the quantum cohomology of the hilbert scheme points of c 2 and the orbifold quantum cohomol ogy the symmetric product of c 2 the results of the paper provide the local gromov witten calculations required for the proofs of these equivalences

algebraic design of anisochronic controllers for time delay systems the prime objective of the paper is to overcome the conventional restriction to the field of rational functions as far as the algebraic design of control systems is concerned by adopting the affine parameterization approach from algebraic control synthesis for the controllers given as the so called rq meromorphic functions an algebraic solution to time delay system control design is opened where neither an introduction of 2d transfer functions nor the assumption of commensurate delays is needed the proposed design results in controllers combining both the integrators and delayers due to this property referred to as anisochronic

on the structure of algebraic cobordism abstract in this paper we investigate the structure of algebraic cobordism of levine morel as a module over the lazard ring with the action of landweber novikov and symmetric operations on it we show that the associated graded groups of algebraic cobordism with respect to the topological filtration r x are unions of finitely presented l modules of very specific structure namely these submodules possess a filtration such that the corresponding factors are either free or isomorphic to cyclic modules l i p n x where deg x p n 1 p 1 as a corollary we prove the syzygies conjecture of vishik on the existence of certain free l resolutions of x and show that algebraic cobordism of a smooth surface can be described in terms of k 0 together with a topological filtration

a degeneration formula of gw invariants this is the sequel to the paper 23 in this paper we construct the virtual moduli cycles of the degeneration of the moduli of stable morphisms constructed in 23 we also construct the virtual moduli cycles of the moduli of relative stable morphisms of a pair of a smooth divisor in a smooth variety based on these we prove a degeneration formula of the gromov witten invariants

algebraic cobordism as a module over the lazard ring in this paper we study the structure of the algebraic cobordism ring of a variety as a module over the lazard ring and show that it has relations in positive codimensions we actually prove the stronger graded version this extends the result of levine and morel algebraic cobordism in springer monographs in mathematics 2007 claiming that this module has generators in non negative codimensions as an application we compute the algebraic cobordism ring of a curve the main tool is symmetric operations in algebraic cobordism vishik symmetric operations for all primes and steenrod operations in algebraic cobordism 2013

gromov witten donaldson thomas correspondence for toric 3 folds we prove the equivariant gromov witten theory of a nonsingular toric 3 fold x with primary insertions is equivalent to the equivariant donaldson thomas theory of x as a corollary the topological vertex calculations by agangic klemm marino and vafa of the gromov witten theory of local calabi yau toric 3 folds are proven to be correct in the full 3 leg setting

algebraic elliptic cohomology theory and flops i we define the algebraic elliptic cohomology theory coming from krichever s elliptic genus as an oriented cohomology theory on smooth varieties over an arbitrary perfect field we show that in the algebraic cobordism ring with rational coefficients the ideal generated by differences of classical flops coincides with the kernel of krichever s elliptic genus this generalizes a theorem of b totaro in the complex analytic setting

nested hilbert schemes on surfaces virtual fundamental class abstract we construct virtual fundamental classes on nested hilbert schemes of points and curves in complex nonsingular projective surfaces these classes recover the virtual classes of seiberg witten theory as well as the reduced stable theory and play a crucial role in the reduced donaldson thomas theory of local surface threefolds that we study in 15 we show that certain integrals against the virtual fundamental classes of punctual nested hilbert schemes are expressed as integrals over the products of the hilbert scheme of points we are able to find explicit formulas for some of these integrals by relating them to carlsson okounkov s vertex operator formulas

brown peterson and ordinary cohomology theories of classifying spaces for compact lie groups the steenrod algebra structures of h bg z p for compact lie groups are studied using these brown peterson cohomology and morava k theory are computed for many concrete cases all these cases have properties similar as torsion free lie groups or finite groups e g bp odd bg 0

geometric representations of the formal affine hecke algebra abstract for any formal group law there is a formal affine hecke algebra defined by hoffnung malagon lopez savage zainoulline coming from this formal group law there is also an oriented cohomology theory we identify the formal affine hecke algebra with a convolution algebra coming from the oriented cohomology theory applied to the steinberg variety as a consequence this algebra acts on the corresponding cohomology of the springer fibers this generalizes the action of classical affine hecke algebra on the k theory of the springer fibers constructed by lusztig we also give a residue interpretation of the formal affine hecke algebra which generalizes the residue construction of ginzburg kapranov vasserot when the formal group law comes from a 1 dimensional algebraic group

on certain hecke rings abstract we examine rings that embed into the smash product of the group algebra of the weyl group with the field of meromorphic functions on the cartan subalgebra and are generated by elements that satisfy braid relations we prove that every such ring is isomorphic to either the hecke algebra the nil hecke ring or the group algebra of the weyl group

a chevalley formula in equivariant k theory abstract the aim of this paper is to give a recursive formula to compute the product of a line bundle with the structure sheaf of a schubert variety in the equivariant k theory of a flag variety

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

zero dimensional donaldson thomas invariants of calabi yau 4 folds abstract we study hilbert schemes of points on a smooth projective calabi yau 4 fold x we define dt 4 invariants by integrating the euler class of a tautological vector bundle l n against the virtual class we conjecture a formula for their generating series which we prove in certain cases when l corresponds to a smooth divisor on x a parallel equivariant conjecture for toric calabi yau 4 folds is proposed this conjecture is proved for smooth toric divisors and verified for more general toric divisors in many examples combining the equivariant conjecture with a vertex calculation we find explicit positive rational weights which can be assigned to solid partitions the weighted generating function of solid partitions is given by exp m q 1 where m q denotes the macmahon function

torsion algebraic cycles and complex cobordism atiyah and hirzebruch gave the first counterexamples to the hodge conjecture with integral coefficients 3 that conjecture predicted that every integral cohomology class of hodge type p p on a smooth projective variety should be the class of an algebraic cycle but atiyah and hirzebruch found additional topological properties which must be satisfied by the integral cohomology class of an algebraic cycle here we provide a more systematic explanation for their results by showing that the classical cycle map from algebraic cycles modulo algebraic equivalence to integral cohomology factors naturally through a topologically defined ring which is richer than integral cohomology the new ring is based on complex cobordism a well developed topological theory which has been used only rarely in algebraic geometry since hirzebruch used it to prove the riemann roch theorem 17 this factorization of the classical cycle map implies the topological restrictions on algebraic cycles found by atiyah and hirzebruch it goes beyond their work by giving a topological method to show that the classical cycle map can be noninjective as well as nonsurjective the kernel of the classical cycle map is called the griffiths group and the topological proof here that the griffiths group can be nonzero is the first proof of this fact which does not use hodge theory the proof here gives nonzero torsion elements in the griffiths group whereas griffiths s hodge theoretic proof gives nontorsion elements 13 this topological argument also gives examples of algebraic cycles in the kernel of various related cycle maps where few or no examples were known before thus answering some questions posed by colliot thelene and schoen 8 p 14 37 p 13 colliot thelene asked in particular whether the map ch2 x n h4 x z n is injective for all smooth complex projective varieties x here ch x is the group of codimension i algebraic cycles modulo rational equivalence the first examples where colliot thelene s map is not injective were found by kollar and van geemen see 4 p 135 very recently bloch and esnault found examples defined over number fields 7 over nonalgebraically closed fields k there are other examples of smooth projective varieties xk with ch2 xk n hg4t xk z n not injective due to colliot thelene and sansuc as reinterpreted by salberger see 9 and 8 remark 7 6 1 and parimala and suresh 31 these elements of ch2 xk n are not shown to remain nonzero in ch2 xc n however here our topological method

elliptic classes of schubert varieties we introduce new notions in elliptic schubert calculus the twisted borisov libgober classes of schubert varieties in general homogeneous spaces g p while these classes do not depend on any choice they depend on a set of new variables for the definition of our classes we calculate multiplicities of some divisors in schubert varieties which were only known for full flag varieties before our approach leads to a simple recursions for the elliptic classes comparing this recursion with r matrix recursions of the so called elliptic weight functions of rimanyi tarasov varchenko we prove that weight functions represent elliptic classes of schubert varieties