poisson structures on complex flag manifolds associated with real forms for a complex semisimple lie group g and a real form go we define a poisson structure on the variety of borel subgroups of g with the property that all go orbits in x as well as all bruhat cells for a suitable choice of a borel subgroup of g are poisson submanifolds in particular we show that every non empty intersection of a go orbit and a bruhat cell is a regular poisson manifold and we compute the dimension of its symplectic leaves

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

spatial localization in heterogeneous systems we study spatial localization in the generalized swift hohenberg equation with either quadratic cubic or cubic quintic nonlinearity subject to spatially heterogeneous forcing different types of forcing sinusoidal or gaussian with different spatial scales are considered and the corresponding localized snaking structures are computed the results indicate that spatial heterogeneity exerts a significant influence on the location of spatially localized structures in both parameter space and physical space and on their stability properties the results are expected to assist in the interpretation of experiments on localized structures where departures from spatial homogeneity are generally unavoidable

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

integrable many body problems and functional equations on etablit des conditions necessaires pour l integrabilite d une classe de problemes a n corps quantiques

t equivariant k theory of generalized flag varieties let g be a kac moody group with borel subgroup b and compact maximal torus t analogous to kostant and kumar kostant b kumar s 1986 proc nati acad sci usa 83 1543 1545 we define a certain ring y purely in terms of the weyl group w associated to g and its action on t by dualizing y we get another ring 1 which we prove is canonically isomorphic with the t equivariant ktheory kt g b of gib now kt g b apart from being an algebra over kt pt a t also has a weyl group action and moreover kt g b admits certain operators dw w similar to the demazure operators defined on a t we prove that these structures on kt g b come naturally from the ring y by evaluating the a t module w at 1 we recover k g b together with the above mentioned structures we believe that many of the results of this paper are new in the finite case i e g is a finite dimensional semisimple group over c

torsion in cohomology of compact lie groups and chow rings of reductive algebraic groups the purpose of the present note is to give a general procedure for calculating the singular cohomology h k ifp with coefficients in an arbitrary finite field ifv of a connected compact lie group k simultaneously we calculate the chow ring a g ifp a g p of the corresponding complex algebraic group g and the degrees of the basic generalized invariants of the weyl group w in arbitrary characteristic p a mysterious connection between the degrees of generators of a g ifp and the kernel of the principal nilpotent element in characteristic p is pointed out the computation of real cohomology of compact lie groups started by e cartan in 1929 was completed for classical groups by r brauer and by l pontryagin in 1935 h hopf and h samelson showed in 1941 that h k f is an exterior algebra on generators of degrees 2 d 1 1 2dr l where l r a n k k but only in 1949 c t yen managed to compute the d for all exceptional groups case by case on the other hand c chevalley and j leray showed that the d i are nothing else but the degrees of the basic invariants over of the weyl group w a simple procedure for calculating the dl was given in the late fifties by a j coleman and by b kostant nowadays this is common knowledge the problem of computing the torsion in h k z proved to be more recalcitrant this was done by several authors c e miller settled so in 1953 23 a borel settled spin ad spin and the quotients of su and sp in 1954 6 p f baum and w browder settled the remaining classical group the semispin group in 1965 4 g 2 and f 4 were settled in 1954 6 and the p 3 torsion for e 6 and p 5tors ion for ev e s in 1961 7 by a borel s araki computed the 2 torsion for e 6 e 7 1 and the 3 torsion for e 7 e s 2 and announced the 2 torsion for e 8 in a joint paper with y shikata in 1961 3 the details of the proof for the last result have appeared only recently 18 there was an error in 1 in the case of ad e 7 which has been fixed in 13

unipotent hecke algebras of gln fq this paper describes a family of hecke algebras h endg indug where u is the subgroup of unipotent upper triangular matrices of g gln fq and is a linear character of u the main results combinatorially index a basis of h provide a large commutative subalgebra of h and after describing the combinatorics associated with the representation theory of h generalize the rsk correspondence that is typically found in the representation theory of the symmetric group

nonlinear scalar field equations with general nonlinearity abstract consider the nonlinear scalar field equation 0 1 u f u in r n u h 1 r n where n 3 and f satisfies the general berestycki lions conditions we are interested in the existence of positive ground states of nonradial solutions and in the multiplicity of radial and nonradial solutions very recently mederski 2017 made a major advance in that direction through the development in an abstract setting of a new critical point theory for constrained functionals in this paper we propose an alternative more elementary approach which permits to recover mederski s results on 0 1 the keys to our approach are an extension to the symmetric mountain pass setting of the monotonicity trick and a new decomposition result for bounded palais smale sequences

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

standing waves of nonlinear schrodinger equations with the gauge field abstract we study standing waves for nonlinear schrodinger equations with the gauge field some existence results of standing waves are established by applying variational methods to the functional which is obtained by representing the gauge field a in terms of complex scalar field we also show that there exists no standing wave for certain range of parameters by establishing a new inequality of sobolev type

on the action of ariki koike algebras on tensor space the action of the ariki koike algebra on tensor space is used to show an isomorphism theorem between the ariki koike algebra and a direct sum of matrix rings over tensorproducts of smaller ariki koike algebras as an application we get schur weyl duality for the action of the ariki koike algebra and the levi subalgebra of the q schur algebra when separation conditions holds mit hilfe der operation der ariki koike algebra auf dem tensorraum wird ein isomorphismus zwischen der ariki koike algebra und einer direkten summe von matrizenringen uber tensorprodukten kleinerer ariki koike algebren gezeigt als anwendung ergibt sich das die ariki koike algebra und die leviunteralgebra der q schur algebra schur weyl dualitat erfullen falls die separationsbedingung erfullt ist

instanton filtering for the stochastic burgers equation we address the question of whether one can identify instantons in direct numerical simulations of the stochastically driven burgers equation for this purpose we first solve the instanton equations using the chernykh stepanov method 2001 phys rev e 64 026306 these results are then compared to direct numerical simulations by introducing a filtering technique to extract prescribed rare events from massive data sets of realizations using this approach we can extract the entire time history of the instanton evolution which allows us to identify the different phases predicted by the direct method of chernykh and stepanov with remarkable agreement

the set of positive solutions of semilinear equations in large balls the exact number of positive solutions of u f u 0 on finite balls in n is determined the assumptions about f u are similar to those imposed by serrin and the second author in a previous study of uniqueness of the positive solution when the spatial domain is all of n see 7 8 for finite balls of sufficiently large radius it is shown here that there are exactly two positive and hence radial solutions to this end we first prove the linear nondegeneracy of the positive solution of n this is obtained by applying the technique of monotone separation of graphs 7 to the linearised equations somewhat sharper estimates are required here see part i section 2

a variational approach for standing waves of fitzhugh nagumo type systems abstract we study the existence of radially symmetric solutions of fitzhugh nagumo type elliptic systems in r n n 2 u g u v in r n d v v u in r n u x v x 0 0 as x we utilize a truncation technique and apply minimax arguments to the corresponding strongly indefinite functional i u v 1 2 r n u 2 d v 2 d x r n g u 2 v 2 u v d x defined on h r 1 r n h r 1 r n to find positive and possibly sign changing solutions of in particular we overcome difficulty related to palais smale condition via our new scaling argument when g 1 0 1 2 we improve the existence result of reinecke sweers 23

block spin transformations in the operator formulation of two dimensional potts models the author shows that statistical mechanical models such as the q state potts model whose transfer matrix may be written in terms of the hecke algebra hn q respond to block spin scale changes at a certain critical point in a way which may be interpreted as an algebra homomorphism part of the structure of the algebra and hence part of the transfer matrix spectrum of the model is preserved provided q 4 cos2 pi r r integer these partial realisations of scale invariance are of a distinct type for each r characterised in some cases by an endomorphism of the algebra

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

sign changing semi classical solutions of the brezis nirenberg problems with jump nonlinearities in high dimensions abstract in this paper we consider the following elliptic equation e 2 u u u u p 2 u u 2 2 u in u 0 on where r n n 4 is a bounded domain u max u 0 0 are constants e 0 is a small parameter and 2 p 2 2 n n 2 we obtain a least energy sign changing solution to this problem for e 0 that is sufficiently small by means of the variational method furthermore by combining the elliptic and local energy estimates we study the concentration behavior of this least energy sign changing solution and the location of the positive and negative spikes as e 0 our results partially complete the studies in 6 2011 in the sense that due to sobolev embedding only subcritical cases are considered in that paper

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

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

large deviations in fast slow systems the incidence of rare events in fast slow systems is investigated via analysis of the large deviation principle ldp that characterizes the likelihood and pathway of large fluctuations of the slow variables away from their mean behavior such fluctuations are rare on short time scales but become ubiquitous eventually classical results prove that this ldp involves an hamilton jacobi equation whose hamiltonian is related to the leading eigenvalue of the generator of the fast process and is typically non quadratic in the momenta in other words the ldp for the slow variables in fast slow systems is different in general from that of any stochastic differential equation sde one would write for the slow variables alone it is shown here that the eigenvalue problem for the hamiltonian can be reduced to a simpler algebraic equation for this hamiltonian for a specific class of systems in which the fast variables satisfy a linear equation whose coefficients depend nonlinearly on the slow variables and the fast variables enter quadratically the equation for the slow variables these results are illustrated via examples inspired by kinetic theories of turbulent flows and plasma in which the quasipotential characterizing the long time behavior of the system is calculated and shown again to be different from that of an sde

existence and concentration result for the kirchhoff type equations with general nonlinearities in this paper we study the existence and concentration behaviors of positive solutions to the kirchhoff type equations varepsilon 2 m left varepsilon 2 n int mathbf r n nabla u 2 mathrm d x right delta u v x u f u quad rm in mathbf r n quad u in h 1 mathbf r n n geqq 1 where m and v are continuous functions under suitable conditions on m and general conditions on f we construct a family of positive solutions u varepsilon varepsilon in 0 tilde varepsilon which concentrates at a local minimum of v after extracting a subsequence ek

localized states in the generalized swift hohenberg equation the swift hohenberg equation with quadratic and cubic nonlinearities exhibits a remarkable wealth of stable spatially localized states the presence of these states is related to a phenomenon called homoclinic snaking numerical computations are used to illustrate the changes in the localized solution as it grows in spatial extent and to determine the stability properties of the resulting states the evolution of the localized states once they lose stability is illustrated using direct simulations in time i introduction ever since the observation that the subcritical complex ginzburg landau equation exhibits stable spatially localized states 1 there has been considerable interest in the properties of these states the presence of these states has important consequences for other systems described by partial differential equations on the line since the ginzburg landau equation describes the behavior of such systems near bifurcation from the trivial state of the system specifically the complex ginzburg landau equation describes the evolution of a long wavelength oscillatory instability as well as oscillatory instabilities at finite wavelength in systems with broken reflection symmetry in contrast near a steady state bifurcation with finite wavelength the evolution of the instability is described by the real ginzburg landau equation and this equation possesses only unstable spatially localized states it is of interest therefore to examine what happens to these unstable states at larger amplitude where the real ginzburg landau equation no longer provides an adequate description of the system in this paper we show that the localized states can become stable at such amplitudes and indeed that there is a large multiplicity of coexisting stable localized states under very general conditions we are able to relate the existence of these states to a phenomenon sometimes called homoclinic snaking that is well known from the theory of reversible systems with 1 1 resonance and use this theory to construct a large number of such states the stability properties of these states are also determined and the evolution of nonstationary localized states is studied by numerical integration in time it is an interesting fact that closely related phenomena have already been described in several areas involving pattern formation the theory was originally developed in the context of water waves where localized states have been studied by moving into a reference frame of the waves and converting the problem into an ordinary differential equation ode the resulting localized states are called solitary waves and in some cases turn out to be solitons kirchgassner 2 has pioneered a successful approach to this type of problem that led to a number of advances in this area specifically the ode is viewed as a dynamical system in space and localized states are sought as homoclinic orbits connecting the trivial state to itself whether such orbits are possible depends in part on the stability properties of the trivial state eigenvalues with positive real part indicate that a nontrivial state can grow from x while eigenvalues with negative real part indicate that such a state may return under appropriate conditions back to the trivial state as x the spectrum of the linearization about the trivial state is influenced by spatial symmetries of the system in many cases and in particular in the case considered here the ode is reversible as a result the bifurcations that are encountered as a parameter is increased are nongeneric in the present case the spatial dynamics of the system near the trivial state turn out to be described by the reversible 1 1 resonance the unfolding of this resonance has been worked out in detail by iooss and peroueme 3 and can be used to understand the appearance of a variety of homoclinic orbits in this system and hence of localized states with different spatial structure

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

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

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

higher algebraic k theory for actions of diagonalizable groups we study the k theory of actions of diagonalizable group schemes on noetherian regular separated algebraic spaces our main result shows how to reconstruct the k theory ring of such an action from the k theory rings of the loci where the stabilizers have constant dimension we apply this to the calculation of the equivariant k theory of toric varieties and give conditions under which the merkurjev spectral sequence degenerates so that the equivariant k theory ring determines the ordinary k theory ring we also prove a very refined localization theorem for actions of this type

singularly perturbed nonlinear dirichlet problems with a general nonlinearity let be a bounded domain in r n n 3 with a boundary c 2 we consider the following singularly perturbed nonlinear elliptic problem on e 2 u u f u 0 u 0 on u 0 on where the nonlinearity f is of subcritical growth under rather strong conditions on f it has been known that for small e 0 there exists a mountain pass solution u e of above problem which exhibits a spike layer near a maximum point of the distance function d from as e 0 in this paper we construct a solution u e of above problem which exhibits a spike layer near a maximum point of the distance function under certain conditions on f which we believe to be almost optimal

schubert polynomials for the affine grassmannian let g be a complex simply connected simple group and k a maximal compact subgroup let f c t denote the field of formal laurent series and o c t the ring of formal power series the quotient qr g f g 0 is known as the affine grassmannian of g it is known that qr is homotopy equivalent to the space of based loops k into the compact group k see 9 30 the multiplication on qk endows the homology h qr and cohomology h qr with structures of dual hopf algebras these hopf algebras were first identified by bott 4 another descrip tion is given by ginzburg 10 which we shall not use the bruhat decomposition of g f induces a stratification of qr by schubert cells ilw indexed by grassman nian elements w of the affine weyl group waffwe denote by aw g h qr and aw g h qr the schubert classes in homology and cohomology when g sl n c bott identifies h qr and h qr with a subring an and a quotient an of the ring of symmetric functions our main theorem theorem 7 1 identifies the schubert classes aw g qr and aw g h qr as explicit symmet ric functions under this isomorphism these symmetric functions are combinatori ally defined in homology the aw g h qr are represented by lascoux lapointe morse s fc schur functions sw x g an and in cohomology the aw g h qr are represented by the dual fc schur functions or affine schur functions fw x g an see 21 24 17 our theorem was originally conjectured by mark shimozono the conjecture was made explicit in the cohomology case by jennifer morse kostant and kumar 14 studied the topology of homogeneous spaces for arbi trary kac moody groups and in particular calculated the structure constants of h qr in the schubert basis using the algebraic construction of the nilhecke ring our connection between the topology and the combinatorics proceeds via the study of three subalgebras of the nilhecke ring the first subalgebra is the nilcoxeter al gebra ao which is the algebra generated by the divided difference operators the second algebra is a certain centralizer algebra aaff s which we call the peterson subalgebra peterson 28 constructs an isomorphism j from the t equivariant ho mology hr qr of the affine grassmannian to z ff s the third algebra b is a combinatorially defined subalgebra of a0 which we call the affine fomin stanley

connection between nonlinear energy optimization and instantons how systems transit between different stable states under external perturbation is an important practical issue we discuss here how a recently developed energy optimization method for identifying the minimal disturbance necessary to reach the basin boundary of a stable state is connected to the instanton trajectory from large deviation theory of noisy systems in the context of the one dimensional swift hohenberg equation which has multiple stable equilibria we first show how the energy optimization method can be straightforwardly used to identify minimal disturbances minimal seeds for transition to specific attractors from the ground state then after generalizing the technique to consider multiple equally spaced in time perturbations it is shown that the instanton trajectory is indeed the solution of the energy optimization method in the limit of infinitely many perturbations provided a specific norm is used to measure the set of discrete perturbations importantly we find that the key features of the instanton can be captured by a low number of discrete perturbations typically one perturbation per basin of attraction crossed this suggests a promising new diagnostic for systems for which it may be impractical to calculate the instanton

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

quantum integrability and generalised quantum schubert calculus abstract we introduce and study a new mathematical structure in the generalised quantum cohomology theory for grassmannians namely we relate the schubert calculus to a quantum integrable system known in the physics literature as the asymmetric six vertex model our approach offers a new perspective on already established and well studied special cases for example equivariant k theory and in addition allows us to formulate a conjecture on the so far unknown case of quantum equivariant k theory

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

mountain pass solutions for singularly perturbed nonlinear dirichlet problems abstract we consider the following singularly perturbed nonlinear elliptic problem 2 u u f u 0 u 0 in u 0 on where the nonlinearity f is of subcritical growth and is a bounded domain in r n under certain conditions of f there exists a mountain pass solution u for above problem and the solution u exhibits a spike layer as 0 to see the location of the spike layer for small 0 some additional assumptions certain nondegeneracy for a limiting problem or monotonicity of f t t have been assumed in this paper we characterize the location of the spike layer without such additional assumptions for n 3

excited young diagrams and equivariant schubert calculus we describe the torus equivariant cohomology ring of isotropic grassmannians by using a localization map to the torus fixed points we present two types of formulas for equivariant schubert classes of these homogeneous spaces the first formula involves combinatorial objects which we call excited young diagrams and the second one is written in terms of factorial schur q or p functions as an application we give a giambelli type formula for the equivariant schubert classes we also give combinatorial and pfaffian formulas for the multiplicity of a singular point in a schubert variety

a mckay correspondence for reflection groups we construct a noncommutative desingularization of the discriminant of a finite reflection group g as a quotient of the skew group ring a s g if g is generated by order 2 reflections then this quotient identifies with the endomorphism ring of the reflection arrangement a g viewed as a module over the coordinate ring sg of the discriminant of g this yields in particular a correspondence between the nontrivial irreducible representations of g to certain maximal cohen macaulay modules over the coordinate ring sg these maximal cohen macaulay modules are precisely the nonisomorphic direct summands of the coordinate ring of the reflection arrangement a g viewed as a module over sg we identify some of the corresponding matrix factorizations namely the so called logarithmic co residues of the discriminant

infinite flag varieties and conjugacy theorems abstract we study the orbit of a highest weight vector in an integrable highest weight module of the group g associated to a kac moody algebra unk a we obtain applications to the geometric structure of the associated flag varieties and to the algebraic structure of unk a in particular we prove conjugacy theorems for cartan and borel subalgebras of unk a so that the cartan matrix a is an invariant of unk a

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

standing waves for nonlinear schrodinger equations with a general nonlinearity for elliptic equations e2 u v x u f u 0 x rn n 3 we develop a new variational approach to construct localized positive solutions which concentrate at an isolated component of positive local minimum points of v as e 0 under conditions on f which we believe to be almost optimal

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

integral motives relative krull schmidt principle and maranda type theorems in the present article we investigate properties of the category of integral grothendieck chow motives over a field we discuss the krull schmidt principle for integral motives provide a complete list of generalized severi brauer varieties with indecomposable integral motive and exploit a relation between the category of motives of twisted flag varieties and integral p adic representations of finite groups

quantum cohomology and the k schur basis we prove that structure constants related to hecke algebras at roots of unity are special cases of k littlewood richardson coefficients associated to a product of k schur functions as a consequence both the 3 point gromov witten invariants appearing in the quantum cohomology of the grassmannian and the fusion coefficients for the wzw conformal field theories associated to su l are shown to be k littlewood richardson coefficients from this mark shimozono conjectured that the k schur functions form the schubert basis for the homology of the loop grassmannian whereas k schur coproducts correspond to the integral cohomology of the loop grassmannian we introduce dual k schur functions defined on weights of k tableaux that given shimozono s conjecture form the schubert basis for the cohomology of the loop grassmannian we derive several properties of these functions that extend those of skew schur functions

a minimum action method for small random perturbations of two dimensional parallel shear flows in this work we develop a parallel minimum action method for small random perturbations of navier stokes equations to solve the optimization problem given by the large deviation theory the freidlin wentzell action functional is discretized by hp finite elements in time direction and spectral methods in physical space a simple diagonal preconditioner is constructed for the nonlinear conjugate gradient solver of the optimization problem a hybrid parallel strategy based on mpi and openmp is developed to improve numerical efficiency both h and p convergence are obtained when the discretization error from physical space can be neglected we also present preliminary results for the transition in two dimensional poiseuille flow from the base flow to a non attenuated traveling wave

a coproduct structure on the formal affine demazure algebra in the present paper we generalize the coproduct structure on nil hecke rings introduced and studied by kostant and kumar to the context of an arbitrary algebraic oriented cohomology theory and its associated formal group law we then construct an algebraic model of the t equivariant oriented cohomology of the variety of complete flags

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