percolation in strongly correlated systems abstract we investigate the threshold percolation density pc in strongly correlated lattice systems the probability distribution of the system is the stationary state for a combined kawasaki and glauber dynamics the voter model with flips when the glauber rate goes to zero the pair correlation function of the system at any density decays in three dimensions as r 1 and does not decay at all in two dimensions using monte carlo calculations and finite size scaling we obtain information about pc as a function of the glauber rate percolation in gaussian fields on a lattice which have similar slow decay is also discussed

middle convolution and harnad duality dettweiler reiter s additive description of katz s middle convolution can be interpreted in terms of the harnad duality using this interpretation together with mumford s geometric invariant theory we generalize the additive middle convolution to have a multi parameter and act on systems of linear ordinary differential equations with irregular singularities we show that the generalized operation holds basic properties of the original one and additionally show that katz s algorithm involving our generalization works well in generic case

stationary distributions for systems with competing creation annihilation dynamics we have developed a simple systematic method to investigate the existence of stationary probability distributions spds for interacting particle or spin lattice systems exhibiting steady non equilibrium states the latter originate in a competition between several creation annihilation spin flip kinetic mechanisms say each acting presuming a different bath temperature particle spin interactions strength or chemical potential magnetic field it follows the existence of spds for a large class of these systems which may thus be studied by simply applying the techniques of equilibrium theory the method is illustrated with several examples bearing a practical interest

module structures on the k theory of graded rings abstract let r be a commutative ring a a0 a1 a graded r algebra and a the graded ideal a1 a2 then ki a ki a0 ki a a we show that the groups ki a a are naturally modules over the ring w r of witt vectors they also have a natural filtration whose associated graded groups are r modules when r contains a field of characteristic zero ki a a is an r module and the filtration is by r submodules

normality of marsden weinstein reductions for representations of quivers we prove that the marsden weinstein reductions for the moment map associated to representations of a quiver are normal varieties we give an application to conjugacy classes of matrices

linear differential equations on the riemann sphere and representations of quivers our interest in this article is a generalization of the additive deligne simpson problem which was originally defined for fuchsian differential equations on the riemann sphere we extend this problem to differential equations having an arbitrary number of unramified irregular singular points and we determine the existence of solutions of the generalized additive deligne simpson problems moreover we apply this result to the geometry of the moduli spaces of stable meromorphic connections of trivial bundles on the riemann sphere namely open embedding of the moduli spaces into quiver varieties and the nonemptiness condition of the moduli spaces furthermore we detail the connectedness of the moduli spaces

arithmetic harmonic analysis on character and quiver varieties ii we study connections between the topology of generic character varieties of fundamental groups of punctured riemann surfaces macdonald polynomials quiver representations hilbert schemes on c x x c x modular forms and multiplicities in tensor products of irreducible characters of finite general linear groups c 2012 elsevier inc all rights reserved

linear differential equations on p1 and root systems abstract in this paper we study the euler transform on linear ordinary differential operators on p 1 the spectral type is the tuple of integers which count the multiplicities of local formal solutions with the same leading terms we compute the changes of spectral types under the action of the euler transform and show that the changes of spectral types generate a transformation group of a z lattice which is isomorphic to a quotient lattice of a kac moody root lattice with the weyl group as the transformation group

a class of algebras similar to the enveloping algebra of sl 2 fix f e c x define r c a b h subject to the relations ha ah a hb bh b ab ba f h we study these algebras for different f and in particular show how they are similar to and different from u sl 2 the enveloping algebra of sl 2 c there is a notion of highest weight modules and a category a for such r for each n 0 if f x x 1 n l xn 1 then r has precisely n simple modules in each finite dimension and every finite dimensional r module is semisimple

noncommutative resolutions of ade fibered calabi yau threefolds in this paper we construct noncommutative resolutions of a certain class of calabi yau threefolds studied by cachazo et al geometric transitions and n 1 quiver theories http arxiv org abs hep th 0108120v2 2001 the threefolds under consideration are fibered over a complex plane with the fibers being deformed kleinian singularities the construction is in terms of a noncommutative algebra introduced by ginzburg calabi yau algebras http arxiv org abs math 0612139v2 2006 which we call the n 1 ade quiver algebra

symmetries for certain coupled nonlinear schrodinger equations generalized symmetries are sought for certain coupled nonlinear schrodinger equations it is shown that all lie backlund symmetries but one are equivalent to lie point symmetries

new similarity reductions of the boussinesq equation some new similarity reductions of the boussinesq equation which arises in several physical applications including shallow water waves and also is of considerable mathematical interest because it is a soliton equation solvable by inverse scattering are presented these new similarity reductions including some new reductions to the first second and fourth painleve equations cannot be obtained using the standard lie group method for finding group invariant solutions of partial differential equations they are determined using a new and direct method that involves no group theoretical techniques

sums of orbits of algebraic groups i abstract let g be a simple algebraic group which is defined and split over a field k and let l be a corresponding lie algebra further let r be the corresponding root system we prove here that if char k 0 or a very good prime for r and k r then there exists an orbit o l l k with respect to the adjoint action of g g k such that l o o this is an analogue of the corresponding result for chevalley groups the thompson problem see e w ellers n gordeev on the conjecture of j thompson and o ore trans amer math soc 350 1998 3657 3671 we also prove that if char k 2 for r b r c r f 4 and char k 3 for r g 2 then the zariski closure of a sum of any r r b r r 3 d r e r f 4 r 1 r a r b 3 g 2 2r r c r g orbits of elements of l z l coincides with l for the group g sl 2 k where k is an algebraically closed field of characteristic zero we list all cases of rational k g modules v which have an orbit o v such that o o v

statistical mechanics of probabilistic cellular automata we investigate the behavior of discrete time probabilistic cellular automata pca which are markov processes on spin configurations on ad dimensional lattice from a rigorous statistical mechanics point of view in particular we exploit whenever possible the correspondence between stationary measures on the space time histories of pcas on d and translation invariant gibbs states for a related hamiltonian on d 1 this leads to a simple large deviation formula for the space time histories of the pca and a proof that in a high temperature regime the stationary states of the pca are gibbsian we also obtain results about entropy fluctuations and correlation inequalities and demonstrate uniqueness of the invariant state and exponential decay of correlations in a high noise regime we discuss phase transitions in the low noise or low temperature regime and review toom s proof of nonergodicity of a certain class of pcas

nonclassical symmetry solutions and the methods of bluman cole and clarkson kruskal nonclassical symmetry solutions of physically relevant partial differential equations are considered via the reduction methods of bluman and cole and clarkson and kruskal consistency conditions will be provided to show that if satisfied these two methods are equivalent in the sense that they lead to the same symmetry solutions the boussinesq equation and burgers equation are used as illustrative examples exact solutions one of which is new will be presented for burgers equation obtained from the bluman and cole method yet not obtainable by clarkson and kruskal s method

on the simplification of the form of lie transformation groups admitted by systems of evolution differential equations abstract for a system of two partial differential equations in two independent variables the lie point symmetry generator has the form t x u v t t x u v x t x u v u t x u v v we consider a system of evolution equations of general class and we provide restrictions on the functional forms of the coefficient functions of the lie generator these restrictions reduce the amount of calculations in determining the lie symmetries of the system under study we show that t except in one special case and we present further restrictions on the remaining coefficient functions

similarity reductions of partial differential equations the author determine the relation between two methods of reduction the similarity reduction method using nonclassical symmetry groups and the direct approach used by clarkson and kruskal 1989 they prove that the solutions which are obtained by similarity in correspondence to nonclassical groups constitute a larger family than the one obtained by the method of clarkson and kruskal the two procedures are equivalent only if the generators zeta x t u tau x t u eta x t u of the nonclassical groups are such that zeta tau is independent of u to explain these results he proves the existence of families of solutions of the burgers equation which are found by means of nonclassical symmetry reduction and which cannot be determined via the general reduction form of clarkson and kruskal

necklace lie algebras and noncommutative symplectic geometry recently v ginzburg proved that calogero phase space is a coadjoint orbit for some infinite dimensional lie algebra coming from noncommutative symplectic geometry 12 in this note we generalize his argument to specific quotient varieties of representations of deformed preprojective algebras this result was also obtained independently by v ginzburg 13 using results of w crawley boevey and m holland 10 8 and 9 we give a combinatorial description of all the relevant couples alpha lambda which are coadjoint orbits we give a conjectural explanation for this coadjoint orbit result and relate it to different noncommutative notions of smoothness

nonequilibrium spin models with ising universal behaviour the properties of a family of nonequilibrium spin models with up down symmetry on a square lattice are determined by a mean field pair approximation and by monte carlo simulation the phase diagram in the parameter space displays a critical line that terminates at a first order critical point it is found that the critical exponents are the same as those of the equilibrium ising model

on rigid tuples in linear groups of odd dimension abstract we construct examples of linearly rigid tuples which lead to regular galois realizations over q for linear and unitary groups gl2m 1 q and u2m 1 q2 where q is odd and m q 1 resp m q 1 the notion of linear rigidity was introduced by strambach and volklein j reine angew math to appear as an analogue of katz physically rigid systems see n m katz rigid local systems annals of mathematics studies vol 139 princeton univ press princeton nj 1996

generalized double affine hecke algebras of rank 1 and quantized del pezzo surfaces abstract let d be a simply laced dynkin diagram of rank r whose affinization has the shape of a star i e d 4 e 6 e 7 e 8 to such a diagram one can attach a group g whose generators correspond to the legs of the affinization have orders equal to the leg lengths plus 1 and the product of the generators is 1 the group g is then a 2 dimensional crystallographic group g z l z 2 where l is 2 3 4 and 6 respectively in this paper we define a flat deformation h t q of the group algebra c g of this group by replacing the relations saying that the generators have prescribed orders by their deformations saying that the generators satisfy monic polynomial equations of these orders with arbitrary roots which are deformation parameters the algebra h t q for d 4 is the cherednik algebra of type c c 1 which was studied by noumi sahi and stokman and controls askey wilson polynomials we prove that h t q is the universal deformation of the twisted group algebra of g and that this deformation is compatible with certain filtrations on c g we also show that if q is a root of unity then for generic t the algebra h t q is an azumaya algebra and its center is the function algebra on an affine del pezzo surface for generic q the spherical subalgebra e h t q e provides a quantization of such surfaces we also discuss connections of h t q with preprojective algebras and painleve vi

nonequilibrium impure lattice systems the authors study d dimensional lattice model systems whose dynamics consist of competing independent processes which may induce nonequilibrium steady states interesting phase transitions and critical phenomena ensue which are influenced even dominated by a sort of dynamical bond disorder the latter which may occur in natural systems reminds one of the disorder in more familiar bond diluted and other impure models for example percolation like phenomena arise in some of the model cases exact solutions for d 1 some exact results for d 1 and a comparison with more standard either quenched or annealed magnetically diluted models are reported

symmetries of the hypergeometric function phantom _ _ 1 in this paper we show that the generalized hypergeometric function m f m 1 has a one parameter group of local symmetries which is a conjugation of a flow of a rational calogero mozer system we use the symmetry to construct fermionic fields on a complex torus which have linear algebraic properties similar to those of the local solutions of the generalized hypergeometric equation the fields admit a nontrivial action of the quaternions based on the above symmetry we use the similarity between the linear algebraic structures to introduce the quaternionic action on the direct sum of the space of solutions of the generalized hypergeometric equation and its dual as a side product we construct a good basis for the monodromy operators of the generalized hypergeometric equation inspired by the study of multiple flag varieties with finitely many orbits of the diagonal action of the general linear group by magyar weyman and zelevinsky as an example of computational effectiveness of the basis we give a proof of the existence of the monodromy invariant hermitian form on the space of solutions of the generalized hypergeometric equation in the case of real local exponents different from the proofs of beukers and heckman and of haraoka as another side product we prove an elliptic generalization of cauchy identity

noncommutative deformations of type a kleinian singularities abstract the type a kleinian singularities are the surfaces of the form c 2 g where g is a finite cyclic subgroup of sl2 c the standard noncommutative analog is the fixed ring of the weyl algebra a1 c under the induced action of g these fixed rings are however only part of a large family of algebras whose associated graded ring is the coordinate ring of such a singularity the structure of these algebras is studied with reference to the structure of the associated singularity and to the structure of primitive factor rings of u s l 2 c

torus orbits on homogeneous varieties and kac polynomials of quivers in this paper we prove that the counting polynomials of certain torus orbits in products of partial flag varieties coincides with the kac polynomials of supernova quivers which arise in the study of the moduli spaces of certain irregular meromorphic connections on trivial bundles over the projective line we also prove that these polynomials can be expressed as a specialization of tutte polynomials of certain graphs providing a new way to compute them we also discuss connections with the gel fand macpherson correspondence

multiple flag varieties of finite type abstract we classify all products of flag varieties with finitely many orbits under the diagonal action of the general linear group we also classify the orbits in each case and construct explicit representatives

the deligne simpson problem a survey abstract the deligne simpson problem dsp respectively the weak dsp is formulated like this give necessary and sufficient conditions for the choice of the conjugacy classes c j gl n c or c j gl n c so that there exist irreducible respectively with trivial centralizer p 1 tuples of matrices m j c j or a j c j satisfying the equality m 1 m p 1 i or a 1 a p 1 0 the matrices m j and a j are interpreted as monodromy operators of regular linear systems and as matrices residua of fuchsian ones on riemann s sphere the present paper offers a survey of the results known up to now concerning the dsp

reflection functors for quiver varieties and weyl group actions we introduce reflectionfunctors on quiver varieties they are hyper kahler isometries between quiver varieties with different parameters related by elements in the weyl group the definition is motivated by the origial reflection functor given by bernstein gelfand ponomarev 1 but they behave much nicely they are isomorphisms and satisfy the weyl group relations as an application we define weyl group representations of homology groups of quiver varieties they are analogues of slodowy s construction of springer representations of the weyl group

multiplicative preprojective algebras middle convolution and the deligne simpson problem abstract we introduce a family of algebras which are multiplicative analogues of preprojective algebras and their deformations as introduced by m p holland and the first author we show that these algebras provide a natural setting for the middle convolution operation introduced by n m katz in his book rigid local systems and put in an algebraic setting by m dettweiler and s reiter and h volklein we prove a homological formula relating the dimensions of hom and ext spaces study varieties of representations of multiplicative preprojective algebras and use these results to study simple representations we apply this work to the deligne simpson problem obtaining a sufficient and conjecturally necessary condition for the existence of an irreducible solution to the equation a 1 a 2 a k 1 with the a i in prescribed conjugacy classes in gl n c

quantized multiplicative quiver varieties abstract beginning with the data of a quiver q and its dimension vector d we construct an algebra d q d q mat d q which is a flat q deformation of the algebra of differential operators on the affine space mat d q the algebra d q is equivariant for an action by a product of quantum general linear groups acting by conjugation at each vertex we construct a quantum moment map for this action and subsequently define the hamiltonian reduction a d q of d q with moment parameter we show that a d q is a flat formal deformation of lusztig s quiver varieties and their multiplicative counterparts for all dimension vectors satisfying a flatness condition of crawley boevey indeed the product on a d q yields a fedosov quantization the of symplectic structure on multiplicative quiver varieties as an application we give a description of the category of representations of the spherical double affine hecke algebra of type a n 1 and its generalization constructed by etingof oblomkov and rains in terms of a quotient of the category of equivariant d q modules by a serre subcategory of aspherical modules

the construction of ale spaces as hyper kahler quotients on decrit la construction d une famille particuliere de 4 varietes hyper kahler les espaces asymptotiquement localement euclidiens ale on decrit une 4 variete de riemann avec juste une extremite qui a l infini ressemble au quotient r 4 de l espace euclidien r 4 par un groupe fini d identifications

k theory of endomorphism rings and of rings of invariants abstract this paper gives the following description ofk0of the endomorphism ring of a finitely generated projective module t heorem let t be a ring and p a finitely generated projective t module let i be the trace ideal of p then k0 end pt is isomorphic to a subgroup of k0 t i if further the natural map k1 t k1 t i is surjective then k0 end pt is isomorphic to the subgroup of k0 t generated by the direct summands of pn for n n as a corollary we can determinek0of the ring of invariants for many free linear actions in particular the following result is proved t heorem let v be a fixed point free linear representation of a finite group g over a field k of characteristic zero and let s v be the symmetric algebra of v let k be any finite dimensional k vector space then k0 s v g ks k s v g ks k similar results are given for suitable noncommutative versions ofs v

symplectic reflection algebras calogero moser space and deformed harish chandra homomorphism to any finite group sp v of automorphisms of a symplectic vector space v we associate a new multi parameter deformation h of the algebra v smash product of with the polynomial algebra on v the parameter runs over points of r where r number of conjugacy classes of symplectic reflections in the algebra h called a symplectic reflection algebra is related to the coordinate ring of a poisson deformation of the quotient singularity v this leads to a symplectic analogue of mckay correspondence which is most complete in case of wreath products if is the weyl group of a root system in a vector space and v then the algebras h are certain rational degenerations of the double affine hecke algebra introduced earlier by cherednik

moduli spaces of meromorphic connections and quiver varieties abstract we describe the moduli spaces of meromorphic connections on trivial holomorphic vector bundles over the riemann sphere with at most one unramified irregular singularity and arbitrary number of simple poles as nakajima s quiver varieties this result enables us to solve partially the additive irregular deligne simpson problem

non reversible stationary measures for infinite interacting particle systems two results concerning the local conditional distributions of a stationary measure for a spin flip process with strictly positive and continuous rates are obtained 1 the local conditional distributions and the rates of the reversed process determine each other 2 either all shift invariant stationary measures are gibbs with the same potential or no shift invariant stationary gibbs measure exist

critical behaviour of nonequilibrium three state systems the phase diagram of two different nonequilibrium three state systems is here studied by means of mfrg and computer simulations critical exponents are obtained by a finite size scaling analysis of the mc data a symmetry argument used by grinstein et al to predict the critical behaviour of two state nonequilibrium systems with up down symmetry is shown to apply to these three state systems with symmetry of interchange between two of those states

geometry and braiding of stokes data fission and wild character varieties a family of new algebraic poisson varieties will be constructed generalising the complex character varieties of riemann surfaces then the well known poisson mapping class group actions on the character varieties will be generalised

monodromy for the hypergeometric function nfn 1 la fonction hypergeometrique le groupe hypergeometrique la forme hermitienne invariante le cas imprimitif theorie de galois differentielle fonctions hypergeometriques algebriques

symplectic manifolds and isomonodromic deformations abstract we study moduli spaces of meromorphic connections with arbitrary order poles over riemann surfaces together with the corresponding spaces of monodromy data involving stokes matrices natural symplectic structures are found and described both explicitly and from an infinite dimensional viewpoint generalising the atiyah bott approach this enables us to give an intrinsic symplectic description of the isomonodromic deformation equations of jimbo miwa and ueno thereby putting the existing results for the six painleve equations and schlesinger s equations into a uniform framework

some explicit solutions of the additive deligne simpson problem and their applications abstract in this paper we construct three infinite series and two extra triples e8 and e 8 of complex matrices b c and a b c of special spectral types associated to simpson s classification in amer math soc proc 1 1992 157 and magyar et al classification in adv math 141 1999 97 this enables us to construct fuchsian systems of differential equations which generalize the hypergeometric equation of gauss riemann in a sense they are the closest relatives of the famous equation because their triples of spectral flags have finitely many orbits for the diagonal action of the general linear group in the space of solutions in all the cases except for e8 we also explicitly construct scalar products such that a b and c are self adjoint with respect to them in the context of fuchsian systems these scalar products become monodromy invariant complex symmetric bilinear forms in the spaces of solutions when the eigenvalues of a b and c are real the matrices and the scalar products become real as well we find inequalities on the eigenvalues of a b and c which make the scalar products positive definite as proved by klyachko spectra of three hermitian or real symmetric matrices b c and a b c form a polyhedral convex cone in the space of triple spectra he also gave a recursive algorithm to generate inequalities describing the cone the inequalities we obtain describe non recursively some faces of the klyachko cone

middle convolution of fuchsian systems and the construction of rigid differential systems abstract in m dettweiler s reiter an algorithm of katz and its application to the inverse galois problem j symbolic comput 30 2000 761 798 a purely algebraic analogon of katz middle convolution functor see n m katz rigid local systems ann of math stud vol 139 princeton university press 1997 is given in this paper we find an explicit riemann hilbert correspondence for this functor this leads to a construction algorithm for differential systems which correspond to rigid local systems on the punctured affine line via the riemann hilbert correspondence

rigid generators of classical groups we introduce the notion of thompson tuples and show they yield rigid generating systems of length 3 of certain classical groups we also give a new construction method for belyi triples and use this to obtain galois realizations over q for all orthogonal and symplectic groups over the field f2 let g1 gr be generators of a finite group g with g1 gr 1 and gi 6 z g the center of g we say they form a quasi rigid generating system if for any generators g 1 g r of g with g 1 g r 1 such that g i is conjugate gi for all i there is g g with g i g 1gig for all i if additionally z g 1 then the system is called rigid we say the system is rational if for each integer m prime to g there is a permutation of r letters such that gm i is conjugate g i for all i one version of the standard rigidity criterion e g v thm 3 25 says if g has a quasi rigid and rational system of r generators then g z g occurs as the galois group of a regular extension of q x with r branch points

the inverse galois problem and rational points on moduli spaces we reduce the regular version of the inverse galois problem for any finite group g to finding one rational point on an infinite sequence of algebraic varieties as a consequence any finite group g is the galois group of an extension l p x with l regular over any pac field p of characteristic zero a special case of this implies that g is a galois group over fp x for almost all primes p

group invariant solutions of differential equations we introduce the concept of a weak symmetry group of a system of partial differential equations that generalizes the nonclassical method introduced by bluman and cole for finding group invariant