global properties of supersymmetric theories and the lens space we compute the supersymmetric partition function on l r 1 times mathbb s 1 the lens space index for 4d gauge theories related by supersymmetric dualities and involving non simply connected groups this computation is sensitive to the global properties of the underlying gauge group and to discrete theta angle parameters and thus distinguishes versions of dualities differing by such we explicitly discuss mathcal n 1 so n c seiberg dualities and mathcal n 4 su n c s dualities

gauge theories labelled by three manifolds we propose a dictionary between geometry of triangulated 3 manifolds and physics of three dimensional n 2 gauge theories under this duality standard operations on triangulated 3 manifolds and various invariants thereof classical as well as quantum find a natural interpretation in field theory for example independence of the sl 2 chern simons partition function on the choice of triangulation translates to a statement that s 3 b partition functions of two mirror 3d n 2 gauge theories are equal three dimensional n 2 field theories associated to 3 manifolds can be thought of as theories that describe boundary conditions and duality walls in four dimensional n 2 scfts thus making the whole construction functorial with respect to cobordisms and gluing

crossed products by endomorphisms of c_0 x algebras abstract in the first part of the paper we develop a theory of crossed products of a c algebra a by an arbitrary not necessarily extendible endomorphism a a we consider relative crossed products c a j where j is an ideal in a and describe up to morita rieffel equivalence all gauge invariant ideals in c a j and give six term exact sequences determining their k theory we also obtain certain criteria implying that all ideals in c a j are gauge invariant and that c a j is purely infinite in the second part we consider a situation where a is a c 0 x algebra and is such that f a f a a a f c 0 x where is an endomorphism of c 0 x pictorially speaking is a mixture of a topological dynamical system x dual to c 0 x and a continuous field of homomorphisms x between the fibers a x x x of the corresponding c bundle for systems described above we establish efficient conditions for the uniqueness property gauge invariance of all ideals and pure infiniteness of c a j we apply these results to the case when x prim a is a hausdorff space in particular if the associated c bundle is trivial we obtain formulas for k groups of all ideals in c a j in this way we constitute a large class of crossed products whose ideal structure and k theory is completely described in terms of x x x x y where y is a closed subset of x

c algebras of toeplitz type associated with algebraic number fields we associate with the ring r of algebraic integers in a number field a c algebra mathfrak t r it is an extension of the ring c algebra mathfrak a r studied previously by the first named author in collaboration with x li in contrast to mathfrak a r it is functorial under homomorphisms of rings it can also be defined using the left regular representation of the ax b semigroup r rtimes r times on ell 2 r rtimes r times the algebra mathfrak t r carries a natural one parameter automorphism group sigma t t in mathbb r we determine its kms structure the technical difficulties that we encounter are due to the presence of the class group in the case where r is not a principal ideal domain in that case for a fixed large inverse temperature the simplex of kms states splits over the class group the partition functions are partial dedekind zeta functions we prove a result characterizing the asymptotic behavior of quotients of such partial zeta functions which we then use to show uniqueness of the beta kms state for each inverse temperature beta in 1 2

the capelli identity the double commutant theorem and multiplicity free actions 0 the capelli identity cal 3 w p 39 is one of the most celebrated and useful formulas of classical invariant theory w d cl z the double commutant theorem w p 91 is likewise a basic result in the general theory of associative algebras both play key roles in weyl s book the classical groups the main purpose of this paper is to demonstrate a close connection between the two in the context of multiplicity free actions ekj of groups on vector spaces the focus of our discussion will be the structure of the differential operators which commute with a multiplicity free action applications include a new derivation of some formulas of shimura shl and rubenthaler and schiffmann rs for b functions associated to hermitian symmetric spaces and a construction of interesting sets of generators for the center of the universal enveloping algebra of gin we also give a detailed discussion of certain aspects of multiplicity free representations here is an overview of the contents of the paper in sect 1 we review the classical capelli identities we observe how their existence is predicted by a double commutant result 1 6 and 1 8 and we show how they can be used to compute b functions we remark that capelli understood that his operators generate the center of the algebra generated by the polarization operators see b1 p 77 also capelli s motivation in introducing his operators was to explain a formula of cayley essentially the computation of a b function capelli s point of view was remarkably modern and structuralist in certain ways more modern even than that of weyl sections 2 through 9 provide a conceptual context for understanding the capelli identities as a feature of multiplicity free actions they contain a general discussion of the structure of g the algebra of polynomial coefficient differential operators which commute with a given group g of linear transformations the results of the discussion are summarized in theorem 9 1 which says in particular that the polynomial coefficient differential operators commuting with a

elliptic hypergeometry of supersymmetric dualities ii orthogonal groups knots and vortices we consider seiberg electric magnetic dualities for 4d mathcal n 1 sym theories with so n gauge group for all such known theories we construct superconformal indices scis in terms of elliptic hypergeometric integrals equalities of these indices for dual theories lead both to proven earlier special function identities and new conjectural relations for integrals in particular we describe a number of new elliptic beta integrals associated with the s confining theories with the spinor matter fields reductions of some dualities from sp 2n to so 2n or so 2n 1 gauge groups are described interrelation of scis and the witten anomaly is briefly discussed possible applications of the elliptic hypergeometric integrals to a two parameter deformation of 2d conformal field theory and related matrix models are indicated connections of the reduced scis with the state integrals of knot theory generalized agt duality for 3 3 d theories and a 2d vortex partition function are described

a kpp road field system with spatially periodic exchange terms we take interest in a reaction diffusion system which has been recently proposed 11 as a model for the effect of a road on propagation phenomena arising in epidemiology and ecology this system consists in coupling a classical fisher kpp equation in a half plane with a line with fast diffusion accounting for a straight road the effect of the line on spreading properties of solutions with compactly supported initial data was investigated in a series of works starting from 11 we recover these earlier results in a more general spatially periodic framework by exhibiting a threshold for road diffusion above which the propagation is driven by the road and the global speed is accelerated we also discuss further applications of our approach which will rely on the construction of a suitable generalized principal eigenvalue and investigate in particular the spreading of solutions with exponentially decaying initial data

shintani barnes zeta and gamma functions abstract we show that shintani s work on multiple zeta and gamma functions can be simplified and extended by exploiting difference equations we re prove many of shintani s formulas and prove several new ones among the latter is a generalization to the shintani barnes gamma functions of raabe s 1843 formula 0 1 log x dx log 2 and a further generalization to the shintani zeta functions these explicit formulas can be interpreted as vanishing period integral side conditions for the ladder of difference equations obeyed by the multiple gamma and zeta functions we also relate barnes triple gamma function to the elliptic gamma function appearing in connection with certain integrable systems

spectroscopy of gauge theories based on exceptional lie groups we generate by computer a basis of invariants for the fundamental representations of the exceptional lie groups e6 and e7 up to degree 18 we discuss the relevance of this calculation for the study of supersymmetric gauge theories and revisit the self dual exceptional models we study the chiral ring of g2 to degree 13 as well as a few classical groups the homological dimension of a ring is a natural estimator of its complexity and provides a guideline for identifying theories that have a good chance to be amenable to a solution

travelling waves spreading and extinction for fisher kpp propagation driven by a line with fast diffusion abstract in an earlier work berestycki et al 2013 we introduced a parabolic system to describe biological invasions in the presence of a line the road with a specific diffusion included in a domain the field subject to fisher kpp propagation we determined the asymptotic spreading speed in the direction of the road w in terms of the various parameters the new result we establish here is the existence of travelling fronts for this system for any speed c w in addition we show that no front can travel with a speed c w we further extend the results to a new model where there is fisher kpp growth on the road embedded in an elsewhere unfavourable plane this is relevant to discuss certain invasions that have been observed along roads we establish sharp conditions for extinction invasion and describe the invasion dynamics

group c algebras as decreasing intersection of nuclear c algebras we prove that for every exact discrete group gamma there is an intermediate rm c ast algebra between the reduced group rm c ast algebra and the intersection of the group von neumann algebra and the uniform roe algebra which is realized as the intersection of a decreasing sequence of isomorphs of the cuntz algebra cal o 2 in particular when gamma has the ap approximation property the reduced group rm c ast algebra is realized in this way we also study extensions of the reduced free group rm c ast algebras and show that any exact absorbing or unital absorbing extension of it by any stable separable nuclear rm c ast algebra is realized in this way

spin chains and gustafson s integrals gustafson s integrals are multidimensional generalizations of the classical mellin barnes integrals we show that some of these integrals arise from relations between matrix elements in sklyanin s representation of separated variables in spin chain models we also present several new integrals

amenable minimal cantor systems of free groups arising from diagonal actions we study amenable minimal cantor systems of free groups arising from the diagonal actions of the boundary actions and certain cantor systems it is shown that every virtually free group admits continuously many amenable minimal cantor systems whose crossed products are mutually non isomorphic kirchberg algebras in the uct class with explicitly determined k theory the technique developed in our study also enables us to compute the k theory of certain amenable minimal cantor systems we apply it to the diagonal actions of the boundary actions and the products of the odometer transformations and determine their k theory then we classify them in terms of the topological full groups continuous orbit equivalence strong orbit equivalence and the crossed products

rarefied elliptic hypergeometric functions abstract two exact evaluation formulae for multiple rarefied elliptic beta integrals related to the simplest lens space are proved they generalize evaluations of the type i and ii elliptic beta integrals attached to the root system c n in a special n 1 case the simplest p 0 limit is shown to lead to a new class of q hypergeometric identities symmetries of a rarefied elliptic analogue of the euler gauss hypergeometric function are described and the respective generalization of the hypergeometric equation is constructed some extensions of the latter function to c n and a n root systems and corresponding symmetry transformations are considered an application of the rarefied type ii c n elliptic hypergeometric function to some eigenvalue problems is briefly discussed

elliptic hypergeometry of supersymmetric dualities we give a full list of known mathcal n 1 supersymmetric quantum field theories related by the seiberg electric magnetic duality conjectures for su n sp 2n and g 2 gauge groups many of the presented dualities are new not considered earlier in the literature for all these theories we construct superconformal indices and express them in terms of elliptic hypergeometric integrals this gives a systematic extension of the related romelsberger and dolan osborn results equality of indices in dual theories leads to various identities for elliptic hypergeometric integrals about half of them were proven earlier and another half represents new challenging conjectures in particular we conjecture a dozen new elliptic beta integrals on root systems extending the univariate elliptic beta integral discovered by the first author

nuclearity of semigroup c algebras and the connection to amenability abstract we study c algebras associated with subsemigroups of groups for a large class of such semigroups including positive cones in quasi lattice ordered groups and left ore semigroups we describe the corresponding semigroup c algebras as c algebras of inverse semigroups groupoid c algebras and full corners in associated group crossed products these descriptions allow us to characterize nuclearity of semigroup c algebras in terms of faithfulness of left regular representations and amenability of group actions moreover we also determine when boundary quotients of semigroup c algebras are uct kirchberg algebras this leads to a unified approach to cuntz algebras and ring c algebras

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

some q beta and mellin barnes integrals on compact lie groups and lie algebras multidimensional generalizations of beta type integrals of bames ramanujan askey wilson and others are evaluated these integrals are analogues of the summation theorems for multilateral hypergeometric series associated to the simple lie algebras of classical type and type g2 many of these integrals can also be written as group integrals over a compact lie group or conjugation invariant integrals over the corresponding lie algebra

gauge theories and macdonald polynomials we study the mathcal n 2 four dimensional superconformal index in various interesting limits such that only states annihilated by more than one supercharge contribute extrapolating from the su 2 generalized quivers which have a lagrangian description we conjecture explicit formulae for all a type quivers of class mathcal s which in general do not have one we test our proposals against several expected dualities the index can always be interpreted as a correlator in a two dimensional topological theory which we identify in each limit as a certain deformation of two dimensional yang mills theory the structure constants of the topological algebra are diagonal in the basis of macdonald polynomials of the holonomies

semigroup crossed products and the toeplitz algebras of nonabelian groups abstract we consider the quasi lattice ordered groups g p recently introduced by nica we realise their universal toeplitz algebra as a crossed product b p p by a semigroup of endomorphisms and show that the toeplitz representation is faithful precisely when an amenability condition is satisfied we then show directly that many interesting free products are amenable the final result contains theorems of cuntz and dinh on the uniqueness of toeplitz cuntz algebras as special cases

elliptic extension of gustafson s q integral of type g2 abstract the evaluation formula for an elliptic beta integral of type g 2 is proved the integral is expressed by a product of ruijsenaars elliptic gamma functions and the formula includes that of gustafson s q beta integral of type g 2 as a special limiting case as p 0 the elliptic beta integral of type b c 1 by van diejen and spiridonov is effectively used in the proof of the evaluation formula

ring c algebras we associate reduced and full c algebras to arbitrary rings and study the inner structure of these ring c algebras as a result we obtain conditions for them to be purely infinite and simple we also discuss several examples originally our motivation comes from algebraic number theory

transformations of elliptic hypergeometric integrals we prove a pair of transformations relating elliptic hypergeometric integrals of different dimensions corresponding to the root systems bc n and a n as a special case we recover some integral identities conjectured by van diejen and spiridonov for bc n we also consider their type ii integral their proof of that integral together with our transformation gives rise to pairs of adjoint integral operators a different proof gives rise to pairs of adjoint difference operators these allow us to construct a family of biorthogonal abelian functions generalizing the koornwinder polynomials and satisfying the analogues of the macdonald conjectures finally we discuss some transformations of type ii style integrals in particular we find that adding two parameters to the type ii integral gives an integral invariant under an appropriate action of the weyl group e 7

first order analytic difference equations and integrable quantum systems we present a new solution method for a class of first order analytic difference equations the method yields explicit minimal solutions that are essentially unique special difference equations give rise to minimal solutions that may be viewed as generalized gamma functions of hyperbolic trigonometric and elliptic type euler s gamma function being of rational type we study these generalized gamma functions in considerable detail the scattering and weight functions u and w functions associated to various integrable quantum systems can be expressed in terms of our generalized gamma functions we obtain detailed information on these u and w functions exploiting the difference equations they satisfy

semigroup c algebras of ax b semigroups we study semigroup c algebras of ax b semigroups over integral domains the goal is to generalize several results about c algebras of ax b semigroups over rings of algebraic integers we prove results concerning k theory and structural properties like the ideal structure or pure infiniteness our methods allow us to treat ax b semigroups over a large class of integral domains containing all noetherian integrally closed domains and coordinate rings of affine varieties over infinite fields

k theoretic duality for hyperbolic dynamical systems the k theoretic analog of spanier whitehead duality for noncommutative c algebras is shown to hold for the ruelle algebras associated to irreducible smale spaces this had previously been proved only for shifts of finite type implications of this result as well as relations to the baum connes conjecture and other topics are also considered hn i x h i s n 1 x given by slant product this was generalized by spanier and whitehead to allow s n 1 x to be replaced by a space of the homotopy type of a finite complex dn x for which the analog of the above relations hold it is in this form that these notions extend naturally to the noncommutative setting in noncommutative topology the roles of homology and cohomology are played by k theory and k homology these have been combined into a bivariant theory by kasparov for which one has k theory k a kk c a and k homology k a kk a c kasparov s theory comes equipped with a product that is the analog of the slant product used above of course noncommutative c algebras play the role of algebras of continuous functions on ordinary spaces with this in hand the generalization of spanier whitehead duality to this setting is easily accomplished let a be a c algebra we will consider indices in k theory to be modulo 2 a c algebra dn a is a spanier whitehead n dual of a or simply a dual when the context

a new approach to recent constructions of c algebras from modular index theory abstract we present a new approach to c algebras recently constructed in the context of modular index theory by carey phillips putnam and rennie it turns out that their constructions can be identified with full corners of ideals in semigroup c algebras this new point of view leads to a systematic analysis of these algebras and their canonical maximal abelian subalgebras

embeddings for the jordan algebra of a bilinear form abstract let k be a field of characteristic zero and let j be a jordan algebra with a formal trace we prove that the algebra j can be embedded into a jordan algebra of a non degenerate symmetric bilinear form over some associative and commutative k algebra c if and only if j satisfies all trace identities of the jordan algebra of a non degenerate symmetric bilinear form over the field k this is an extension of results of procesi and berele concerning the analogous problem for the associative matrix algebras and the matrix algebras with involution as a consequence of these results we also prove that the ideal of all trace identities of the jordan algebra of a non degenerate symmetric bilinear form over k satisfies the specht property

pure infiniteness and ideal structure of c algebras associated to fell bundles abstract we investigate structural properties of the reduced cross sectional algebra c r b of a fell bundle b over a discrete group g conditions allowing one to determine the ideal structure of c r b are studied notions of aperiodicity paradoxicality and b infiniteness for the fell bundle b are introduced and investigated by themselves and in relation to the partial dynamical system dual to b several criteria of pure infiniteness of c r b are given it is shown that they generalize and unify corresponding results obtained in the context of crossed products by the following duos laca spielberg 34 jolissaint robertson 21 sierakowski rordam 47 giordano sierakowski 18 and ortega pardo 39 for exact separable fell bundles satisfying the residual intersection property primitive ideal space of c r b is determined the results of the paper are shown to be optimal when applied to graph c algebras applications to a class of exel larsen crossed products are presented

simple purely infinite c algebras and n filling actions let n be a positive integer we introduce a concept which we call the n filling property for an action of a group on a separable unital c algebra a if a c is a commutative unital c algebra and the action is induced by a group of homeomorphisms of then the n filling property reduces to a weak version of hyperbolicity the n filling property is used to prove that certain crossed product c algebras are purely infinite and simple a variety of group actions on boundaries of symmetric spaces and buildings have the n filling property an explicit example is the action of sln z on the projective n space

the speed of propagation for kpp type problems ii general domains this paper is devoted to nonlinear propagation phenomena in general unbounded domains of rn for reaction diffusion equations with kolmogorov petrovsky piskunov kpp type nonlinearities this article is the second in a series of two and it is the follow up of the paper 7 which dealt which the case of periodic domains this paper is concerned with general domains and we give various definitions of the spreading speeds at large times for solutions with compactly supported initial data we study the relationships between these new notions and analyze their dependence on the geometry of the domain and on the initial condition some a priori bounds are proved for large classes of domains the case of exterior domains is also discussed in detail lastly some domains which are very thin at infinity and for which the spreading speeds are infinite are exhibited the construction is based on some new heat kernel estimates in such domains

spreading in a cone for the fisher kpp equation abstract in this paper we consider the spreading phenomena in the fisher kpp equation in a high dimensional cone with dirichlet boundary condition we show that any solution starting from a nonnegative and compact supported initial data spreads and converges to the unique positive steady state moreover the asymptotic spreading speeds of the front in all directions pointing to the opening are c 0 which is the minimal speed of the traveling wave solutions of the 1 dimensional fisher kpp equation surprisingly they do not depend on the shape of the cone the propagating directions and the boundary condition

k theory of certain purely infinite crossed products it was shown by rordam and the second named author that a countable group g admits an action on a compact space such that the crossed product is a kirchberg algebra if and only if g is exact and non amenable this construction allows a certain amount of choice we show that different choices can lead to different algebras at least with the free group

asymptotic spreading in heterogeneous diffusive excitable media abstract we establish propagation and spreading properties for nonnegative solutions of nonhomogeneous reaction diffusion equations of the type t u a t x u q t x u f t x u with compactly supported initial conditions at t 0 here a q f have a general dependence in t r and x r n we establish properties of families of propagation sets which are defined as families of subsets s t t 0 of r n such that lim inf t inf x s t u t x 0 the aim is to characterize such families as sharply as possible in particular we give some conditions under which 1 a given path t t 0 where t r n forms a family of propagation sets or 2 one can find such a family with s t x r n x r t and lim t r t this second property is called here complete spreading furthermore in the case q 0 and inf t x r r n f u t x 0 0 as well as under some more general assumptions we show that there is a positive spreading speed that is r t can be chosen so that lim inf t r t t 0 in the general case we also show the existence of an explicit upper bound c 0 such that lim sup t r t t c on the other hand we provide explicit examples of reaction diffusion equations such that for an arbitrary e 0 any family of propagation sets s t t 0 has to satisfy s t x r n x e t for large t in connection with spreading properties we derive some new uniqueness results for the entire solutions of this type of equations lastly in the case of space time periodic media we develop a new approach to characterize the largest propagation sets in terms of eigenvalues associated with the linearized equation in the neighborhood of zero

bc_n symmetric abelian functions we construct a family of bc n symmetric biorthogonal abelian functions generalizing koornwinder s orthogonal polynomials and prove a number of their properties most notably analogues of macdonald s conjectures the construction is based on a direct construction for a special case generalizing okounkov s interpolation polynomials we show that these interpolation functions satisfy a collection of generalized hypergeometric identities including new multivariate elliptic analogues of jackson s summation and bailey s transformation

laplace operator and polynomial invariants abstract let a be a finite dimensional simple algebra not necessarily associative over the field of complex numbers c and let g denote the automorphism group aut a suppose that a has a symmetric nondegenerate associative g invariant bilinear form x y and a compact real form i e a subalgebra b over r of dimension dim r b dim c a where a is equal to the span of b over c and the restriction of x y to b is positive definite we describe generators of the algebra of polynomial g invariants of a system of several vectors from a in terms of x y and laplace operators in particular we give generators of the algebra of polynomial invariants of the adjoint representation of a simple linear algebraic group of any exceptional type e 6 as a consequence we get the first main theorem on matrix invariants invariants of minimal representation of g 2 and f 4

new elliptic solutions of the yang baxter equation we consider finite dimensional reductions of an integral operator with the elliptic hypergeometric kernel describing the most general known solution of the yang baxter equation with a rank 1 symmetry algebra the reduced r operators reproduce at their bottom the standard baxter s r matrix for the 8 vertex model and sklyanin s l operator the general formula has a remarkably compact form and yields new elliptic solutions of the yang baxter equation based on the finite dimensional representations of the elliptic modular double the same result is also derived using the fusion formalism

the shape of expansion induced by a line with fast diffusion in fisher kpp equations we establish a new property of fisher kpp type propagation in a plane in the presence of a line with fast diffusion we prove that the line enhances the asymptotic speed of propagation in a cone of directions past the critical angle given by this cone the asymptotic speed of propagation coincides with the classical fisher kpp invasion speed several qualitative properties are further derived such as the limiting behaviour when the diffusion on the line goes to infinity

extended global symmetries for 4d mathcal n 1 sqcd theories in spiridonov and vartanov 2010 nucl phys b 824 192 the authors using the superconformal index technique found that four dimensional sqcd theory with su 2 gauge group and four flavors has 72 dual representations recently in dimofte and gaiotto 2012 j high energy phys jhep10 2012 129 the authors showed that these dual theories when coupled to five dimensional 5d hypermultiplets with specific boundary conditions have an extended e7 global symmetry in this work we find that for a reduced theory with three flavors the explicit su 6 global symmetry is enhanced to an e6 symmetry in the presence of 5d hypermultiplets we also show connections between indices of different theories in three and four dimensions

the 2 adic ring c algebra of the integers and its representations abstract we study the 2 adic version of the ring c algebra of the integers first we work out the precise relation between the cuntz algebra o 2 and our 2 adic ring c algebra in terms of representations secondly we prove a 2 adic duality theorem identifying the crossed product arising from 2 adic affine transformations on the 2 adic numbers with the analogous crossed product algebra over the real numbers and finally as an outcome of this duality result we construct an explicit imprimitivity bimodule and prove that it transports one canonical representation into the other

a class of c algebras generalizing both graph algebras and homeomorphism c algebras iv pure infiniteness this is the final one in the series of papers where we introduce and study the c algebras associated with topological graphs in this paper we get a sufficient condition on topological graphs so that the associated c algebras are simple and purely infinite using this result we give one method to construct all kirchberg algebras as c algebras associated with topological graphs

boundary quotients and ideals of toeplitz c algebras of artin groups abstract we study the quotients of the toeplitz c algebra of a quasi lattice ordered group g p which we view as crossed products by a partial actions of g on closed invariant subsets of a totally disconnected compact hausdorff space the nica spectrum of g p our original motivation and our main examples are drawn from right angled artin groups but many of our results are valid for more general quasi lattice ordered groups we show that the nica spectrum has a unique minimal closed invariant subset which we call the boundary spectrum and we define the boundary quotient to be the crossed product of the corresponding restricted partial action the main technical tools used are the results of exel laca and quigg on simplicity and ideal structure of partial crossed products which depend on amenability and topological freeness of the partial action and its restriction to closed invariant subsets when there exists a generalised length function or controlled map defined on g and taking values in an amenable group we prove that the partial action is amenable on arbitrary closed invariant subsets the topological freeness of the boundary action depends on topological freeness of the restriction to a certain lattice subgroup of g the core of g p which often turns out to be trivial our main results are obtained for right angled artin groups with trivial centre that is those with no cyclic direct factor they include a presentation of the boundary quotient in terms of generators and relations that generalises cuntz s presentation of o n a proof that the boundary quotient is purely infinite and simple and a parametrisation of the ideals of the toeplitz c algebra in terms of subsets of the standard generators of the artin group

velocity enhancement of reaction diffusion fronts by a line of fast diffusion we study the velocity of travelling waves of a reaction diffusion system coupling a standard reaction diffusion equation in a strip with a one dimensional diffusion equation on a line we show that it grows like the square root of the diffusivity on the line this generalises a result of berestycki roquejoffre and rossi in the context of fisher kpp propagation where the question could be reduced to algebraic computations thus our work shows that this phenomenon is a robust one the ratio between the asymptotic velocity and the square root of the diffusivity on the line is characterised as the unique admissible velocity for fronts of an hypoelliptic system which is shown to admit a travelling wave profile

inversions of integral operators and elliptic beta integrals on root systems we prove a novel type of inversion formula for elliptic hypergeometric integrals associated to a pair of root systems using the a c inversion formula to invert one of the known c type elliptic beta integrals we obtain a new elliptic beta integral for the root system of type a validity of this integral is established by a different method as well

the importance of the selberg integral it has been remarked that a fair measure of the impact of atle selberg s work is the number of mathematical terms that bear his name one of these is the selberg integral an n dimensional generalization of the euler beta integral we trace its sudden rise to prominence initiated by a question to selberg from enrico bombieri more than thirty years after its initial publication in quick succession the selberg integral was used to prove an outstanding conjecture in random matrix theory and cases of the macdonald conjectures it further initiated the study of q analogues which in turn enriched the macdonald conjectures we review these developments and proceed to exhibit the sustained prominence of the selberg integral as evidenced by its central role in random matrix theory calogero sutherland quantum many body systems knizhnik zamolodchikov equations and multivariable orthogonal polynomial theory

strongly multiplicity free modules for lie algebras and quantum groups abstract let u be either the universal enveloping algebra of a complex semisimple lie algebra g or its drinfel d jimbo quantisation over the field c z of rational functions in the indeterminate z we define the notion of strongly multiplicity free smf for a finite dimensional u module v and prove that for such modules the endomorphism algebras end u v r are generic in the sense that in the classical unquantised case they are quotients of kohno s infinitesimal braid algebra t r while in the quantum case they are quotients of the group ring c z b r of the r string braid group b r in the classical case the generators are generalisations of the quadratic casimir operator c of u while in the quantum case they arise from r matrices which may be thought of as square roots of a quantum analogue of c in a completion of u r this unifies many known results and brings some new cases into their context these include the irreducible 7 dimensional module in type g 2 and arbitrary irreducibles for sl 2 the work leads naturally to questions concerning non semisimple deformations of the relevant endomorphism algebras which arise when the ground rings are varied

non simple purely infinite c algebras a c algebra a is defined to be purely infiniteif there are no characters on a a nd if for every pair of positive elements a b in a such that b lies in the closed two sided ideal generated by a there exists a sequence rn in a such that rn arn b this definition agrees with the usual definition by j cuntz when a is simple it is shown that the property of being purely infinite is preserved under extensions morita equivalence inductive limits and it passes to quotients and to hereditary sub c algebras it is shown that a is purely infinite for every c algebra a purely infinite c algebras admit no traces and conversely an approximately divisible exact c algebra is purely infinite if it admits no nonzero trace

the elliptic gamma function and sl 3 z z3 abstract the elliptic gamma function is a generalization of the euler gamma function and is associated to an elliptic curve its trigonometric and rational degenerations are the jackson q gamma function and the euler gamma function respectively the elliptic gamma function appears in baxter s formula for the free energy of the eight vertex model and in the hypergeometric solutions of the elliptic qkzb equations in this paper the properties of this function are studied in particular we show that elliptic gamma functions are generalizations of automorphic forms of g sl 3 z z 3 associated to a non trivial class in h3 g z

the structure ofc algebras associated with hyperbolic dynamical systems we consider the stable unstablec algebras and the ruelle algebras associated to a mixing smale space in the case of a shift of finite type these are the af algebras studied by w krieger and the stabilized cuntz krieger algebras in the general case we show that the stable and unstable algebras are simple and amenable we also show that the ruelle algebras are simple amenable and purely infinite

families of type iii kms states on a class of c algebras containing on and qn abstract we construct a family of purely infinite c algebras q for 0 1 that are classified by their k groups there is an action of the circle t with a unique kms state on each q for 1 n q 1 n o n with its usual t action and kms state for p q rational in lowest terms q o n n q p 1 with uhf fixed point algebra of type p q for any n 1 q o n for infinitely many with distinct kms states and uhf fixed point algebras for any 0 1 q o for irrational the fixed point algebras are not af and the q are usually not cuntz algebras for transcendental k 1 q k 0 q z so that q is cuntz q n cuntz 2008 16 if and 1 are both algebraic integers the only o n which appear are those for which n 3 mod 4 for each the representation of q defined by the kms state generates a type iii factor these algebras fit into the framework of modular index theory twisted cyclic theory of carey et al 2010 8 carey et al 2009 12 carey et al in press 5

semigroup c algebras and amenability of semigroups we construct reduced and full semigroup c algebras for left cancellative semigroups our new construction covers particular cases already considered by a nica and also toeplitz algebras attached to rings of integers in number fields due to j cuntz moreover we show how left amenability of semigroups can be expressed in terms of these semigroup c algebras in analogy to the group case