rational singularities and almost split sequences the main aim of this paper is to relate almost split sequences to singularity theory by showing that the mckay quiver built from the finite dimen sional representations of a finite subgroup g of gl 2 c where c is the complex numbers is isomorphic to the ar quiver of the reflexive modules of the quotient singularity associated with g over the past decade almost split sequences have been playing an increasingly important role in the representation theories of finite dimensional algebras and classical orders see 5 and 3 8 for basic existence theorems in these contexts while they have been known for some time to exist in higher dimensional situations 3 it has not been at all clear how they related to singularity theory if at all the main aim of this paper is to relate almost split sequences to singularity theory by showing that the mckay quiver built from the fimte dimensional complex represen tations of a finite subgroup g of gl 2 c where c is the complex numbers is isomorphic to the ar quiver of the reflexive modules over the quotient singularity r associated with g as in the case of finite dimensional algebras the ar quiver of reflexive r modules is defined in terms of the almost split sequences of reflexive r modules in the case g c sl 2 c mckay observed that the underlying graph of the mckay quiver with the trivial module removed is isomorphic to the desingulariza tion graph of the associated singularity various explanations of this phenomenon have been given by knorrer gonzalez sprinberg verdier 6 and artin verdier 1 which along the way have established most explicitly in 1 a natural one to one correspondence between the indecomposable reflexive r modules and the nodes of the desingularization graph but why the almost split sequences describe the edge of the desingularization graph still remains to be explained an effort has been made to make this paper as self contained as possible in particular no prior knowledge of almost split sequences is required before describing the contents of the six sections of this paper we fix some notation throughout this paper g is a finite group k an algebraically closed field of characteristic not dividing the order of g and v a two dimensional k representation of g setting s k x y the k algebra of formal power series the two dimen sional representation v gives a linear action of g on s as a group of k algebra automorphisms we denote by s g the skew group ring given by this action

maximum principles and the method of upper and lower solutions for time periodic problems of the telegraph equations abstract this paper deals with the existence of bounded time periodic solutions for the nonlinear telegraph equation u t t u x x c u t f t x u t x r 2 where c 0 is a constant f c r 3 r is 2 periodic in t we build a maximum principle for the time periodic solutions of the corresponding linear telegraph equation using this maximum principle we develop a method of the upper and lower solutions for the time periodic problem of the nonlinear telegraph equation and obtain some existence and uniqueness results

general quantum surface of section method a new method for exact quantization of general bound hamiltonian systems is presented it is the quantum analogue of the classical poincare surface of section sos reduction of classical dynamics the quantum poincare mapping is shown to be the product of the two generalized non unitary but compact on shell scattering operators of the two scattering hamiltonians which are obtained from the original bound one by cutting the f dimensional configuration space cs the along the f 1 dimensional configurational sos and attaching the flat quasi one dimensional waveguides instead the quantum poincare mapping has fixed points at the eigenenergies of the original bound hamiltonian the energy dependent quantum propagator e h 1 can be decomposed in terms of the four energy dependent propagators which propagate from and or to cs to and or from configurational sos which may generally be composed of many disconnected parts i show that in the semiclassical limit h cross to 0 the quantum poincare mapping converges to the bogomolny propagator and explain how the higher order semiclassical corrections can be obtained systematically

quantum maps from transfer operators abstract the selberg zeta function s s yields an exact relationship between the periodic orbits of a fully chaotic hamiltonian system the geodesic flow on surfaces of constant negative curvature and the corresponding quantum system the spectrum of the laplace beltrami operator on the same manifold it was found that for certain manifolds s s can be exactly ewritten as the fredholm grothendieck determinant det 1 t s where ts is a generalization of the ruelle perron frobenius transfer operator we present an alternative derivation of this result yielding a method to find not only the spectrum but also the eigenfunctions of the laplace beltrami operator in terms of eigenfunctions of ts various properties of the transfer operator are investigated both analytically and numerically for several systems

large amplitude time periodic solutions of a semilinear wave equation abstract this paper studies the existence of periodic solutions for a family of semilinear wave equations where the restoring force is independent of time monotone and grows at a more rapid rate than linear near infinity with appropriate technical assumptions it is shown that there is an unbounded sequence of such free vibrations i e there are solutions of arbitrarily large amplitude if the restoring force is independent of x the monotonicity assumption can be omitted

vanishing of ext cluster tilting modules and finite global dimension of endomorphism rings let r be a cohen macaulay ring and m a maximal cohen macaulay r module inspired by recent striking work by iyama burban iyama keller reiten and van den bergh we study the question of when the endomorphism ring of m has finite global dimension via certain conditions about vanishing of ext modules we are able to strengthen certain results by iyama on connections between a higher dimension version of auslander correspondence and existence of non commutative crepant resolutions we also recover and extend to positive characteristics a recent theorem by burban iyama keller reiten on cluster tilting objects in the category of maximal cohen macaulay modules over reduced 1 dimensional hypersurfaces

on a cluster category of infinite dynkin type and the relation to triangulations of the infinity gon let k be a field and let d be a k linear algebraic triangulated category with split idempotents let be the suspension functor of d and let s be a 2 spherical object of d that is the morphism space d s i s is k for i 0 and i 2 and vanishes otherwise assume that s classically generates d that is each object of d can be built from s using de suspensions direct sums direct summands and distinguished triangles it was proved in 15 thm 2 1 that d is uniquely determined by these properties as we will explain d is a good candidate for a cluster category of dynkin type a for instance we show that there is a bijection between the cluster tilting subcategories of d and certain triangulations of the gon we use this to give an example of a subcategory a which is weak cluster tilting that is satisfies a 1a a but fails to be functorially finite perpendicular

bogomolny s semiclassical transfer operator for rotationally invariant integrable systems the transfer operator due to bogomolny provides a convenient method for obtaining a semiclassical approximation to the energy eigenvalues of a quantum system no matter what the nature of the analogous classical system in this paper the method is applied to integrable systems which are rotationally invariant in two and three dimensions in two dimensions the transfer operator is expanded in a fourier series in the angle variable while in three dimensions it is expanded in spherical harmonics in both cases when the fourier coefficients are evaluated using the stationary phase approximation we arrive at the einstein brillouin keller quantization conditions the associated maslov indices are shown to agree with the results calculated by well known simple rules the theory is applied to several rotationally invariant systems including the hydrogen atom and the isotropic harmonic oscillator in two and three dimensions the circle billiard a billiard inside a spherical cavity and a harmonic potential with a singular magnetic flux line

auslander gorenstein algebras and precluster tilting abstract we generalize the notions of n cluster tilting subcategories and selfinjective algebras into n precluster tilting subcategories and n selfinjective algebras where we show that a subcategory naturally associated to n precluster tilting subcategories has a higher auslander reiten theory furthermore we give a bijection between n precluster tilting subcategories and n minimal auslander gorenstein algebras which is a higher dimensional analog of auslander solberg correspondence 8 as well as a gorenstein analog of n auslander correspondence 22 the auslander reiten theory associated to an n precluster tilting subcategory is used to classify the n minimal auslander gorenstein algebras into four disjoint classes our method is based on relative homological algebra due to auslander solberg

double periodic boundary value problem for non linear dissipative hyperbolic equations let z and r be the set of all integers and real numbers respectively and let q 0 27c x 0 2711 let l q be the space of measurable real valued functions u sz r which are lebesgue integrable over 52 with usual norm ii ji l1 let l2 q be the space of measurable real valued functions u sz r which are lebesgue square integrable over q with usual inner product and usual norm i ljl2 and let l q be the space of measurable real valued functions u q r which are essentially bounded with the norm 1 u 11 lm ess sup 1 u t x i i x e r let c q be the space of all continuous functions u 52 f r such that the partial derivatives up to order k with respect to both variables are continuous on 52 while c q is used for co q with the usual norm 11 iloo

periodicity of d cluster tilted algebras abstract it is well known that any maximal cohen macaulay module over a hypersurface has a periodic free resolution of period 2 auslander reiten 1996 4 and buchweitz 1998 8 have used this periodicity to explain the existence of periodic projective resolutions over certain finite dimensional algebras which arise as stable endomorphism rings of cohen macaulay modules these algebras are in fact periodic meaning that they have periodic projective resolutions as bimodules and thus periodic hochschild cohomology as well the goal of this article is to generalize this construction of periodic algebras to the context of iyama s higher ar theory let c be a maximal d 1 orthogonal subcategory of an exact frobenius category b and start by studying the projective resolutions of finitely presented functors on the stable category c over both c and c under the assumption that c is fixed by d we show that d induces the 2 d th syzygy on mod c if c has finite type i e if c add t for a d cluster tilting object t then we show that the stable endomorphism ring of t has a quasi periodic resolution over its enveloping algebra moreover this resolution will be periodic if some power of d is isomorphic to the identity on c it follows in particular that 2 c y tilted algebras arising as stable endomorphism rings of cohen macaulay modules over curve singularities as in the work of burban iyama keller and reiten 2008 9 have periodic bimodule resolutions of period 4

almost split sequences in subcategories let r be a commutative artin ring and let i be an r algebra which is a finitely generated r module in 5 auslander and reiten introduced the notion of an almost split sequence in mod a the category of finitely generated modules over a and the existence and uniqueness of such short exact sequences were established there the theory of almost split sequences developed further and similar results for certain subcategories of mod i were proved by bautista and martinez 8 and roggenkamp 9 the main purpose of this paper is to develop a more general theory for subcategories of mod i having almost split sequences which has the previous examples as special cases notions closely related to that of almost split sequences are those of minimal left and right almost split morphisms these notions were introduced in 61 for mod i and further developed in 7 for subcategories of mod ii this paper is based upon 7 and we therefore use the same notations and conventions as used there we now give some definitions and then proceed to describe the content of the paper section by section let c be a subcategory of mod ii then a morphism g b c in c is said to be a right almost split morphism in c if i g is not a splittable epimorphism and ii whenever there is a nonsplittable epimorphism h c c in c there exists an h c b such that gh h dually a morphismfi a f b in c is said to be left almost split if i f is not a splittable monomorphism and ii whenever there is a nonsplittable monomorphism h a a there exists an h b a such that h f h we say that c has right almost split morphisms if for all indecomposable objects

unstable periodic orbits and semiclassical quantisation the bohr sommerfeld quantisation condition has a meaningful extension to classically chaotic systems whose periodic unstable orbits are isolated it provides a semiclassical euler factorisation for the functional determinant of the quantal hamiltonian in contrast to the hadamard infinite product over the eigenvalues by which the exact determinant is defined

higher nakayama algebras i construction we introduce higher dimensional analogues of the nakayama algebras from the viewpoint of iyama s higher auslander reiten theory more precisely for each nakayama algebra a and each positive intege

quantum section method for the soft stadium abstract the soft stadium is defined by a monomial potential with exponent as a parameter such that corresponds to the billiard the practical use of the quantum section method depends only on the partial separability of the system on both sides of the section which holds for all s in particular for 1 0 the system becomes globally separable allowing for a general test of the method for various values of the parameter we also tested the use of the asymptotic wkb type approximation in the construction of green s functions and asymptotic overlap integrals to obtain higher energy eigenvalues we show these approximations to be reliable

nonlinear boundary value problems at resonance soit un espace de mesure fini et h l 2 l espace des fonctions a valeurs reelles de carre integrable mesurables de lebesgue sur on considere l equation lu r n u g u h ou h h l dom l h h est un operateur lineaire auto adjoint densement defini r n appartient au spectre l g r r est une fonction qui peut etre non lineaire on donne des resultats d existence generaux

almost split sequences in dimension two a does not split b if h x r c is a morphism in c which is not a splittable epimorphism in c then there is a t x c such that gt h c if j ci y is a morphism in c which is not a splittable monomorphism in c then there is an s c y in c such that j sf almost split sequences were first introduced by us in the case a is the category of finitely generated modules over an artin algebra i and c a see 6 71 the reader is referred to the expository articles 11 121 for an account of the role almost split sequences have played in the theory of the representation theory of artin algebras that almost split sequences exist for various subcategories of module categories over a much wider class of rings than artin algebras was first shown in 2 31 where various existence theorems for almost split sequences were given these results seemed to indicate that there might be connections between the structure and existence of almost split sequences and algebraic geometry recent developments show that this is indeed the case in 4 it is shown that the structure of the almost split sequences in the category of reflexive modules of a complete rational double point over the complex numbers determines the desingularization graph of the singularity also it has been shown in s that a complete cohen macaulay local ring s is an isolated singularity if and only if the category of cohen macaulay

numerical accuracy of bogomolny s semiclassical quantization scheme in quantum billiards we use the semiclassical quantization scheme of bogomolny to calculate eigenvalues of the limacon quantum billiard corresponding to a conformal map of the circle billiard we use the entire billiard boundary as the chosen surface of section and use a finite approximation for the transfer operator in coordinate space computation of the eigenvalues of this matrix combined with a quantization condition determines a set of semiclassical eigenvalues which are compared with those obtained by solving the schrodinger equation the classical dynamics of this billiard system undergoes a smooth transition from integrable circle to completely chaotic motion thus providing a test of bogomolny s semiclassical method in coordinate space in terms of the morphology of the wavefunction we analyse the results for billiards which exhibit both soft and hard chaos

auslander reiten sequences and t structures on the homotopy category of an abelian category we use t structures on the homotopy category kb r mod for an artin algebra r and watts representability theorem to give an existence proof for auslander reiten sequences of r modules this framework naturally leads to a notion of generalized or higher auslander reiten sequences

higher auslander reiten sequences and t structures abstract let r be an artin algebra and c an additive subcategory of mod r we construct a t structure on the homotopy category k c and argue that its heart h c is a natural domain for higher auslander reiten ar theory in the paper 5 we showed that k mod r is a natural domain for classical ar theory here we show that the abelian categories h mod r and h c interact via various functors if c is functorially finite then h c is a quotient category of h mod r we illustrate our theory with two examples when c is a maximal n orthogonal subcategory iyama developed a higher ar theory see 10 in this case we show that the simple objects of h c correspond to iyama s higher ar sequences and derive his higher ar duality from the existence of a serre functor on the derived category d b h c the category o of a complex semi simple lie algebra fits into higher ar theory in the situation when r is the coinvariant algebra of the weyl group

the hodge theory of soergel bimodules we prove soergel s conjecture on the characters of indecomposable soergel bimodules we deduce that kazhdan lusztig polynomials have positive coecients for arbitrary coxeter systems using results of soergel one may deduce an algebraic proof of the kazhdan lusztig conjecture

a note on abelian quotient categories abstract let c be a triangulated category with a serre functor s and x a non zero contravariantly finite rigid subcategory of c then x is cluster tilting if and only if the quotient category c x is abelian and s x x 2 as an application this result generalizes the work by beligiannis

the classification of τ tilting modules over nakayama algebras abstract in this paper we study tilting modules over nakayama algebras we establish bijections between tilting modules triangulations of a polygon with a puncture and certain integer sequences moreover we give an algorithm to construct the hasse quiver of support tilting modules by using drozd kirichenko rejection

a rule for quantizing chaos the authors find a real function delta e whose zeros approximate the quantum energy levels of a system with chaotic classical trajectories delta e is a finite sum over combinations of classical periodic orbits it is obtained from gutzwiller s infinite and divergent sum 1982 representing the spectral density in terms of periodic orbits by means of a resummation conjectured by analogy with a derivation of the riemann siegel formula for the riemann zeros they assess the practicality of the quantization condition

auslander type conditions and cotorsion pairs we study the properties of rings satisfying auslander type conditions if an artin algebra satisfies the auslander condition that is is an gorenstein artin algebra then we construct two kinds of subcategories which form functorially finite cotorsion pairs

higher dimensional auslander reiten theory on maximal orthogonal subcategories abstract we introduce the concept of maximal n 1 orthogonal subcategories over artin algebras and orders and develop n 1 dimensional auslander reiten theory on them we give the n auslander reiten translation and the n auslander reiten duality then show the existence of n almost split sequences and n fundamental sequences we give some examples

derived counterparts of fusion categories of quantum groups abstract in this paper we study derived versions of the fusion category associated to lusztig s quantum group u q the categories that arise in this way are not semisimple but recovers the usual fusion ring when passing to grothendieck rings on the derived level it turns out that it is possible to define fusion for u q without using the notion of tilting modules hence we arrive at a definition of the fusion ring that makes sense in any spherical category we apply this new definition to the small quantum group and we relate it with some rings appearing in 22

d representation finite self injective algebras abstract in this paper we initiate the study of higher dimensional auslander reiten theory of self injective algebras we give a systematic construction of weakly d representation finite self injective algebras as orbit algebras of the repetitive categories of algebras of finite global dimension satisfying a certain finiteness condition for the serre functor the condition holds in particular for all fractionally calabi yau algebras of global dimension at most d this generalizes riedtmann s classical construction of representation finite self injective algebras our method is based on an adaptation of gabriel s covering theory for k linear categories to the setting of higher dimensional auslander reiten theory applications include n fold trivial extensions and classical and higher preprojective algebras which are shown to be d representation finite in many cases we also get a complete classification of all d representation finite self injective nakayama algebras for arbitrary d

τ2 stable tilting complexes over weighted projective lines abstract let x be a weighted projective line and coh x the associated category of coherent sheaves we classify the tilting complexes t in d b coh x such that 2 t t where is the auslander reiten translation in d b coh x as an application of this result we classify the 2 representation finite algebras which are derived equivalent to a canonical algebra this complements iyama oppermann s classification of the iterated tilted 2 representation finite algebras by passing to 3 preprojective algebras we obtain a classification of the selfinjective cluster tilted algebras of canonical type this complements ringel s classification of the selfinjective cluster tilted algebras

on algebras of finite cohen macaulay type abstract we study artin algebras and commutative noetherian complete local rings r in connection with the following decomposition property of gorenstein projective modules any gorenstein projective module is a direct sum of finitely generated modules we show that the class of algebras enjoying coincides with the class of virtually gorenstein algebras of finite cohen macaulay type introduced in beligiannis and reiten 2007 27 beligiannis 2005 24 thus we solve the problem stated in chen 2008 33 this is proved by characterizing when a resolving subcategory is of finite representation type in terms of decomposition properties of its closure under filtered colimits thus generalizing a classical result of auslander 1976 9 and ringel and tachikawa 1974 63 in the commutative case if r admits a non free finitely generated gorenstein projective module then we show that r is of finite cohen macaulay type iff r is gorenstein and satisfies we also generalize a result of yoshino 2005 68 by characterizing when finitely generated modules without extensions with the ring are gorenstein projective finally we study the stable relative auslander algebra of a virtually gorenstein algebra of finite cohen macaulay type and under the presence of a cluster tilting object we give descriptions of the stable category of gorenstein projective modules in terms of the cluster category associated to the quiver of the stable relative auslander algebra in this setting we show that the cluster category is invariant under derived equivalences

n angulated categories we dene n angulated categories by modifying the axioms of triangulated categories in a natural way we show that heller s parametrization of pre triangulations extends to pre n angulations we obtain a large class of examples of n angulated cate gories by considering n 2 cluster tilting subcategories of triangulated categories which are stable under the n 2 nd power of the suspension functor as an application we show how n angulated calabi yau categories yield triangulated calabi yau categories of higher calabi yau dimension finally we sketch a link to algebraic geometry and string theory

application of bogomolny s transfer operator to a circular harmonic oscillator plus potential bogomolny s transfer operator has been used to find an analytical solution for the semiclassical energy eigenvalues of a simple two dimensional integrable system the system studied consists of a particle moving in an isotropic harmonic oscillator potential plus a potential the classical trajectories are used to construct the transfer matrix and an expression is derived for the eigenvalues of this matrix as a function of the energy these eigenvalue curves yield the semiclassical energy eigenvalues for the quantum system which turn out to be exactly the same as the results obtained by solving the schrodinger equation some insight into this unexpected agreement is provided by considering an exact transfer operator we show that when this operator is expanded in powers of planck s constant the leading term in the expansion is bogomolny s transfer operator