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

the classification of homogeneous cohen macaulay rings of finite representation type in this paper we classify the homogeneous cohen macaulay rings which are of finite representation type that is the cohen macaulay rings which are positively graded and generated in degree 1 with an algebraically closed field of characteristic 0 in degree 0 and which have up to isomorphisms and shifts in the grading only a finite number of indecomposable maximal cohen macaulay modules mcm modules our main contribution is to show that a homogeneous cohen macaulay ring of finite representation type and dimension 2 must have minimal multiplicity putting this together with previous results of auslander auslander reiten buchweitz greuel schreyer greuel kn6rrer and solberg we obtain

indecomposable cohen macaulay modules and their multiplicities the main aim of this paper is to find a large class of rings for which there are indecomposable maximal cohen macaulay modules of arbitrary high multiplicity or rank in the case of domains

comparative lagrangian formulations for localized buckling an energy functional for a strut on a nonlinear softening foundation is worked into two different lagrangian forms in fast and slow space respectively the developments originate independently of the underlying differential equation and carry some quite general features in each case the kinetic energy is an indefinite quadratic form in fast space this leads to an escape phenomenon with fractal properties in slow space kinetic energy is added to a potential contribution that is familiar from modal formulations together and in conjunction with a recent set of numerical experiments they illustrate the extra complexities of localized as opposed to distributed periodic buckling

local rings of finite cohen macaulay type abstract let r m be a local cohen macaulay ring whose m adic completion r has an isolated singularity we verify the following conjecture of f o schreyer r has finite cohen macaulay type if and only if r has finite cohen macaulay type we also show that the hypersurface k x 0 x d f has finite cohen macaulay type if and only if k s x 0 x d f has finite cohen macaulay type where k s is the separable closure of the field k

monomorphism categories cotilting theory and gorenstein projective modules abstract the monomorphism category s n x is introduced where x is a full subcategory of the module category a mod of an artin algebra a the key result is a reciprocity of the monomorphism operator s n and the left perpendicular operator for a cotilting a module t there is a canonical construction of a cotilting module m t over the upper triangular matrix algebra t n a such that s n t m t as applications s n x is a resolving contravariantly finite subcategory in t n a mod with s n x t n a mod if and only if x is a resolving contravariantly finite subcategory in a mod with x a mod for a gorenstein algebra a the category t n a g proj of gorenstein projective t n a modules can be explicitly determined as s n a also self injective algebras a can be characterized by the property t n a g proj s n a finally we obtain a characterization of those categories s n a which have finite representation type in terms of auslander s representation dimension

the acyclic closure of an exact category and its triangulation abstract for any exact category a with splitting idempotents a maximal exact category t a containing a as a biresolving subcategory is constructed important types of exact categories including n tilting torsion classes categories of cohen macaulay modules over a cohen macaulay order or categories of gorenstein projectives are shown to be of the form t a the quotient category t a a in the sense of grothendieck always exists and carries a triangulated structure more generally it is proved that any biresolving subcategory a of an exact category m gives a triangulated localization m a for example the unbounded derived category d a of an exact category a is obtained directly with no passage through a homotopy category of complexes as applications some recent developments related to gorenstein projectivity non commutative crepant resolutions singularity categories and cohen macaulay representations are extended and improved in the new framework for example the concept of non commutative resolution of a noetherian frobenius category is extended to arbitrary exact categories which leads to an overarching connection with representation dimension of exact categories and n tilting

surface singularities of finite buchsbaum representation type the purpose of this paper is to determine the structure of surface singularities of finite buchsbaum representation type and the main results are summarized into the following let r be a noetherian complete local ring of dim r 2 and assume that the residue class field of r is algebraically closed let e r and v r denote respectively the multiplicity and the embedding dimension of r then the following three conditions are equivalent 1 r has a finite buchsbaum representation type that is r possesses only finitely many isomorphism classes of indecomposable maximal buchsbaum r modules 2 e r 1 and v r 3 3 r p xi where p is a three dimensional complete regular local ring with maximal ideal n x n n 2 and i is an ideal of p such that ht p i 2

homogeneous acm bundles on a grassmannian abstract in this paper we characterize homogeneous arithmetically cohen macaulay acm for short bundles on grassmannians gr k n and we construct irreducible families of indecomposable acm bundles on a grassmannian gr k n of arbitrary high rank and dimension as a consequence we prove that all grassmann varieties gr k n are of wild representation type unless the projective spaces gr 0 n and gr n 1 n and the hyperquadric gr 1 3 in p 5 which are of finite representation type

acm sheaves of pure rank two on reducible hyperquadrics abstract we classify a special type of arithmetically cohen macaulay sheaves of rank two on reducible and reduced quadric hypersurfaces as a consequence we show that a reducible and reduced quadric surface is of wild type

indecomposable generalized cohen macaulay modules the aim of this paper is to study the indecomposable modules which are cohen macaulay on the punctured spectrum of a local ring and to describe some of their invariants such as their local cohomology groups and ranks one of our main concerns is to find indecomposable quasi buchsbaum modules of high rank with prescribed cohomology over a regular local ring

algorithms for projecting points onto conics we study the problem of projecting 2d points onto quadratic curves ellipses hyperbolas parabolas we investigate various projection algorithms focusing on those that are mathematically proven to produce or converge to correct results in all cases our tests demonstrate that those may be still unfit for practical use due to large computational errors we present two new algorithms that are not only theoretically proven to converge but achieve nearly perfect accuracy

the hilbert kunz functions of two dimensional rings of type ade abstract we compute the hilbert kunz functions of two dimensional rings of type ade by using representations of their indecomposable maximal cohen macaulay modules in terms of matrix factorizations and as first syzygy modules of homogeneous ideals

regularity and cohomological splitting conditions for vector bundles on multiprojective spaces abstract we give a definition of regularity on multiprojective spaces which is different from the definitions of hoffman and wang and costa and miro roig using this notion we prove some splitting criteria for vector bundles

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 the periodic orbits of the fourth order differential equation u qu u ef u u u u we provide sufficient conditions for the existence of periodic solutions of the fourth order differential equation u qu u ef u u u u where q and e are real parameters e is small and f is a nonlinear function

the representation type of segre varieties abstract the goal of this work is to prove that all segre varieties n 1 n s p n n i 1 s n i 1 1 unless the quadric surface in p 3 support families of arbitrarily large dimension and rank of simple ulrich and hence acm vector bundles therefore they are of wild representation type

cohomological characterization of vector bundles on grassmannians of lines abstract we introduce a notion of regularity for coherent sheaves on grassmannians of lines we use this notion to prove some extension of evans griffith criterion to characterize direct sums of line bundles we also give in the line of previous results by costa and miro roig a cohomological characterization of exterior and symmetric powers of the universal bundles of the grassmannian

gorenstein biliaison and acm sheaves abstract let x be a normal arithmetically gorenstein scheme in p n we give a criterion for all codimension two acm subschemes of x to be in the same gorenstein biliaison class on x in terms of the category of acm sheaves on x these are sheaves that correspond to the graded maximal cohen macaulay modules on the homogeneous coordinate ring of x using known results on mcm modules we are able to determine the gorenstein biliaison classes of codimension two subschemes of certain varieties including the nonsingular quadric surface in p 3 and the cone over it in p 4 as an application we obtain a new proof of some theorems of lesperance about curves in p 4 and answer some questions he raised

on the limit cycles for a class of fourth order differential equations we provide sufficient conditions for the existence of periodic solutions of the fourth order differential equation where p is a rational number different from 0 e is small and f is a nonlinear function

d brane charges in gepner models we construct gepner models in terms of coset conformal field theories and compute their twisted equivariant k theories these classify the d brane charges on the associated geometric backgrounds and therefore agree with the topological k theories we show this agreement for various cases in particular the fermat quintic

structural localization phenomena and the dynamical phase space analogy the localization of buckle patterns in elastic structures is reviewed from three complementary viewpoints a from a modal perspective b from a formulation which allows the amplitude to modulate in an asymptotically defined slow space and c from a dynamical analogy in phase space suggested by the form of the underlying differential equation a simple strut on an asymmetric nonlinear foundation provides a typical illustrative example the three approaches emphasize different features of the localization phenomenon the modal view illustrates the distinctive effects of boundary conditions the modulated approach generates a convenient second order differential equation in the amplitude function and the dynamical phase space analogy suggests a useful interpretation of localization as a homoclinic connection comparisons are also made with nonlinear numerical solutions as the strut length approaches infinity it is shown that the fully localized solution represents the unstable post buckled state with the lowest energy allowing evaluation of the minimum energy barrier relevant to dynamical impact studies attention is drawn to the possibility of spatial chaos becoming manifest as a randomly spaced sequence of localizations caused by a regular sinusoidal spatial imperfection

gorenstein singularity categories abstract the aim of this paper is to introduce gorenstein singularity category d g p s g b a as the verdier quotient of the gorenstein derived category d g p b a by the triangulated subcategory k b gp a the main result shows that if a is cm contravariantly finite abelian category then the gorenstein singularity category d g p s g b a is triangle equivalent to the gorenstein defect category d defect b a in the sense of 7 we point out several classes of algebras with non zero singularity categories but their gorenstein singularity categories become zero we also prove that a gorenstein derived equivalence between cm contravariantly finite abelian categories with infinite direct sums induces a gorenstein singular equivalence

ascent of finite cohen macaulay type in 1987 f o schreyer conjectured that a local ring r has finite cohen macaulay type if and only if the completion r has finite cohen macaulay type we prove the conjecture for excellent cohen macaulay local rings and also show by example that it can fail in general

relative singularity categories i auslander resolutions abstract let r be an isolated gorenstein singularity with a non commutative resolution a end r r m in this paper we show that the relative singularity category r a of a has a number of pleasant properties such as being hom finite moreover it determines the classical singularity category d s g r of buchweitz and orlov as a certain canonical quotient category if r has finite cm type which includes for example kleinian singularities then we show the much more surprising result that d s g r determines r aus r where aus r is the corresponding auslander algebra the proofs of these results use dg algebras a koszul duality and the new concept of dg auslander algebras which may be of independent interest

a construction of gorenstein projective modules abstract we determine all the gorenstein projective modules over the t 2 extension of a gorenstein algebra and over a m 0 b where a and b are self injective algebras and m is an a b bimodule with m a and m b projective using this we obtain a criterion for the cm finiteness of the t 2 extensions of cm finite gorenstein algebras as an application we get non trivial examples of cm finite gorenstein algebras

low dimensional orders of finite representation type in this paper we study noncommutative surface singularities arising from orders the singularities we study are mild in the sense that they have finite representation type or equivalently are log terminal in the sense of the mori minimal model program for orders chan and ingalls in invent math 161 2 427 452 2005 these were classified independently by artin in terms of ramification data and reiten van den bergh in terms of their ar quivers the first main goal of this paper is to connect these two classifications by going through the finite subgroups g subset mathrm gl 2 explicitly computing h 2 g k and then matching these up with artin s list of ramification data and reiten van den bergh s ar quivers this provides a semi independent proof of their classifications and extends the study of canonical orders in chan et al proc lond math soc 3 98 1 83 115 2009 to the case of log terminal orders a secondary goal of this paper is to study noncommutative analogues of plane curves which arise as follows let b k zeta llbracket x y rrbracket be the skew power series ring where zeta is a root of unity or more generally a terminal order over a complete local ring we consider rings of the form a b f where f in z b which we interpret to be the ring of functions on a noncommutative plane curve we classify those noncommutative plane curves which are of finite representation type and compute their ar quivers

periodic solutions for extended fisher kolmogorov and swift hohenberg equations by truncature techniques abstract combining truncature techniques with a variational approach we establish an existence result for nontrivial periodic solutions for a class of fourth order ordinary differential equations involving extended fisher kolmogorov and swift hohenberg equations

skew group algebras in the representation theory of artin algebras then the corresponding crossed product algebra i g or i x g for short has an elements cgieg gi a li i addition is componentwise and multiplication is given by g a g and g g2 y g g2 m in this paper we assume that the values of y lie in the center z a of i hence the action of g on i is given by a group homomorphism g p aut i and 3 can be left out in the special case that y is the trivial map we write 1g instead of i g and the elements as c c li gi ag is then called a skew group ring there is a lot of literature on skew group algebras and crossed product algebras and on the relationship with the ring ac whose elements are those elements of a left fixed by g much work has been done on which properties of li are inherited by n g or ac some of the work on the relationship between these rings has its roots in trying to develop a galois theory for noncommutative rings we refer to 3 7 13 15 19 21 23 25 27 281 and their references in this paper we study these constructions when i is an artin algebra and g a finite group of order n such that n is invertible in 1 under these assumptions the construction preserves central properties of interest in the

tensor products of modules and the rigidity of tor let r be a hypersurface domain that is a ring of the form s f where s is a regular local ring and f is a prime element suppose m and n are finitely generated r modules we prove two rigidity theorems on the vanishing of tor in the first theorem we assume that the regular local ring s is unramified that m r n has finite length and that dim m dim n dim r with these assumptions if tor r m n 0 for some j 0 then tor r m n 0 for all i j the second rigidity theorem states that if m r n is reflexive then tor r m n 0 for all i 1 we use these theorems to prove the following theorem valid even if s is ramified if m r n is a maximal cohen macaulay r module then both m and n are maximal cohen macaulay modules and at least one of them is free

global branches of multi bump periodic solutions of the swift hohenberg equation in this paper we study the existence of single and multi bump periodic solutions of a class of fourth order ordinary differential equations arising in problems of pattern formation measuring the tendency to form patterns by a parameter q we view the problem as a nonlinear eigenvalue problem with the use of analytical as well as numerical methods branches of periodic solutions are investigated both locally and globally

a generalization of the eckart young mirsky matrix approximation theorem abstract the eckart young mirsky theorem solves the problem of approximating a matrix by one of lower rank however the approximation generally differs from the original in all its elements in this paper it is shown how to obtain a best approximation of lower rank in which a specified set of columns of the matrix remains fixed the paper concludes with some applications of the generalization

deformations of rational surface singularities and reflexive modules with an application to flops abstract blowing up a rational surface singularity in a reflexive module gives a any partial resolution dominated by the minimal resolution the main theorem shows how deformations of the pair singularity module relates to deformations of the corresponding pair of partial resolution and locally free strict transform and to deformations of the underlying spaces the results imply some recent conjectures on small resolutions and flops

the cohen macaulay type of cohen macaulay rings let r be an integrally closed complete local commutative noetherian cohen macaulay domain with maximal ideal m and r m k an algebraically closed field we abbreviate cohen macaulay to cm and denote by cm r the category of finitely generated cm modules in this paper we deal with the question of when r is of finite cm type that is has only a linite number of indecomposable cm modules if dim r 2 and r is a c algebra with r m the complex numbers then the r of finite cm type are exactly the fixed rings x y where x y are indeterminates and g c gl 2 c is a finite group acting on x y 17 3 151 for hypersurfaces in characteristic zero the r of finite cm type are exactly the simple hypersurface singularities in the sense of arnold 18 133 so in this case there is a nice connection with algebraic geometry when dim r 3 and r is not a hypersurface we know only two examples of finite cm type and they both have dimension 3 these are r kccxo x x y y iliwo xoy x y x y x yc and r2 k x y z lz where the generator of z acts by sending each variable to its negative and the characteristic of k is different from 2 each of these rings belongs naturally to a larger class r is a scroll of type 2 1 we define more generally a scroll of type m m see 141 and show that when dim r 2 3 then all other scrolls except type 1 l which is a hypersurface are of infinite cm type r is among the rings kccj xjig where the order of g c gl n k is invertible in k and n 2 3 and is the only one of finite cm type these are the main results of this paper to prove that the scroll of type 2 1 is of finite cm type we use the 1 oool 8708 89 7 50

remarks on non commutative crepant resolutions of complete intersections abstract we study obstructions to existence of non commutative crepant resolutions in the sense of van den bergh over local complete intersections

gorenstein algebras of finite cohen macaulay type abstract an artin algebra a is said to be cm finite if there are only finitely many isomorphism classes of indecomposable finitely generated gorenstein projective a modules inspired by auslander s idea on representation dimension we prove that for 2 n a is a cm finite n gorenstein algebra if and only if there is a resolving gorenstein projective a module e such that gl dim end a e op n

super stretched and graded countable cohen macaulay type abstract we define what it means for a cohen macaulay ring to be super stretched and show that cohen macaulay rings of graded countable cohen macaulay type are super stretched we use this result to show that rings of graded countable cohen macaulay type and positive dimension have possible h vectors 1 1 n or 1 n 1 further one dimensional standard graded gorenstein rings of graded countable type are shown to be hypersurfaces this result is not known in higher dimensions in the non gorenstein case rings of graded countable cohen macaulay type of dimension larger than 2 are shown to be of minimal multiplicity

noncommutative knorrer periodicity and noncommutative kleinian singularities abstract we establish a version of knorrer s periodicity theorem in the context of noncommutative invariant theory namely let a be a left noetherian as regular algebra let f be a normal and regular element of a of positive degree and take b a f then there exists a bijection between the set of isomorphism classes of indecomposable non free maximal cohen macaulay modules over b and those over a noncommutative analog of its second double branched cover b our results use and extend the study of twisted matrix factorizations which was introduced by the first three authors with cassidy these results are applied to the noncommutative kleinian singularities studied by the second and fourth authors with chan and zhang

homoclinic orbits in reversible systems and their applications in mechanics fluids and optics abstract this survey article reviews the theory and application of homoclinic orbits to equilibria in reversible continuous time dynamical systems where the homoclinic orbit and the equilibrium are both reversible the focus is on even order reversible systems in four or more dimensions local theory generic argument and global existence theories are examined for each qualitatively distinct linearisation several recent results such as coalescence caused by non transversality and the reversible orbit flip bifurcation are covered a number of open problems are highlighted applications are reviewed to systems arising from a variety of disciplines with the aid of numerical methods three examples are presented in detail one of which is infinite dimensional

rank 3 acm bundles on general hypersurfaces in p5 abstract we prove that a general hypersurface in p 5 of degree d 3 does not support an indecomposable rank 3 arithmetically cohen macaulay acm bundle this settles the base case of a generic version of a conjecture of buchweitz greuel and schreyer

deformations of maximal cohen macaulay modules let r m be a complete cohen macaulay isolated singularity over a field k which is either perfect or k k p 0 according to dieterich di and yo ch 6 see also pol pr or more generally chp there exists a system of parameters x xl xr of r such that the base change functor r x rdefines an injection v preserving the indecomposability from the set of isomorphism classes of maximal cohen macaulay r modules to the set of isomorphism classes of r x modules in this case the ideal generated by x is called after di a reduction ideal trying to describe the image of v we noticed in po2 that a finitely generated r x module is in imv if and only if it has the form xl x p for a finitely generated r2 r x 2 x module satisfying

an example of a non fourier mukai functor between derived categories of coherent sheaves orlov s famous representability theorem asserts that any fully faithful exact functor between the bounded derived categories of coherent sheaves on smooth projective varieties is a fourier mukai functor in this paper we show that this result is false without the fully faithfulness hypothesis we also show that our functor does not lift to the homotopy category of spectral categories if the ground field is mathbb q

existence of perturbed solitary wave solutions to a model equation for water waves abstract we prove the existence of travelling wave solutions to a fifth order partial differential equation which is a formal asymptotic approximation for water waves with surface tension these travelling waves are arbitrarily small perturbations of solitary waves but are not solitary waves themselves because they approach small amplitude oscillations for large values of the independent variable this result suggests that for bond numbers less than one third there are branches of travelling wave solutions to the water wave equations which are perturbations of supercritical elevation solitary waves and which bifurcate from froude number one and bond number one third

arithmetically cohen macaulay bundles and sets of points on general surfaces abstract the first chern class of an initialized rank 2 bundle determines its level of stability we determine bounds for numbers c 1 0 such that on a general surface x p 3 of degree d 4 there exists a rank 2 indecomposable stable acm bundle e on x with first chern class c 1 e c 1

some extensions of theorems of knorrer and herzog popescu abstract a construction due to knorrer shows that if n is a maximal cohen macaulay module over a hypersurface defined by f y 2 then the first syzygy of n y n decomposes as the direct sum of n and its own first syzygy this was extended by herzog popescu to hypersurfaces f y n replacing n y n by n y n 1 n we show in the same setting as herzog popescu that the first syzygy of n y k n is always an extension of n by its first syzygy and moreover that this extension has useful approximation properties we give two applications first we construct a ring over which every finitely generated module has an eventually 2 periodic projective resolution prompting us to call it a non commutative hypersurface ring second we give upper bounds on the dimension of the stable module category a k a the singularity category of a hypersurface defined by a polynomial of the form x 1 a 1 x d a d

brauer thrall for totally reflexive modules let r be a commutative noetherian local ring that is not gorenstein it is known that the category of totally reflexive modules over r is representation infinite provided that it contains a non free module the main goal of this paper is to understand how complex the category of totally reflexive modules can be in this situation local rings r m with m3 0 are commonly regarded as the structurally simplest rings to admit diverse categorical and homological characteristics for such rings we obtain conclusive results about the category of totally reflexive modules modeled on the brauer thrall conjectures starting from a non free cyclic totally reflexive module we construct a family of indecomposable totally reflexive r modules that contains for every n n a module that is minimally generated by n elements moreover if the residue field r m is algebraically closed then we construct for every n n an infinite family of indecomposable and pairwise non isomorphic totally reflexive r modules each of which is minimally generated by n elements the modules in both families have periodic minimal free resolutions of period at most 2

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

quivers from matrix factorizations we discuss how matrix factorizations offer a practical method of computing the quiver and associated superpotential for a hypersurface singularity this method also yields explicit geometrical interpretations of d branes i e quiver representations on a resolution given in terms of grassmannians as an example we analyze some non toric singularities which are resolved by a single mathbb p 1 but have length greater than one these examples have a much richer structure than conifolds a picture is proposed that relates matrix factorizations in landau ginzburg theories to the way that matrix factorizations are used in this paper to perform noncommutative resolutions

the matrix factorizations of the d model the fundamental matrix factorizations of the d model superpotential are found and identified with the boundary states of the corresponding conformal field theory the analysis is performed for both gso projections we also comment on the relation of this analysis to the theory of surface singularities and their orbifold description

noncommutative resolution f blowups and d modules abstract we explain the isomorphism between the g hilbert scheme and the f blowup from the noncommutative viewpoint after van den bergh in doing this we immediately and naturally arrive at the notion of d modules we also find as a byproduct a canonical way to construct a noncommutative resolution at least for a few classes of singularities in positive characteristic

hyperspheres and hyperplanes fitted seamlessly by algebraic constrained total least squares abstract for each finite set of points in a euclidean space of any dimension the algorithm presented here determines all the algebraically best fitting circles or lines spheres or planes or hyperspheres or hyperplanes in a seamless manner from spherical through affine manifolds in particular affine submanifolds of any dimensions are not singularities of the algorithm to this end the algorithm combines projective geometry coope s and gander et al s layouts of the equations and golub et al s generalization of the schmidt mirsky matrix approximation theorem to solve the equations the resulting best fitting manifolds remain invariant under rigid transformations moreover if the best fitting manifold is affine then it coincides with golub and van loan s affine manifold of total least squares thus the algorithm can also fit hyperspheres in a manner that remains robust with data lying near a hyperplane furthermore an analysis with a theorem of wedin s shows that the fitted hyperspheres sensitivity to perturbations of the data increases as the colinearity of the data increases

simplicity of rings of differential operators in prime characteristic let w be a finite dimensional representation of a linearly reductive group g over a field k motivated by their work on classical rings of invariants levasseur and stafford asked whether the ring of invariants under g of the symmetric algebra of w has a simple ring of differential operators in this paper we show that this is true in prime characteristic indeed if r is a graded subring of a polynomial ring over a perfect field of characteristic p 0 and if the inclusion r hookrightarrow s splits then d k r is a simple ring in the last section of the paper we discuss how one might try to deduce the characteristic zero case from this result as yet however this is a subtle problem and the answer to the question of levasseur and stafford remains open in characteristic zero http www luc ac be research algebra 1991 mathematics subject classification 16s32 16g60 13a35

fitting conics of specific types to data abstract for each finite set of points in the euclidean plane and for each type of conic section elliptic hyperbolic or parabolic the algorithm presented here determines all the algebraically best fitting conics of the selected type the best ellipse the best hyperbola or the best parabola the supporting theory expands on golub hoffman and stewart s generalization of the schmidt mirsky matrix approximation theorem on bookstein s and pratt s methods to fit conics and on gander golub and strebel s method to fit ellipses because neither the set of ellipses nor the set of hyperbolae is closed the algorithm and its supporting theory must accommodate their boundary which consists of the parabolic conics the corresponding optimization problem consists in minimizing a quadratic form with two quadratic constraints which an orthogonal change of variables transforms into a least squares problem with one quadratic constraint hence analogies with geodetic coordinates identify geometric causes of numerical instability for each type of conic the resulting best fitting conic remains invariant under euclidean transformations applications include the theory and use of sundials in archaeology and astronomy

matrix factorizations and representations of quivers ii type ade case abstract we study a triangulated category of graded matrix factorizations for a polynomial of type ade we show that it is equivalent to the derived category of finitely generated modules over the path algebra of the corresponding dynkin quiver also we discuss a special stability condition for the triangulated category in the sense of t bridgeland which is naturally defined by the grading