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

local solution for the kadomtsev petviashvili equation in r2 abstract in this paper we obtain results about local existence uniqueness regularity and continuous dependence on the initial data for the solution of the kadomtsev petviashvili equation with positive dispersion in r 2 and initial data in sobolev spaces of order s 3 our method uses an approximated problem where the theory of quasi linear evolution equations is applicable

tilting and cluster tilting for quotient singularities we shall show that the stable categories of graded cohen macaulay modules over quotient singularities have tilting objects in particular these categories are triangle equivalent to derived categories of finite dimensional algebras our method is based on higher dimensional auslander reiten theory which gives cluster tilting objects in the stable categories of ungraded cohen macaulay modules

the poincare bendixson theorem for monotone cyclic feedback systems with delay abstract we consider cyclic nearest neighbor systems of differential delay equations in which the coupling between neighbors possesses a monotonicity property using a discrete integer valued lyapunov function we prove that the poincare bendixson theorem holds for such systems we also obtain results on piecewise monotonicity and stability of periodic solutions of such systems

computational fixed point theory for differential delay equations with multiple time lags abstract we introduce a general computational fixed point method to prove existence of periodic solutions of differential delay equations with multiple time lags the idea of such a method is to compute numerical approximations of periodic solutions using newton s method applied on a finite dimensional projection to derive a set of analytic estimates to bound the truncation error term and finally to use this explicit information to verify computationally the hypotheses of a contraction mapping theorem in a given banach space the fixed point so obtained gives us the desired periodic solution we provide two applications the first one is a proof of coexistence of three periodic solutions for a given delay equation with two time lags and the second one provides rigorous computations of several nontrivial periodic solutions for a delay equation with three time lags

generalized strichartz inequalities for the wave equation strichartz inequalities for the wave equation u f are estimates of the solution u of the cauchy problem for that equation in the form of space time integral norms in terms of similar norms of the inhomogeneity f and of suitable norms of the initial data

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

singular derived categories of factorial terminalizations and maximal modification algebras abstract let x be a gorenstein normal 3 fold satisfying elf with local rings which are at worst isolated hypersurface e g terminal singularities by using the singular derived category d sg x and its idempotent completion d sg x we give necessary and sufficient categorical conditions for x to be q factorial and complete locally q factorial respectively we then relate this information to maximal modification algebras mmas introduced in 20 by showing that if an algebra is derived equivalent to x as above then x is q factorial if and only if is an mma thus all rings derived equivalent to q factorial terminalizations in dimension three are mmas as an application we extend some of the algebraic results in 6 and 14 using geometric arguments

endpoint strichartz estimates we prove an abstract strichartz estimate which implies previously unknown endpoint strichartz estimates for the wave equation in dimension n 4 and the schrodinger equation in dimension n 3 three other applications are discussed local existence for a nonlinear wave equation and strichartz type estimates for more general dispersive equations and for the kinetic transport equation

recent advances about the uniqueness of the slowly oscillating periodic solutions of wright s equation abstract an old conjecture in delay equations states that wright s equation y t y t 1 1 y t r has a unique slowly oscillating periodic solution sops for every parameter value 2 we reformulate this conjecture and we use a method called validated continuation to rigorously compute a global continuous branch of sops of wright s equation using this method we show that a part of this branch does not have any fold point partially answering the new reformulated conjecture

vanishing theorems for complete intersections the starting point for this work is auslander s seminal paper a the main focus of his paper was to understand when the tensor product m rn of finitely generated modules m and n over a regular local ring r is torsion free this condition forces the vanishing of a certain tor module associated to m and n which in turn by auslander s famous rigidity theorem implies that tori m n 0 for all i 1 the vanishing of tor carries a great deal of information for example it implies the depth formula depth m depth n dim r depth m r n from this formula one can deduce for example the highly non trivial fact that if m r n is torsion free and non zero then m and n must both be torsion free in hw huneke and wiegand generalized some of auslander s results to hypersurfaces though with some unavoidable extra hypotheses in particular they proved a rigidity theorem hw 2 4 and the following vanishing theorem

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

saddle node bifurcation of periodic orbits for a delay differential equation abstract we consider the scalar delay differential equation x t x t f k x t 1 with a nondecreasing feedback function f k depending on a parameter k and we verify that a saddle node bifurcation of periodic orbits takes place as k varies the nonlinearity f k is chosen so that it has two unstable fixed points hence the dynamical system has two unstable equilibria and these fixed points remain bounded away from each other as k changes the generated periodic orbits are of large amplitude in the sense that they oscillate about both unstable fixed points of f k

regular articlegeneralized strichartz inequalities for the wave equation we make a synthetic exposition of the generalized strichartz inequalities for the wave equation obtained in 6 together with the limiting cases recently obtained in 13 with as simple proofs as possible the proofs combine stationary phase estimates dyadic decompositions the hardy littlewood inequality or the holder inequality in time and abstract duality and interpolation arguments

non commutative desingularization of determinantal varieties i we show that determinantal varieties defined by maximal minors of a generic matrix have a non commutative desingularization in that we construct a maximal cohen macaulay module over such a variety whose endomorphism ring is cohen macaulay and has finite global dimension in the case of the determinant of a square matrix this gives a non commutative crepant resolution

long wave approximations for water waves in this paper we obtain new nonlinear systems describing the interaction of long water waves in both two and three dimensions these systems are symmetric and conservative rigorous convergence results are provided showing that solutions of the complete free surface euler equations tend to associated solutions of these systems as the amplitude becomes small and the wavelength large using this result as a tool a rigorous justification of all the two dimensional approximate systems recently put forward and analysed by bona chen and saut is obtained in the two dimensional context our methods also allows a significant improvement of the convergence estimate obtained by schneider and wayne in their justification of the decoupled korteweg de vries approximation of the two dimensional euler equations it also follows from our theory that coupled models provide a better description than the decoupled ones over short time scales results are obtained both on an unbounded domain for solutions that evanesce at infinity as well as for solutions that are spatially periodic

configurations of periodic orbits for equations with delayed positive feedback abstract we consider scalar delay differential equations of the form x t x t f x t 1 where 0 and f is a nondecreasing c 1 function if is a fixed point of f r u f u r with f 1 then 1 0 s r is an unstable equilibrium a periodic solution is said to have large amplitude if it oscillates about at least two fixed points of f with f 1 and f 1 we investigate what type of large amplitude periodic solutions may exist at the same time when the number of such fixed points and hence the number of unstable equilibria is an arbitrary integer n 2 it is shown that the number of different configurations equals the number of ways in which n symbols can be parenthesized the location of the floquet multipliers of the corresponding periodic orbits is also discussed

secular growth estimates for hyperbolic systems abstract we consider a model problem for the secular growth which covers all the cases likely to happen in multi scales bkw expansions for nonlinear hyperbolic systems this model problem consists in studying the growth in time of the solution of a nonhomogeneous hyperbolic system whose source term is a product of various functions which solve homogeneous hyperbolic systems the secular growth is due to resonances that we try to control when this is not possible other tools such as decay properties or strichartz estimates must be used

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

asymptotic behavior of waves abstract the asymptotic conjugation relation lim t g t m e itp m g m e itp 2 0 is established for all l 2 r n under mild assumptions on and g where m h f 1 h f denotes fourier multiplication the asymptotic estimate lim t j u t u j 2 0 for finite energy solutions u of the wave equation is deduced from along with generalizations to a class of first order symmetric hyperbolic systems of partial differential equations that are homogeneous and constant coefficient and a weakened version for the klein gordon equation also deduced from is the fact that for a free schrodinger particle the probability of being in the set ta at time t tends to the probability that the velocity is in a as t

flops and clusters in the homological minimal model programme suppose that f x rightarrow mathrm spec r is a minimal model of a complete local gorenstein 3 fold where the fibres of f are at most one dimensional so by van den bergh duke math j 122 3 423 455 2004 there is a noncommutative ring lambda derived equivalent to x for any collection of curves above the origin we show that this collection contracts to a point without contracting a divisor if and only if a certain factor of lambda is finite dimensional improving a result of donovan and wemyss contractions and deformations arxiv 1511 00406 we further show that the mutation functor of iyama and wemyss invent math 197 3 521 586 2014 6 is functorially isomorphic to the inverse of the bridgeland chen flop functor in the case when the factor of lambda is finite dimensional these results then allow us to jump between all the minimal models of mathrm spec r in an algorithmic way without having to compute the geometry at each stage we call this process the homological mmp this has several applications in git approaches to derived categories and also to birational geometry first using mutation we are able to compute the full git chamber structure by passing to surfaces we say precisely which chambers give the distinct minimal models and also say which walls give flops and which do not enabling us to prove the craw ishii conjecture in this setting second we are able to precisely count the number of minimal models and also give bounds for both the maximum and the minimum numbers of minimal models based only on the dual graph enriched with scheme theoretic multiplicity third we prove a bijective correspondence between maximal modifying r module generators and minimal models and for each such pair in this correspondence give a further correspondence linking the endomorphism ring and the geometry this lifts the auslander mckay correspondence to dimension three

complex interpolation and fourier multipliers for the spacesb p q s andf p q s of besov hardy sobolev type the case 0 p 0 q

long wave short wave resonance for nonlinear geometric optics the aim of this paper is to study oscillatory solutions of nonlinear hyperbolic systems in the framework developped during the last decade by joly metivier and rauch here we mainly focus on rectification effects that is the interaction of oscillations with a mean field created by the nonlinearity a real interaction can occur only under some geometric conditions described in jmr1 and l1 that are generally not satisfied by the physical models except in the 1 d case we introduce here a new type of ansatz that allows us to obtain rectification effects under weaker asumptions we obtain a new class of profile equations and construct solutions for a subclass finally the stability of the asymptotic expansion is proved in the context of maxwell bloch type systems

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

uniform decay estimates and the lorentz invariance of the classical wave equation on etudie le comportement asymptotique des solutions de u 0 ou t 2 1 2 n 2 pour des conditions initiales u 0 u t g x en t 0 avec g reguliere a support compact dans r n

the long wave limit for the water wave problem i the case of zero surface tension the korteweg de vries equation boussinesq equation and many other equations can be formally derived as approximate equations for the two dimensional water wave problem in the limit of long waves here we consider the classical problem concerning the validity of these equations for the water wave problem in an infinitely long canal without surface tension we prove that the solutions of the water wave problem in the long wave limit split up into two wave packets one moving to the right and one to the left where each of these wave packets evolves independently as a solution of a korteweg de vries equation our result allows us to describe the nonlinear interaction of solitary waves 2000 john wiley sons inc

systems of differential delay equations floquet multipliers and discrete lyapunov functions abstract we define a discrete integer valued lyapunov function v for cyclic nearest neighbor systems of differential delay equations possessing a feedback condition this extends analogous definitions for cyclic systems of ode s and for scalar differential delay equations we relate the values of v to the real parts of the floquet multipliers for such linear periodic systems and thereby prove all floquet subspaces are at most two dimensional

the long wave limit for a boussinesq equation we consider the long wave limit for a boussinesq equation it is shown that for spatially localized initial conditions the dynamics are under a korteweg de vries kdv regime on a large time interval i e in this limit the solutions split up into two counterpropagating wave packets where each of the wave packets evolve independently and approximately as a solution of a kdv equation for the proof exact estimates between the long wave solutions of the boussinesq equation and the approximations obtained via the decoupled set of kdv equations are demonstrated we expect that this result holds for the water wave problem too

reduction of triangulated categories and maximal modification algebras for ca_n singularities in this paper we define and study triangulated categories in which the homspaces have krull dimension at most one over some base ring hence they have a natural 2 step filtration and each factor of the filtration satisfies some calabi yau type property if c is such a category we say that c is calabi yau with dim c 1 we extend the notion of calabi yau reduction to this setting and prove general results which are an analogue of known results in cluster theory such categories appear naturally in the setting of gorenstein singularities in dimension three as the stable categories cm r of cohen macaulay modules we explain the connection between calabi yau reduction of cm r and both partial crepant resolutions and q factorial terminalizations of spec r and we show under quite general assumptions that calabi yau reductions exist in the remainder of the paper we focus on complete local can singularities r by using a purely algebraic argument based on calabi yau reduction of cm r we give a complete classification of maximal modifying modules in terms of the symmetric group generalizing and strengthening results in 7 10 where we do not need any restriction on the ground field we also describe the mutation of modifying modules at an arbitrary not necessarily indecomposable direct summand as a corollary when k d c we obtain many autoequivalences of the derived category of the q factorial terminalizations of spec r