deformation theory of objects in homotopy and derived categories iii abelian categories abstract this is the third paper in a series in part i we developed a deformation theory of objects in homotopy and derived categories of dg categories here we show how this theory can be used to study deformations of objects in homotopy and derived categories of abelian categories then we consider examples from noncommutative algebraic geometry in particular we study noncommutative grassmanians that are true noncommutative moduli spaces of structure sheaves of projective subspaces in projective spaces

non commutative crepant resolution of minimal nilpotent orbit closures of type a and mukai flops abstract in this article we construct a non commutative crepant resolution nccr of a minimal nilpotent orbit closure b 1 of type a and study relations between an nccr and crepant resolutions y and y of b 1 more precisely we show that the nccr is isomorphic to the path algebra of the double beilinson quiver with certain relations and we reconstruct the crepant resolutions y and y of b 1 as moduli spaces of representations of the quiver we also study the kawamata namikawa s derived equivalence between crepant resolutions y and y of b 1 in terms of an nccr we also show that the p twist on the derived category of y corresponds to a certain operation of the nccr which we call multi mutation and that a multi mutation is a composition of iyama wemyss s mutations

stability conditions and crepant small resolutions in this paper we describe the spaces of stability conditions on the triangulated categories associated to three dimensional crepant small resolutions the resulting spaces have chamber structures such that each chamber corresponds to a birational model together with a special fourier mukai transform we observe that these spaces are covering spaces over certain open subsets of finite dimensional vector spaces and determine their deck transformations

the classification of special cohen macaulay modules in this paper we completely classify all the special cohen macaulay cm modules corresponding to the exceptional curves in the dual graph of the minimal resolutions of all two dimensional quotient singularities in every case we exhibit the specials explicitly in a combinatorial way our result relies on realizing the specials as those cm modules whose first ext group vanishes against the ring r thus reducing the problem to combinatorics on the ar quiver such possible ar quivers were classified by auslander and reiten we also give some general homological properties of the special cm modules and their corresponding reconstruction algebras

fourier mukai transforms for gorenstein schemes we extend to singular schemes with gorenstein singularities or fibered in schemes of that kind bondal and orlov s criterion for an integral functor to be fully faithful we also prove that the original condition of characteristic zero cannot be removed by providing a counterexample in positive characteristic we contemplate a criterion for equivalence as well in addition we prove that for locally projective gorenstein morphisms a relative integral functor is fully faithful if and only if its restriction to each fibre is also fully faithful these results imply the invertibility of the usual relative fourier mukai transform for an elliptic fibration as a direct corollary

deformation theory of objects in homotopy and derived categories ii pro representability of the deformation functor abstract this is the second paper in a series in part i we developed deformation theory of objects in homotopy and derived categories of dg categories here we extend these derived deformation functors to an appropriate bicategory of artinian dg algebras and prove that these extended functors are pro representable in a strong sense

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

geometry of pde s iv navier stokes equation and integral bordism groups abstract following our previous results on this subject r p agarwal a prastaro geometry of pde s iii i webs on pde s and integral bordism groups the general theory adv math sci appl 17 2007 239 266 r p agarwal a prastaro geometry of pde s iii ii webs on pde s and integral bordism groups applications to riemannian geometry pde s adv math sci appl 17 2007 267 285 a prastaro geometry of pde s and mechanics world scientific singapore 1996 a prastaro quantum and integral co bordism in partial differential equations acta appl math 5 3 1998 243 302 a prastaro co bordism groups in pde s acta appl math 59 2 1999 111 201 a prastaro quantized partial differential equations world scientific publishing co singapore 2004 500 pp a prastaro geometry of pde s i integral bordism groups in pde s j math anal appl 319 2006 547 566 a prastaro geometry of pde s ii variational pde s and integral bordism groups j math anal appl 321 2006 930 948 a prastaro th m rassias ulam stability in geometry of pde s nonlinear funct anal appl 8 2 2003 259 278 i stakgold boundary value problems of mathematical physics i the macmillan company new york 1967 i stakgold boundary value problems of mathematical physics ii collier macmillan canada ltd toronto ontario 1968 integral bordism groups of the navier stokes equation are calculated for smooth singular and weak solutions respectively then a characterization of global solutions is made on this ground enough conditions to assure existence of global smooth solutions are given and related to nullity of integral characteristic numbers of the boundaries stability of global solutions are related to some characteristic numbers of the space like cauchy data global solutions of variational problems constrained by ns are classified by means of suitable integral bordism groups too

gopakumar vafa invariants do not determine flops two 3 fold flops are exhibited both of which have precisely one flopping curve one of the two flops is new and is distinct from all known algebraic d 4 flops it is shown that the two flops are neither algebraically nor analytically isomorphic yet their curve counting gopakumar vafa invariants are the same we further show that the contraction algebras associated to both are not isomorphic so the flops are distinguished at this level this shows that the contraction algebra is a finer invariant than various curve counting theories and it also provides more evidence for the proposed analytic classification of 3 fold flops via contraction algebras

the mckay correspondence as an equivalence of derived categories the classical mckay correspondence relates representations of a finite subgroup g sl 2 c to the cohomology of the well known minimal resolution of the kleinian singularity c2 g gonzalez sprinberg and verdier 10 interpreted the mckay correspondence as an isomorphism on k theory observing that the representation ring of g is equal to the g equivariant k theory of c2 more precisely they identify a basis of the k theory of the resolution consisting of the classes of certain tautological sheaves associated to the irreducible representations of g

deformation theory of objects in homotopy and derived categories i general theory abstract this is the first paper in a series we develop a general deformation theory of objects in homotopy and derived categories of dg categories namely for a dg module e over a dg category we define four deformation functors def h e codef h e def e codef e the first two functors describe the deformations and co deformations of e in the homotopy category and the last two in the derived category we study their properties and relations these functors are defined on the category of artinian not necessarily commutative dg algebras

orthogonally spherical objects and spherical fibrations abstract we introduce a relative version of the spherical objects of seidel and thomas 32 define an object e in the derived category d z x to be spherical over z if the corresponding functor from d z to d x gives rise to autoequivalences of d z and d x in a certain natural way most known examples come from subschemes of x fibred over z this categorifies to the notion of an object of d z x orthogonal over z we prove that such an object is spherical over z if and only if it possesses certain cohomological properties similar to those in the original definition of a spherical object we then interpret this geometrically in the case when our objects are actual flat fibrations in x over z

sensor diagnosis and state estimation for a class of skew symmetric time varying systems in this contribution we investigate the problem of simultaneous observer based sensor fault reconstruction and state estimation of a class of linear time varying ltv systems that are skew symmetric models the main features concern the use of a bank of observers to detect and isolate faulty sensors and in the same time provide unbiased state estimation on the other hand we introduced a switching gain technique to deal with singular points stability analysis is achieved thanks to the barbalat s lemma and without solving the well known time varying sylvester equation the proposed approach is extended to more general ltv systems of any order

three dimensional flops and noncommutative rings limburgs univ ctr dept math b 3590 diepenbeek belgium van den bergh m limburgs univ ctr dept math univ campus b 3590 diepenbeek belgium vdbergh luc ac be

flops and equivalences of derived categories for threefolds with only terminal gorenstein singularities the main purpose of this paper is to show that bridgeland s moduli space of perverse point sheaves for certain flopping contractions gives the flops and the fourier mukai transform given by the birational correspondence of the flop is an equivalence between bounded derived categories

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

deformation theory of abelian categories in this paper we develop the basic infinitesimal deformation theory of abelian categories this theory yields a natural generalization of the well known deformation theory of algebras developed by gerstenhaber as part of our deformation theory we define a notion of flatness for abelian categories we show that various basic properties are preserved under flat deformations and we construct several equivalences between deformation problems

contraction algebra and singularity of three dimensional flopping contraction donovan and wemyss duke math j 165 8 1397 1474 2016 introduced the contraction algebra of flopping curves in threefold they conjectured that the contraction algebra determines the formal neighborhood of the underlying singularity of the contraction in this paper we prove that the contraction algebra together with its natural a infty structure constructed in hua and toda int math res not rnw333 2017 determine the formal neighborhood of the singularity

braid group actions on derived categories of coherent sheaves this paper gives a construction of braid group actions on the derived category of coherent sheaves on a variety x the motivation for this is m kontsevich s homological mirror conjecture together with the occurrence of certain braid group actions in symplectic geometry one of the main results is that when dim x geq 2 our braid group actions are always faithful we describe conjectural mirror symmetries between smoothings and resolutions of singularities which lead us to find examples of braid group actions arising from crepant resolutions of various singularities relations with the mckay correspondence and with exceptional sheaves on fano manifolds are given moreover the case of an elliptic curve is worked out in some detail

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

maximal modifications and auslander reiten duality for non isolated singularities we first generalize classical auslander reiten duality for isolated singularities to cover singularities with a one dimensional singular locus we then define the notion of ct modules for non isolated singularities and we show that these are intimately related to noncommutative crepant resolutions nccrs when r has isolated singularities ct modules recover the classical notion of cluster tilting modules but in general the two concepts differ then wanting to generalize the notion of nccrs to cover partial resolutions of operatorname spec r in the main body of this paper we introduce a theory of modifying and maximal modifying modules under mild assumptions all the corresponding endomorphism algebras of the maximal modifying modules for three dimensional gorenstein rings are shown to be derived equivalent we then develop a theory of mutation for modifying modules which is similar but different to mutations arising in cluster tilting theory our mutation works in arbitrary dimension and in dimension three the behavior of our mutation strongly depends on whether a certain factor algebra is artinian

the a deformation theory of a point and the derived categories of local calabi yaus abstract let a be an augmented algebra over a semi simple algebra s we show that the ext algebra of s as an a module enriched with its natural a infinity structure can be used to reconstruct the completion of a at the augmentation ideal we use this technical result to justify a calculation in the physics literature describing algebras that are derived equivalent to certain non compact calabi yau three folds since the calculation produces superpotentials for these algebras we also include some discussion of superpotential algebras and their invariants

the tilting theory of contraction algebras abstract to every minimal model of a complete local isolated cdv singularity donovan wemyss associate a finite dimensional symmetric algebra known as the contraction algebra we construct the first known standard derived equivalences between these algebras and then use the structure of an associated hyperplane arrangement to control the compositions obtaining a faithful group action on the bounded derived category further we determine precisely those standard equivalences which are induced by two term tilting complexes and show that any standard equivalence between contraction algebras up to algebra isomorphism can be viewed as the composition of our constructed functors thus for a contraction algebra we obtain a complete picture of its derived equivalence class and in particular of its derived autoequivalence group

noncommutative deformations and flops we prove that the functor of noncommutative deformations of every flipping or flopping irreducible rational curve in a 33 fold is representable and hence we associate to every such curve a noncommutative deformation algebra aconacon this new invariant extends and unifies known invariants for flopping curves in 33 folds such as the width of reid and the bidegree of the normal bundle it also applies in the settings of flips and singular schemes we show that the noncommutative deformation algebra aconacon is finite dimensional and give a new way of obtaining the commutative deformations of the curve allowing us to make explicit calculations of these deformations for certain 3 1 3 1 curves we then show how our new invariant aconacon also controls the homological algebra of flops for any flopping curve in a projective 33 fold with only gorenstein terminal singularities we construct an autoequivalence of the derived category of the 33 fold by twisting around a universal family over the noncommutative deformation algebra aconacon and prove that this autoequivalence is an inverse of bridgeland s flop flop functor this demonstrates that it is strictly necessary to consider noncommutative deformations of curves in order to understand the derived autoequivalences of a 33 fold and thus the bridgeland stability manifold