derived category automorphisms from mirror symmetry inspired by the homological mirror symmetry conjecture of kontsevich 30 we construct new classes of automorphisms of the bounded derived category of coherent sheaves on a smooth quasi projective variety msc 2000 18e30 14j32

mckay correspondence for symplectic quotient singularities let v be a finite dimensional complex vector space and g sl v a finite subgroup the quotient variety x v g is usually singular if dim v 2 then there exists a canonical smooth projective resolution y x of the singular variety x this resolution is crepant which in this case means that the manifold y has trivial canonical bundle we recall the precise definitions in definition 1 2 the fiber 1 0 y over the singular point 0 x v g is a rational curve whose components are numbered by the nontrivial conjugacy classes in g the homology classes of these components freely generate h2 y q all the other homology groups of y are trivial except for h0 y q q this situation was described by j mckay in mck it is known as the mckay correspondence in higher dimensions the picture splits into two parts one can consider two separate questions 1 when does the quotient x v g admit a smooth crepant resolution y x 2 assuming there exists a smooth crepant resolution y x what can one say about the homology h y q

the grothendieck duality theorem via bousfield s techniques and brown representability grothendieck proved that if f x y is a proper morphism of nice schemes then rf has a right adjoint which is given as tensor product with the relative canonical bundle the original proof was by patching local data deligne proved the existence of the adjoint by a global argument and verdier showed that this global adjoint may be computed locally in this article we show that the existence of the adjoint is an immediate consequence of brown s representability theorem lit follows almost as imme diately by smashing arguments that the adjoint is given by tensor product with a dualising complex verdier s base change theorem is an easy conse quence department of mathematics university of virginia charlottesville virginia 22903 e mail address an3rqvirginia edu this content downloaded from 157 55 39 27 on wed 07 sep 2016 06 26 19 utc all use subject to http about jstor org terms

noncommutative resolutions of ade fibered calabi yau threefolds in this paper we construct noncommutative resolutions of a certain class of calabi yau threefolds studied by cachazo et al geometric transitions and n 1 quiver theories http arxiv org abs hep th 0108120v2 2001 the threefolds under consideration are fibered over a complex plane with the fibers being deformed kleinian singularities the construction is in terms of a noncommutative algebra introduced by ginzburg calabi yau algebras http arxiv org abs math 0612139v2 2006 which we call the n 1 ade quiver algebra

transforms associated to square integrable group representations i general results let g be a locally compact group which need not be unimodular let x u x x g be an irreducible unitary representation of g in a hilbert space h u assume that u is square integrable i e that there exists in h u at least one nonzero vector g such that u x g g 2 dx we give here a reasonably self contained analysis of the correspondence associating to every vector f h u the function u x g f on g discussing its isometry characterization of the range inversion and simplest interpolation properties this correspondence underlies many properties of generalized coherent states

duality theorems for etale gerbes on orbifolds abstract let g be a finite group and y a g gerbe over an orbifold b a disconnected orbifold y and a flat u 1 gerbe c on y is canonically constructed from y motivated by a proposal in physics we study a mathematical duality between the geometry of the g gerbe y and the geometry of y twisted by c we prove several results verifying this duality in the contexts of non commutative geometry and symplectic topology in particular we prove that the category of sheaves on y is equivalent to the category of c twisted sheaves on y when y is symplectic we show by a combination of techniques from non commutative geometry and symplectic topology that the chen ruan orbifold cohomology of y is isomorphic to the c twisted orbifold cohomology of y as graded algebras

a composite fourier wavelet transform and square integrable group representations a formula for a composite fourier wavelet transform is suggested from which the noncommutative fourier and wavelet transforms can be obtained as particular examples as well as several other fourier type transforms it is shown how this composite transform can be used to extend certain results originally obtained for square integrable group representations to more general measures arising in quantum probability theory

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

moduli stacks of serre stable representations in tilting theory abstract we introduce a new moduli stack called the serre stable moduli stack which corresponds to studying families of point objects in an abelian category with a serre functor this allows us in particular to re interpret the classical derived equivalence between most concealed canonical algebras and weighted projective lines by showing they are induced by the universal sheaf on the serre stable moduli stack we explain why the method works by showing that the serre stable moduli stack is the tautological moduli problem that allows one to recover certain nice stacks such as weighted projective lines from their moduli of sheaves as a result this new stack should be of interest in both representation theory and algebraic geometry

rational cherednik algebras and hilbert schemes let h c be the rational cherednik algebra of type a n 1 with spherical subalgebra u c e h c e then u c is filtered by order of differential operators with associated graded ring mbox gr u c mathbb c mathfrak h oplus mathfrak h where w is the n th symmetric group we construct a filtered mathbb z algebra b such that under mild conditions on c the category b qgr of graded noetherian b modules modulo torsion is equivalent to u c mod the associated graded mathbb z algebra mbox gr b has mbox gr b lqgr simeq coh hilb n this can be regarded as saying that u c simultaneously gives a non commutative deformation of mathfrak h oplus mathfrak h w and of its resolution of singularities hilb n rightarrow mathfrak h oplus mathfrak h as we show elsewhere this result is a powerful tool for studying the representation theory of h c and its relationship to hilb n

remarks on the derived mckay correspondence for hilbert schemes of points and tautological bundles we study the images of tautological bundles on hilbert schemes of points on surfaces and their wedge powers under the derived mckay correspondence the main observation of the paper is that using a derived equivalence differing slightly from the standard one considerably simplifies both the results and their proofs as an application we obtain shorter proofs for known results as well as new formulae for homological invariants of tautological sheaves in particular we compute the extension groups between wedge powers of tautological bundles associated to line bundles on the surface

vertex operators grassmannians and hilbert schemes we approximate the infinite grassmannian by finite dimensional cutoffs and define a family of fermionic vertex operators as the limit of geometric correspondences on the equivariant cohomology groups with respect to a one dimensional torus action we prove that in the localization basis these are the well known fermionic vertex operators on the infinite wedge representation furthermore the boson fermion correspondence locality and intertwining properties with the virasoro algebra are the limits of relations on the finite dimensional cutoff spaces which are true for geometric reasons we then show that these operators are also almost by definition the vertex operators defined by okounkov and the author in carlsson and okounkov http arxiv org abs 0801 2565v2 math ag 2009 on the equivariant cohomology groups of the hilbert scheme of points on mathbb c 2 with respect to a special torus action

tilting generators via ample line bundles abstract it is known that a tilting generator on an algebraic variety x gives a derived equivalence between x and a certain non commutative algebra in this paper we present a method to construct a tilting generator from an ample line bundle and construct it in several examples

knot homology via derived categories of coherent sheaves i the mathfrak sl 2 case using derived categories of equivariant coherent sheaves we construct a knot homology theory which categorifies the quantum mathfrak sl m knot polynomial our knot homology naturally satisfies the categorified moy relations and is conjecturally isomorphic to khovanov rozansky homology our construction is motivated by the geometric satake correspondence and is related to manolescu s by homological mirror symmetry

knot homology via derived categories of coherent sheaves ii mathfrak sl _ m case using derived categories of equivariant coherent sheaves we construct a knot homology theory which categorifies the quantum mathfrak sl m knot polynomial our knot homology naturally satisfies the categorified moy relations and is conjecturally isomorphic to khovanov rozansky homology our construction is motivated by the geometric satake correspondence and is related to manolescu s by homological mirror symmetry

mathbb c 2 mathbb z _ n fractional branes and monodromy we construct geometric representatives for the mathbb c 2 mathbb z n fractional branes in terms of branes wrapping certain exceptional cycles of the resolution in the process we use large radius and conifold type monodromies and also check some of the orbifold quantum symmetries we find the explicit seiberg duality which connects our fractional branes to the ones given by the mckay correspondence we also comment on the harvey moore bps algebras

the wavelet transform time frequency localization and signal analysis two different procedures for effecting a frequency analysis of a time dependent signal locally in time are studied the first procedure is the short time or windowed fourier transform the second is the wavelet transform in which high frequency components are studied with sharper time resolution than low frequency components the similarities and the differences between these two methods are discussed for both schemes a detailed study is made of the reconstruction method and its stability as a function of the chosen time frequency density finally the notion of time frequency localization is made precise within this framework by two localization theorems

moduli of mckay quiver representations ii grobner basis techniques abstract in this paper we introduce several computational techniques for the study of moduli spaces of mckay quiver representations making use of grobner bases and toric geometry for a finite abelian group g gl n k let y be the coherent component of the moduli space of stable representations of the mckay quiver our two main results are as follows we provide a simple description of the quiver representations corresponding to the torus orbits of y and in the case where y equals nakamura s g hilbert scheme we present explicit equations for a cover by local coordinate charts the latter theorem corrects the first result from nakamura i nakamura hilbert schemes of abelian group orbits j algebraic geom 10 4 2001 757 779 the techniques introduced here allow experimentation in this subject and give concrete algorithmic tools to tackle further open questions to illustrate this point we present an example of a nonnormal g hilbert scheme thereby answering a question raised by nakamura

exact finite approximations of average cost countable markov decision processes for a countable state markov decision process we introduce an embedding which produces a finite state markov decision process the finite state embedded process has the same optimal cost and moreover it has the same dynamics as the original process when restricting to the approximating set the embedded process can be used as an approximation which being finite is more convenient for computation and implementation

terminal quotient singularities in dimension three via variation of git abstract a 3 fold terminal quotient singularity x c 3 g admits the economic resolution y x which is close to being crepant this paper proves that the economic resolution y is isomorphic to a distinguished component of a fine moduli space of stable g constellations for a suitable git parameter