the cr geometry of weighted extremal kahler and sasaki metrics we establish an equivalence between conformally einstein maxwell kahler 4 manifolds recently studied in apostolov et al j reine angew math 721 109 147 2016 apostolov and maschler j eur math soc 21 1319 1360 2019 futaki and ono j math soc jpn 70 1493 1521 2018 koca et al ann glob anal geom 50 29 46 2016 lebrun einstein maxwell equations extremal kahler metrics and seiberg witten theory in the many facets of geometry a tribute to nigel hitchin oxford university press oxford pp 17 33 2009 lebrun j geom phys 91 171 2015 and lebrun commun math phys 344 621 653 2016 and extremal kahler 4 manifolds in the sense of calabi extremal kahler metrics seminar on differential geometry annals of mathematics studies vol 102 pp 259 290 princeton university press princeton 1982 with nowhere vanishing scalar curvature the corresponding pairs of kahler metrics arise as transversal kahler structures of sasaki metrics compatible with the same cr structure and having commuting sasaki reeb vector fields this correspondence extends to higher dimensions using the notion of a weighted extremal kahler metric apostolov et al in levi kahler reduction of cr structures product of spheres and toric geometry arxiv 1708 05253 apostolov et al in weighted extremal kahler metrics and the einstein maxwell geometry of projective bundles arxiv 1808 02813 lahdili in j geom anal 29 542 568 2019 lahdili in int math res not arxiv 1710 00235 lahdili in proc lond math soc 119 1065 1114 2019 illuminating and uniting several explicit constructions in kahler and sasaki geometry it also leads to new existence and non existence results for extremal sasaki metrics suggesting a link between the notions of relative weighted k stability of a polarized variety introduced in apostolov et al weighted extremal kahler metrics and the einstein maxwell geometry of projective bundles arxiv 1808 02813 and lahdili proc lond math soc 119 1065 1114 2019 and relative k stability of the kahler cone corresponding to a sasaki polarization studied in boyer and van coevering math res lett 25 1 19 2018 and collins and szekelyhidi j differ geom 109 81 109 2018

complex symplectomorphisms and pseudo kahler islands in the quantization of toric manifolds let p be a delzant polytope we show that the quantization of the corresponding toric manifold x p in toric kahler polarizations and in the toric real polarization are related by analytic continuation of hamiltonian flows evaluated at time t sqrt 1 s we relate the quantization of x p in two different toric kahler polarizations by taking the time sqrt 1 s hamiltonian flow of strongly convex functions on the moment polytope p by taking s to infinity we obtain the quantization of x p in the singular real toric polarization recall that x p has an open dense subset which is biholomorphic to mathbb c n the quantization of x p in a toric kahler polarization can also be described by applying the complexified hamiltonian flow of the abreu guillemin symplectic potential g at time t sqrt 1 to an appropriate finite dimensional subspace of quantum states in the quantization of t mathbb t n in the vertical polarization by taking other imaginary times t k sqrt 1 k in mathbb r we describe toric kahler metrics with cone singularities along the toric divisors in x p for convex hamiltonian functions and sufficiently negative imaginary part of the complex time we obtain degenerate kahler structures which are negative definite in some regions of x p we show that the pointwise and l 2 norms of quantum states are asymptotically vanishing on negative definite regions

toric surfaces k stability and calabi flow let x be a toric surface and u be a normalized symplectic potential on the corresponding polygon p suppose that the riemannian curvature is bounded by a constant c 1 and int partial p u d sigma 0 depending only on c 1 c 2 and p such that donaldson s m condition holds for u as an application we show that if x p is analytic relative k stable then the modified calabi flow converges to an extremal metric exponentially fast by assuming that the calabi flow exists for all time and the riemannian curvature is uniformly bounded along the calabi flow

remarks on the extended characteristic uncertainty relations three remarks concerning the form and the range of validity of the state extended characteristic uncertainty relations urs are presented a more general definition of the uncertainty matrix for pure and mixed states is suggested some new urs are provided

tropical counting from asymptotic analysis on maurer cartan equations let be a toric surface and let be its landau ginzburg lg mirror where is the hori vafa potential as shown in their preprint we apply asymptotic analysis to study the extended deformation theory of the lg model and prove that semi classical limits of fourier modes of a specific class of maurer cartan solutions naturally give rise to tropical disks in with maslov index 0 or 2 the latter of which produces a universal unfolding of for our construction reproduces gross perturbed potential adv math 224 2010 pp 169 245 which was proven to be the universal unfolding of written in canonical coordinates we also explain how the extended deformation theory can be used to reinterpret the jumping phenomenon of across walls of the scattering diagram formed by maslov index 0 tropical disks originally observed by gross in the same work in the case of

ricci flat kahler metrics on crepant resolutions of kahler cones we prove that a crepant resolution y x of a ricci flat kahler cone x admits a complete ricci flat kahler metric asymptotic to the cone metric in every kahler class in h 2 c y mathbb r a kahler cone x bar g is a metric cone over a sasaki manifold s g i e x c s s times mathbb r 0 with bar g dr 2 r 2 g and x bar g is ricci flat precisely when s g einstein of positive scalar curvature this result contains as a subset the existence of ale ricci flat kahler metrics on crepant resolutions pi y rightarrow x mathbb c n gamma with gamma subset sl n mathbb c due to p kronheimer n 2 and d joyce n 2 we then consider the case when x c s is toric it is a result of a futaki h ono and g wang that any gorenstein toric kahler cone admits a ricci flat kahler cone metric it follows that if a toric kahler cone x c s admits a crepant resolution y x then y admits a tn invariant ricci flat kahler metric asymptotic to the cone metric x bar g in every kahler class in h 2 c y mathbb r a crepant resolution in this context is a simplicial fan refining the convex polyhedral cone defining x we then list some examples which are easy to construct using toric geometry

toric partial density functions and stability of toric varieties let l h rightarrow x omega denote a polarized toric kahler manifold fix a toric submanifold y and denote by hat rho tk x rightarrow mathbb r the partial density function corresponding to the partial bergman kernel projecting smooth sections of l k onto holomorphic sections of l k that vanish to order at least tk along y for fixed t 0 such that tk in mathbb n we prove the existence of a distributional expansion of hat rho tk as k rightarrow infty including the identification of the coefficient of k n 1 as a distribution on x this expansion is used to give a direct proof that if omega has constant scalar curvature then x l must be slope semi stable with respect to y cf ross and thomas in j differ geom 72 3 429 466 2006 similar results are also obtained for more general partial density functions these results have analogous applications to the study of toric k stability of toric varieties

lagrangian floer theory on compact toric manifolds i we introduced the notion of weakly unobstructed lagrangian submanifolds and constructed their potential function po purely in terms of a model data in fooo3 in this article we carry out explicit calculations involving po on toric manifolds and study the relationship between this class of lagrangian submanifolds with the earlier work of givental g1 which advocates that the quantum cohomology ring is isomorphic to the jacobian ring of a certain function called the landau ginzburg superpotential combining this study with the results from fooo3 we also apply the study to various examples to illustrate its implications to symplectic topology of lagrangian fibers of toric manifolds in particular we relate it to the hamiltonian displacement property of lagrangian fibers and to entov polterovich s symplectic quasi states

scalar curvature and stability of toric varieties we define a stability condition for a polarised algebraic variety and state a conjecture relating this to the existence of a kahler metric of constant scalar curvature the main result of the paper goes some way towards verifying this conjecture in the case of toric surfaces we prove that under the stability hypothesis the mabuchi functional is bounded below on invariant metrics and that minimising sequences have a certain convergence property in the reverse direction we give new examples of polarised surfaces which do not admit metrics of constant scalar curvature the proofs use a general framework developed by guillemin and abreu in which invariant kahler metrics correspond to convex functions on the moment polytope of a toric variety this paper is a step towards the solution of the general problem of finding conditions under which a complex projective variety admits a kahler metric of constant scalar curvature the pattern of the answer one expects is that this differential geometric condition should be equivalent to some notion of stability in the sense of geometric invariant theory this expectation is probably now an item of folklore going back to suggestions put forward by yau in the case of kahlereinstein metrics and the many results of tian and others in this case reinforced by a detailed formal picture which makes clear the analogy with the well established relation between the stability of vector bundles and yang mills connections 5 here we begin the investigation of

hamiltonian 2 forms in kahler geometry ii global classification we present a classification of compact kahler manifolds admitting a hamiltonian 2 form which were classified locally in part i of this work this involves two components of independent interest the first is the notion of a rigid hamiltonian torus action this natural condition for torus actions on a kahler manifold was introduced locally in part i but such actions turn out to be remarkably well behaved globally leading to a fairly explicit classification up to a blow up compact kahler manifolds with a rigid hamiltonian torus action are bundles of toric kahler manifolds the second idea is a special case of toric geometry which we call orthotoric we prove that orthotoric kahler manifolds are diffeomorphic to complex projective space but we extend our analysis to orthotoric orbifolds where the geometry is much richer we thus obtain new examples of kahler einstein 4 orbifolds combining these two themes we prove that compact kahler manifolds with hamiltonian 2 forms are covered by blow downs of projective bundles over kahler products and we describe explicitly how the kahler metrics with a hamiltonian 2 form are parameterized we explain how this provides a context for constructing new examples of extremal kahler metrics in particular a subclass of such metrics which we call weakly bochner flat we also provide a self contained treatment of the theory of compact toric kahler manifolds since we need it and find the existing literature incomplete

toric self dual einstein metrics as quotients we use the quaternion kahler reduction technique to study old and new self dual einstein metrics of negative scalar curvature with at least a two dimensional isometry group and relate the quotient construction to the hyperbolic eigenfunction ansatz we focus in particular on the semi quaternion kahler quotients of semi quaternion kahler hyperboloids analysing the completeness and topology and relating them to the self dual einstein hermitian metrics of apostolov gauduchon and bryant

an integral inequality for constant scalar curvature metrics on kahler manifolds abstract we present in this note a lower bound for the calabi functional in a given kahler class this yields an integral inequality for constant scalar curvature metrics which can be viewed as a refinement of s t yau s chern number inequality from the einstein metric case to constant scalar curvature metric case

toric hypersymplectic quotients we study the hypersymplectic spaces obtained as quotients of flat hypersymplectic space r 4d by the action of a compact abelian group these 4n dimensional quotients carry a multi hamilitonian action of an n torus the image of the hypersymplectic moment map for this torus action may be described by a configuration of solid cones in r 3n we give precise conditions for smoothness and non degeneracy of such quotients and show how some properties of the quotient geometry and topology are constrained by the combinatorics of the cone configurations examples are studied including non trivial structures on r 4n and metrics on complements of hypersurfaces in compact manifolds

characteristic uncertainty relations new uncertainty relations for n observables are established the relations take the invariant form of inequalities between the characteristic coefficients of order r rd 1 2 n of the uncertainty matrix and the matrix of mean commutators of the observables it is shown that the second and third order characteristic inequalities for the three generators of su 1 1 and su 2 are minimized in the corresponding group related coherent states with maximal symmetry

k energy on polarized compactifications of lie groups abstract in this paper we study mabuchi s k energy on a compactification m of a reductive lie group g which is a complexification of its maximal compact subgroup k we give a criterion for the properness of k energy on the space of k k invariant kahler potentials in particular it turns to give an alternative proof of delcroix s theorem for the existence of kahler einstein metrics in case of fano manifolds m we also study the existence of minimizers of k energy for general kahler classes of m

semi classical weights and equivariant spectral theory abstract we prove inverse spectral results for differential operators on manifolds and orbifolds invariant under a torus action these inverse spectral results involve the asymptotic equivariant spectrum which is the spectrum itself together with very large weights of the torus action on eigenspaces more precisely we show that the asymptotic equivariant spectrum of the laplace operator of any toric metric on a generic toric orbifold determines the equivariant biholomorphism class of the orbifold we also show that the asymptotic equivariant spectrum of a t n invariant schrodinger operator on r n determines its potential in some suitably convex cases in addition we prove that the asymptotic equivariant spectrum of an s 1 invariant metric on s 2 determines the metric itself in many cases finally we obtain an asymptotic equivariant inverse spectral result for weighted projective spaces as a crucial ingredient in these inverse results we derive a surprisingly simple formula for the asymptotic equivariant trace of a family of semi classical differential operators invariant under a torus action

kahler metrics on toric orbifolds a theorem of e lerman and s tolman generalizing a result of t delzant states that compact symplectic toric orbifolds are classified by their moment polytopes together with a positive integer label attached to each of their facets in this paper we use this result and the existence of global actionangle coordinates to give an effective parametrization of all compatible toric complex structures on a compact symplectic toric orbifold by means of smooth functions on the corresponding moment polytope this is equivalent to parametrizing all toric kahler metrics and generalizes an analogous result for toric manifolds a simple explicit description of interesting families of extremal kahler metrics arising from recent work of r bryant is given as an application of the approach in this paper the fact that in dimension four these metrics are self dual and conformally einstein is also discussed this gives rise in particular to a one parameter family of self dual einstein metrics connecting the well known eguchi hanson and taub nut metrics

on the position uncertainty measure on the circle new position uncertainty delocalization measures for a particle on the circle are proposed and illustrated in several examples where the previous measures based on 2 periodic position operators appear to be unsatisfactory the new measures are suitably constructed using the standard multiplication angle operator variances they are shown to depend solely on the state of the particle and to obey uncertainty relations of the schrodinger robertson type

two multi dimensional uncertainty relations two multi dimensional uncertainty relations one related to the probability density and the other one related to the probability density current are derived and discussed both relations are stronger than the usual uncertainty relations for the coordinates and momentum

toric generalized kahler ricci solitons with hamiltonian 2 form we show that the generalized kahler ricci soliton equation on 4 dimensional toric kahler orbifolds reduces to odes assuming there is a hamiltonian 2 form this leads to an explicit resolution of this equation on labelled triangles and convex labelled quadrilaterals in particular we give the explicit expression of the kahler ricci solitons of weighted projective planes as well as new examples

kahler ricci solitons on toric fano orbifolds we prove the existence of kahler ricci solitons on toric fano orbifolds hence extend the theorem of wang and zhu adv math 188 87 103 2004 to the orbifold case

maximum rank of a legendrian web abstract we propose the legendrian web in a contact three manifold as a second order generalization of the planar web an abelian relation for a legendrian web is analogously defined as an additive equation among the first integrals of its foliations for a class of legendrian d webs defined by simple second order ode s we give an algebraic construction of d d 1 d 2 2 d 3 6 linearly independent abelian relations we then employ the method of local differential analysis and the theory of linear differential systems to show that d is the maximum rank of a legendrian d web in the complex analytic category we give a possible projective geometric interpretation of d as an analogue of castelnuovo bound for degree 2 d surfaces in the 3 quadric q 3 p 4 via the duality between p 3 and q 3 associated with the rank two complex simple lie group sp 2 c the legendrian 3 webs of maximum rank three are analytically characterized and their explicit local normal forms are found for an application we give an alternative characterization of a darboux super integrable metric as a two dimensional riemannian metric g which admits a mate metric g such that a legendrian 3 web naturally associated with the geodesic foliations of the pair g has maximum rank

ale scalar flat kahler metrics on non compact weighted projective spaces we construct new explicit toric scalar flat kahler ale metrics with orbifold singularities which we use to obtain smooth extremal kahler metrics on appropriate resolutions of orbifolds

generalized intelligent states and squeezing the robertson schrodinger uncertainty relation for two observables a and b is shown to be minimized in the eigenstates of the operator a ib being a complex number such states called generalized intelligent states gis can exhibit arbitrarily strong squeezing of a or b the time evolution of gis is stable for hamiltonians which admit linear in a and b invariants systems of gis for the su 1 1 and su 2 groups are constructed and discussed it is shown that su 1 1 gis contain all the perelomov coherent states cs and the barut and girardello cs while the spin cs are a subset of su 2 gis cs for an arbitrary semisimple lie group can be considered as a gis for the quadratures of the weyl generators

calabi yau metrics on canonical bundles of complex flag manifolds abstract in the present paper we provide a description of complete calabi yau metrics on the canonical bundle of generalized complex flag manifolds by means of lie theory we give an explicit description of complete ricci flat kahler metrics obtained through the calabi ansatz technique we use this approach to provide several explicit examples of noncompact complete calabi yau manifolds these examples include canonical bundles of non toric flag manifolds e g grassmann manifolds and full flag manifolds

interior estimates for the n dimensional abreuʼs equation abstract we study the abreu s equation in n dimensional polytopes and derive interior estimates of solutions under the assumption of the uniform k stability

four dimensional indefinite kahler osserman manifolds it is shown that a four dimensional kahler metric is pointwise osserman if and only if it is either of constant holomorphic sectional curvature or a ricci flat complex surface examples of kahler osserman metrics with nilpotent jacobi operators of all possible degrees are given

robertson intelligent states the diagonalization of the uncertainty matrix and the minimization of robertson inequality for n observables are considered it is proved that for even n this relation is minimized in states which are eigenstates of n 2 independent complex linear combinations of the observables in the case of canonical observables this eigenvalue condition is also necessary such minimizing states are called robertson intelligent states ris the group related coherent states cs with maximal symmetry for semisimple lie groups are a particular case of ris for the quadratures of weyl generators explicit constructions of ris are considered for operators of su 1 1 su 2 and sp n r algebras unlike the group related cs ris can exhibit strong squeezing of group generators multimode squared amplitude squeezed states are naturally introduced as sp n r ris it is shown that the uncertainty matrices for quadratures of q deformed boson operators q 0 and of any k power of are positive definite and can be diagonalized by symplectic linear transformations

a dynamical uncertainty principle in von neumann algebras by operator monotone functions suppose that a1 an are observables selfadjoint matrices and is a state density matrix in this case the standard uncertainty principle proved by robertson gives a bound for the quantum generalized variance namely for det cov aj ak using the commutators aj ak this bound is trivial when n is odd recently a dierent inequality of robertson type has been proved by the authors with the help of the theory of operator monotone functions in this case the bound makes use of the commutators a j and is non trivial for any n in the present paper we generalize this new result to the von neumann algebra case nevertheless the proof appears to simplify all the existing ones

rank based selection strategies for the random walk process abstract in many decision situations such as hiring a secretary selling an asset or seeking a job the value of each offer applicant or choice is assumed to be an independent identically distributed random variable in this paper we consider a special case where the observations are auto correlated as in the random walk model for stock prices for a given random walk process of n observations we explicitly compute the probability that the j th observation in the sequence is the maximum or minimum among all n observations based on the probability distribution of the rank we derive several distribution free selection strategies under which the decision maker s expected utility of selecting the best choice is maximized we show that unlike in the classical secretary problem evaluating more choices in the random walk process does not increase the likelihood of successfully selecting the best

state extended uncertainty relations a scheme for construction of uncertainty relations for n observables and m states is presented several lowest order inequalities are displayed and briefly discussed for two states and and canonical observables the entangled extension of the heisenberg relation reads p 2 q 2 p 2 q 2

uncertainty relations in deformation quantization robertson and hadamard robertson theorems on non negative definite hermitian forms are generalized to an arbitrary ordered field these results are then applied to the case of formal power series fields and the heisenberg robertson robertson schrodinger and trace uncertainty relations in deformation quantization are found some conditions under which the uncertainty relations are minimized are also given

categorical mirror symmetry on cohomology for a complex genus 2 curve abstract motivated by observations in physics mirror symmetry is the concept that certain manifolds come in pairs x and y such that the complex geometry on x mirrors the symplectic geometry on y it allows one to deduce symplectic information about y from known complex properties of x strominger yau zaslow 61 described how such pairs arise geometrically as torus fibrations with the same base and related fibers known as syz mirror symmetry kontsevich 43 conjectured that a complex invariant on x the bounded derived category of coherent sheaves should be equivalent to a symplectic invariant of y the fukaya category see 9 29 49 1 this is known as homological mirror symmetry in this project we first use the construction of generalized syz mirrors for hypersurfaces in toric varieties following abouzaid auroux katzarkov 6 in order to obtain x and y as manifolds the complex manifold is the genus 2 curve 2 so of general type c 1 0 as a hypersurface in its jacobian torus its generalized syz mirror is a landau ginzburg model y v 0 equipped with a holomorphic function v 0 y c which we put the structure of a symplectic fibration on we then describe an embedding of a full subcategory of d b c o h 2 into a cohomological fukaya seidel category of y as a symplectic fibration while our fibration is one of the first nonexact non lefschetz fibrations to be equipped with a fukaya category the main geometric idea in defining it is the same as in seidel s construction for fukaya categories of lefschetz fibrations in 55 and in abouzaid seidel 3

numerical kahler einstein metric on the third del pezzo the third del pezzo surface admits a unique kahler einstein metric which is not known in closed form the manifold s toric structure reduces the einstein equation to a single monge ampere equation in two real dimensions we numerically solve this nonlinear pde using three different algorithms and describe the resulting metric the first two algorithms involve simulation of ricci flow in complex and symplectic coordinates respectively the third algorithm involves turning the pde into an optimization problem on a certain space of metrics which are symplectic analogues of the algebraic metrics used in numerical work on calabi yau manifolds our algorithms should be applicable to general toric manifolds using our metric we compute various geometric quantities of interest including laplacian eigenvalues and a harmonic 1 1 form the metric and 1 1 form can be used to construct a klebanov tseytlin like supergravity solution

the geometric dual of a maximisation for toric sasaki einstein manifolds we show that the reeb vector and hence in particular the volume of a sasaki einstein metric on the base of a toric calabi yau cone of complex dimension n may be computed by minimising a function z on mathbb r n which depends only on the toric data that defines the singularity in this way one can extract certain geometric information for a toric sasaki einstein manifold without finding the metric explicitly for complex dimension n 3 the reeb vector and the volume correspond to the r symmetry and the a central charge of the ads cft dual superconformal field theory respectively we therefore interpret this extremal problem as the geometric dual of a maximisation we illustrate our results with some examples including the yp q singularities and the complex cone over the second del pezzo surface

a robertson type uncertainty principle and quantum fisher information let a1 an be complex self adjoint matrices and let be a density matrix the robertson uncertainty principle det cov ah aj det i 2 tr a h aj gives a bound for the quantum generalized variance in terms of the commutators ah aj the right side matrix is antisymmetric and therefore the bound is trivial equal to zero in the odd case n 2m 1 let f be an arbitrary normalized symmetric operator monotone function and let h i f be the associated quantum fisher information we have conjectured the inequality det cov ah aj det f 0 hi a h i a j i f

the cauchy problem for the homogeneous monge ampere equation ii legendre transform abstract we continue our study of the cauchy problem for the homogeneous real and complex monge ampere equation hrma hcma in the prequel y a rubinstein and s zelditch 27 a quantum mechanical approach for solving the hcma was developed and was shown to coincide with the well known legendre transform approach in the case of the hrma in this article that uses tools of convex analysis and can be read independently we prove that the candidate solution produced by these methods ceases to solve the hrma even in a weak sense as soon as it ceases to be differentiable at the same time we show that it does solve the equation on its dense regular locus and we derive an explicit a priori upper bound on its monge ampere mass the technique involves studying regularity of legendre transforms of families of non convex functions

moments of non gaussian wigner distributions and a generalized uncertainty principle i the single mode case the non negativity of the density operator of a state is faithfully coded in its wigner distribution and this coding places on the moments of the wigner distribution constraints arising from the non negativity of the density operator working in a monomial basis for the algebra of operators on the hilbert space of a bosonic mode we formulate these constraints in a canonically covariant form which is both concise and explicit since the conventional uncertainty relation is such a constraint on the first and second moments our result constitutes a generalization of the same to all orders the structure constants of in the monomial basis are shown to be essentially the su 2 clebsch gordan coefficients our results have applications in quantum state reconstruction using optical homodyne tomography and when generalized to the n mode case which will be done in the second part of this work will have applications also for continuous variable quantum information systems involving non gaussian states

t duality and homological mirror symmetry for toric varieties abstract let x be a complete toric variety the coherent constructible correspondence of fang et al 2011 14 equates p erf t x with a subcategory sh c c m r of constructible sheaves on a vector space m r the microlocalization equivalence of nadler and zaslow 2009 27 and nadler 2009 25 relates these sheaves to a subcategory fuk t m r of the fukaya category of the cotangent t m r when x is nonsingular taking the derived category yields an equivariant version of homological mirror symmetry dcoh t x dfuk t m r which is an equivalence of triangulated tensor categories the nonequivariant coherent constructible correspondence of treumann preprint 33 embeds p erf x into a subcategory s h c t r of constructible sheaves on a compact torus t r when x is nonsingular the composition of and microlocalization yields a version of homological mirror symmetry dcoh x dfuk t t r which is a full embedding of triangulated tensor categories when x is nonsingular and projective the composition is compatible with t duality in the following sense an equivariant ample line bundle l has a hermitian metric invariant under the real torus whose connection defines a family of flat line bundles over the real torus orbits this data produces a t dual lagrangian brane l on the universal cover t m r of the dual real torus fibration we prove l l in fuk t m r thus equivariant homological mirror symmetry is determined by t duality

canonical sasakian metrics let m be a closed manifold of sasaki type a polarization of m is defined by a reeb vector field and for any such polarization we consider the set of all sasakian metrics compatible with it on this space we study the functional given by the square of the l2 norm of the scalar curvature we prove that its critical points or canonical representatives of the polarization are sasakian metrics that are transversally extremal we define a sasaki futaki invariant of the polarization and show that it obstructs the existence of constant scalar curvature representatives for a fixed cr structure of sasaki type we define the sasaki cone of structures compatible with this underlying cr structure and prove that the set of polarizations in it that admit a canonical representative is open we use our results to describe fully the case of the sphere with its standard cr structure showing that each element of its sasaki cone can be represented by a canonical metric we compute their sasaki futaki invariant and use it to describe the canonical metrics that have constant scalar curvature and to prove that only the standard polarization can be represented by a sasaki einstein metric

examples of asymptotically conical ricci flat kahler manifolds previously the author has proved that a crepant resolution y x of a ricci flat kahler cone x admits a complete ricci flat kahler metric asymptotic to the cone metric in every kahler class in h 2 c y mathbb r these manifolds can be considered to be generalizations of the ricci flat ale kahler spaces known by the work of p kronheimer d joyce and others this article considers further the problem of constructing examples we show that every 3 dimensional gorenstein toric kahler cone admits a crepant resolution for which the above theorem applies this gives infinitely many examples of asymptotically conical ricci flat manifolds then other examples are given of which are crepant resolutions hypersurface singularities which are known to admit ricci flat kahler cone metrics by the work of c boyer k galicki j kollar and others we concentrate on 3 dimensional examples two families of hypersurface examples are given which are distinguished by the condition b3 y 0 or b3 y 0

selfdual einstein metrics with torus symmetry it is well known that any 4 dimensional hyperkahler metric with two commuting killing fields may be obtained explicitly via the gibbons hawking ansatz from a harmonic function invariant under a killing field on 3 in this paper we find all selfdual einstein metrics of nonzero scalar curvature with two commuting killing fields they are given explicitly in terms of a local eigenfunction of the laplacian on the hyperbolic plane we discuss the relation of this construction to a class of selfdual spaces found by joyce and some einstein weyl spaces found by ward and then show that certain multipole hyperbolic eigenfunctions yield explicit formulae for the quaternion kahler quotients of pm 1 by an m 2 torus studied by galicki and lawson as a consequence we are able to place the well known cohomogeneity one metrics the quaternion kahler quotients of p2 and noncompact analogues and the more recently studied selfdual einstein hermitian metrics in a unified framework and give new complete examples

bi isotropic decompositions of semisimple lie algebras and homogeneous bi lagrangian manifolds let g be a real semisimple lie algebra with killing form b and k a b nondegenerate subalgebra of g of maximal rank we give a description of all adk invariant decompositions g k m m such that b m 0 b k m m 0 and k m are subalgebras it is reduced to a description of parabolic subalgebras of g with given reductive part k this is obtained in terms of crossed satake diagrams as an application we get a classification of invariant bi lagrangian or equivalently para kahler structures on homogeneous manifolds g k of a semisimple group g

dynamic programming formulation of the group interview problem with a general utility function abstract in many managerial decision situations such as buying an electronic appliance several groups of alternatives are presented sequentially and an accept or reject decision is made immediately after evaluating the alternatives in each group if each group contains only one alternative this optimal selection problem is known as the secretary problem which has a long and rich history of research devoted to developing solution strategies we propose a more generalized version of the secretary problem called the group interview problem in which each group contains more than one alternative and each group is presented and evaluated sequentially over time using a dynamic programming approach we derive a backward recursive equation for solving the group interview problem in which a decision maker s utility of selecting a certain choice is expressed as a general function depending on the specific form of this function we derive optimal selection strategies for various types of group interview problems such as minimum rank maximum utility best choice and one out of the p best choice problems