linear free resolutions and minimal multiplicity let s k x x be a polynomial ring over a field and let a4 a m be a finitely generated graded module in the most interesting case a4 is an ideal of s for a given natural number p there is a great interest in the question can m be generated by homogeneous elements of degree p no simple answer say in terms of the local cohomology of m is known but somewhat surprisingly the stronger question can the jth syzygy of m be generated by elements of degree qp j for all j 0 l n does admit a simple response the following is neither new nor difficult to prove though it seems not well known

dynamical systems associated to separated graphs graph algebras and paradoxical decompositions abstract we attach to each finite bipartite separated graph e c a partial dynamical system e c f where e c is a zero dimensional metrizable compact space f is a finitely generated free group and is a continuous partial action of f on e c the full crossed product c algebra o e c c e c f is shown to be a canonical quotient of the graph c algebra c e c of the separated graph e c similarly we prove that for any field k the algebraic crossed product l k ab e c c k e c alg f is a canonical quotient of the leavitt path algebra l k e c of e c the monoid v l k ab e c of isomorphism classes of finitely generated projective modules over l k ab e c is explicitly computed in terms of monoids associated to a canonical sequence of separated graphs using this we are able to construct an action of a finitely generated free group f on a zero dimensional metrizable compact space z such that the type semigroup s z f k is not almost unperforated where k denotes the algebra of clopen subsets of z finally we obtain a characterization of the separated graphs e c such that the canonical partial action of f on e c is topologically free

eigenvalues invariant factors highest weights and schubert calculus we describe recent work of klyachko totaro knutson and tao that characterizes eigenvalues of sums of hermitian matrices and decomposition of tensor products of representations of gln c we explain related applications to invariant factors of products of matrices intersections in grassmann varieties and singular values of sums and products of arbitrary matrices recent breakthroughs primarily by a klyachko b totaro a knutson and t tao with contributions by p belkale and c woodward have led to complete solutions of several old problems involving the various notions in the title our aim here is to describe this work and especially to show how these solutions are derived from it along the way we will see that these problems are also related to other areas of mathematics including geometric invariant theory symplectic geometry and combinatorics in addition we present some related applications to singular values of arbitrary matrices although many of the theorems we state here have not appeared elsewhere their proofs are mostly soft algebra based on the hard geometric or combinatorial work of others indeed this paper emphasizes concrete elementary arguments we do give some new examples and counterexamples and raise some new open questions we have attempted to point to the sources and to some of the key partial results that had been conjectured or proved before however there is a very large literature particularly for linear algebra problems about eigenvalues singular values and invariant factors we have listed only a few of these articles from whose bibliographies we hope an interested reader can trace the history we apologize to the many whose work is not cited directly we begin in the first five sections by describing each of the problems together with some of their early histories and we state as theorems the new solutions to these problems in section 6 we describe the steps toward these solutions that were carried out before the recent breakthroughs then we discuss the recent solutions and explain how these theorems follow from the work of the above mathematicians sections 7 8 9 and 10 also contain variations and generalizations of some of the theorems stated in the first five sections as well as attributions of the theorems to their authors one of our fascinations with this subject even now that we have proofs of the theorems is the challenge to understand in a deeper way why all these subjects are received by the editors in july 1999 and in revised form january 3 2000 2000 mathematics subject classification primary 15a42 22e46 14m15 secondary 05e15 13f10 14c17 15a18 47b07 the author was partly supported by nsf grant dms9970435 c 2000 american mathematical society

stochastic gradient algorithm with random truncations abstract let f r d r d r be a borel measurable function which satisfies r d f x r d where q 0 is a probability measure on r d b d the problem of minimization of the function f 0 r d x q 0 d r d is considered for the case when the probability measure q 0 is unknown but a realization of a non stationary random process x n n 1 whose single probability measures in a certain sense tend to q 0 is available the random process x n n 1 is defined on a common probability space r valued correlated and satisfies certain uniform mix conditions the function f is completely known a stochastic gradient algorithm with random truncations is used for the minimization of f 0 and its almost sure convergence is proved

conformal blocks and generalized theta functions let yp be the moduli space of rank r vector bundles with trivial determinant on a riemann surface x this space carries a natural line bundle the determinant line bundle s we describe a canonical isomorphism of the space of global sections of sk with the space of conformal blocks defined in terms of representations of the lie algebra 5 c z it follows in particular that the dimension of h yk is given by the verlinde formula

infinite grassmannians and moduli spaces of g bundles let c be a smooth projective irreducible algebraic curve over c of any genus and g a connected simply connected simple affine algebraic group over c in this paper we elucidate the relationship between 1 the space of vacua conformal blocks defined in conformal field theory using an integrable highest weight representation of the affine kac moody algebra associated to g and 2 the space of regular sections generalised theta functions of a line bundle on the moduli space ta of semistable principal g bundles on c fix a point p in c and let p resp kp be the completion of the local ring p of c at p resp the quotient field of c p let g kp the kp rational points of the algebraic group g be the loop group of g and let g p be the standard maximal parahoric subgroup of then the generalised flag variety x n is an inductive limit of projective varieties in fact of generalised schubert varieties one has a natural r line bundle s on x cf sect 2 2 and the picard group pic x is isomorphic to z which is generated by s proposition 2 3 where f is the universal central extension of by the multiplicative group c cf sect 2 2 by an analogue of the borel weil theorem proved in the kac moody setting by kumar and also by mathieu the space h 162 s of the regular sections of the line bundle y dxo y xo for any d 0 is canonically isomorphic with the full vector space dual v dxo of the integrable highest weight irreducible module v dxo of the affine kacmoody group or of the associated affine kac moody algebra which is a certain one dimensional central extension of the loop algebra lie g with highest weight dxo cf sect 6 1

on quantum gauge theories in two dimensions two dimensional quantum yang mills theory is studied from three points of view i by standard physical methods ii by relating it to the largek limit of three dimensional chern simons theory and two dimensional conformal field theory iii by relating its weak coupling limit to the theory of reidemeister ray singer torsion the results obtained from the three points of view agree and give formulas for the volumes of the moduli spaces of representations of fundamental groups of two dimensional surfaces

the cone of type a level 1 conformal blocks divisors abstract we prove that the type a level 1 conformal blocks divisors on m 0 n span a finitely generated full dimensional subcone of the nef cone each such divisor induces a morphism from m 0 n and we identify its image as a git quotient parameterizing configurations of points supported on a flat limit of veronese curves we show how scaling git linearizations gives geometric meaning to certain identities among conformal blocks divisor classes this also gives modular interpretations in the form of git constructions to the images of the hyperelliptic and cyclic trigonal loci in m g under an extended torelli map

discrete time stochastic adaptive control this paper establishes global convergence of a stochastic adaptive control algorithm for discrete time linear systems it is shown that with probability one the algorithm will ensure the system inputs and outputs are sample mean square bounded and the conditional mean square output tracking error achieves its global minimum possible value for linear feedback control thus asymptotically the adaptive control algorithm achieves the same performance as could be achieved if the system parameters were known

the quantization conjecture revisited a strong version of the quantization conjecture of guillemin and sternberg is proved for a reductive group action on a smooth compact polarized variety x l the cohomologies of l over the git quotient x g equal the invariant part of the cohomologies over x this generalizes the theorem of gs on global sections and strengthens its subsequent extensions jk li to riemannroch numbers remarkable by products are the invariance of cohomology of vector bundles over x g under a small change in the defining polarization or under shift desingularization as well as a new proof of boutot s theorem also studied are equivariant holomorphic forms and the equivariant hodgeto de rham spectral sequences for x and its strata whose collapse is shown one application is a new proof of the borel weil bott theorem of ti for the moduli stack of c bundles over a curve and of analogous statements for the moduli stacks and spaces of bundles with parabolic structures collapse of the hodge to de rham sequences for these stacks is also shown

optimization of the transient and steady state behavior of discrete event systems we present a general framework for applying simulation to optimize the behavior of discrete event systems our approach involves modeling the discrete event system under study as a general state space markov chain whose distribution depends on the decision parameters we then show how simulation and the likelihood ratio method can be used to evaluate the performance measure of interest and its gradient and we present conditions that guarantee that the robbins monro stochastic approximation algorithm will converge almost surely to the optimal values of the decision parameters both transient and steady state performance measures are considered for steady state performance measures we consider both the case when the markov chain of interest is regenerative in the standard sense as well as the case when this markov chain is harris recurrent and thereby regenerative in a wider sense

on the asymptotic accuracy of pseudo linear regression algorithms the accuracy properties of a general pseudo linear regression plr method are examined both off line and on line algorithms are considered assuming the parameter estimates converge they will under weak conditions be asymptotically gaussian distributed expressions for the corresponding covariance matrices are given it is shown that the asymptotic covariance matrix of the off line plr algorithm is bounded from above by the matrix corresponding to the on line plr algorithm and from below by that corresponding to the prediction error method some simple numerical illustrations of the theoretical results are also included

a stochastic approximation algorithm with varying bounds many optimization problems that are intractable with conventional approaches will yield to stochastic approximation algorithms this is because these algorithms can be used to optimize functions that cannot be evaluated analytically but have to be estimated for instance through simulation or measured thus stochastic approximation algorithms can be used for optimization in simulation unfortunately the classical stochastic approximation algorithm sometimes diverges because of unboundedness problems we study the convergence of a variant of stochastic approximation defined over a growing sequence of compact sets we show that this variant converges under more general conditions on the objective function than the classical algorithm while maintaining the same asymptotic convergence rate we also present empirical evidence that shows that this algorithm sometimes converges much faster than the classical algorithm

borel weil bott theory on the moduli stack of g bundles over a curve let g be a semi simple group and m the moduli stack of g bundles over a smooth complex projective curve using representation theoretic methods i prove the pure dimensionality of sheaf cohomology for certain evaluation vector bundles over m twisted by powers of the fundamental line bundle this result is used to prove a borel weil bott theorem conjectured by g segal for certain generalized flag varieties of loop groups along the way the homotopy type of the group of algebraic maps from an affine curve to g and the homotopy type the hodge theory and the picard group of m are described one auxiliary result in appendix a is the alexander cohomology theorem conjectured in gro2 a self contained account of the uniformization theorem of ls for the stack m is given including a proof of a key result of drinfeld and simpson in characteristic 0 a basic survey of the simplicial theory of stacks is outlined in appendix b

a manufacturing system with general stationary failure process stability and ipa of hedging control policies this paper concerns a new methodology for the adaptive optimization of piecewise deterministic non markovian systems via a simple example of interest in manufacturing this methodology takes into account the fact that piecewise deterministic systems are rarely markovian and that classical control theory based on dynamic programming of markovian systems cannot provide quantitative answers in most realistic situations the example considered is that of a single machine single part production system from the point of view of the control theory by kimemia and gershwin 1983 we obtain stochastic gradient estimates via the perturbation analysis of ho and cao 1991 by the regenerative method of konstantopoulos and zazanis 1992 in view of stochastic optimization its mathematical justification requires a careful study of the regenerative structure of the process in particular we give the necessary and sufficient conditions of stability of this system via the method of loynes 1962

factorisation of generalised theta functions i we prove a version of factorisation relating the space of sections of theta bundles on the moduli spaces of parabolic rank 2 vector bundles on curves of genusg andg 1

quantum generalization of the horn conjecture recall that the lie algebra of the special unitary group su n is isomorphic to the real vector space of traceless hermitian matrices as representations of su n and hence the terminology additive eigenvalue problem for su n in 1962 horn 17 gave a conjectural solution to a problem equivalent to the above additive eigenvalue problem for su n by a recursively determined system of inequalities in 19 klyachko gave a solution to the additive eigenvalue problem for su n in terms of a certain system of inequalities to write down this system we need to know which structure constants in the cohomology of gr r n written in the schubert basis 0 2 be a positive integer characterize the possible eigenvalues a1 a2 as of matrices a a 2 a e su n which satisfy a a a 1

convergence of adaptive minimum variance algorithms via weighting coefficient selection weighted least squares and related stochastic approximation algorithms are studied for parameter estimation adaptive state estimation adaptive n step ahead prediction and adaptive control in both white and colored noise environments for the fundamental algorithm which is the basis for the various applications the step size in the stochastic approximation versions and the weighting coefficient in the weighted least squares schemes are selected according to a readily calculated stability measure associated with the estimator the selection is guided by the convergence theory in this way strong global convergence of the parameter estimates state estimates and prediction or tracking errors is not only guaranteed under the appropriate noise passivity and stability or minimum phase conditions but the convergence is also as fast as it appears reasonable to achieve given the simplicity of the adaptive scheme

scaling of conformal blocks and generalized theta functions over overline mathcal m _ g n by way of intersection theory on overline mathcal m g n we show that geometric interpretations for conformal blocks as sections of ample line bundles over projective varieties do not have to hold at points on the boundary we show such a translation would imply certain recursion relations for first chern classes of these bundles while recursions can fail geometric interpretations are shown to hold under certain conditions

estimating the false discovery rate using the stochastic approximation algorithm testing of multiple hypotheses involves statistics that are strongly dependent in some applications but most work on this subject is based on the assumption of independence we propose a new method for estimating the false discovery rate of multiple hypothesis tests in which the density of test scores is estimated parametrically by minimizing the kullback leibler distance between the unknown density and its estimator using the stochastic approximation algorithm and the false discovery rate is estimated using the ensemble averaging method our method is applicable under general dependence between test statistics numerical comparisons between our method and several competitors conducted on simulated and real data examples show that our method achieves more accurate control of the false discovery rate in almost all scenarios

geometric proof of a conjecture of fulton we give a geometric proof of a conjecture of fulton on the multiplicities of irreducible representations in a tensor product of irreducible representations for gl r

convergence and rate of convergence of a recursive identification and adaptive control method which uses truncated estimators a stochastic approximation like method is used for the recursive identification of the coefficients in y n sum min 1 max l 1 a i y n i sum min 0 max l 2 b i u n i sum min 1 max l 3 c i w n i w n where w n is a sequence of mutually independent and bounded random variables and is independent of the bounded u n such methods normally require the recursive estimation of the residuals or the w n and the algorithms for doing this can be unstable if the parameter estimates are not close enough to their true values the problem is solved here by use of a simple truncated estimator which is probably what would be used in implementation in any case then under a stability and strict positive real type condition with probability 1 w p 1 convergence is proved and the rate of convergence is obtained an associated adaptive control problem is also treated

adjoining a universal inner inverse to a ring element abstract let r be an associative unital algebra over a field k let p be an element of r and let r r q p q p p we obtain normal forms for elements of r and for elements of r modules arising by extension of scalars from r modules the details depend on where in the chain p r r p p r r p p r r p r the unit 1 of r first appears this investigation is motivated by a hoped for application to the study of the possible forms of the monoid of isomorphism classes of finitely generated projective modules over a von neumann regular ring but that goal remains distant we end with a normal form result for the algebra obtained by tying together a k algebra r given with a nonzero element p satisfying 1 p r r p and a k algebra s given with a nonzero q satisfying 1 q s s q via the pair of relations p p q p q q p q

budget dependent convergence rate of stochastic approximation convergence rate results are derived for a stochastic optimization problem where a performance measure is minimized with respect to a vector parameter t assuming that a gradient estimator is available and that both the bias and the variance of the estimator are known functions of the budget devoted to its computation the gradient estimator is employed in conjunction with a stochastic approximation sa algorithm our interest is to figure out how to allocate the total available computational budget to the successive sa iterations the effort is devoted to solving the asymptotic version of this problem by finding the convergence rate of sa toward the optimizer first as a function of the number of iterations and then as a function of the total computational effort as a result the optimal rate of increase of the computational budget per iteration can be found explicit expressions for the case where the computational budget devoted to an iteration is a polynomial in the iteration number and where the bias and variance of the gradient estimator are polynomials of the computational budget are derived applications include the optimization of steady state simulation models with likelihood ratio perturbation analysis or finite difference gradient estimators optimization of infinite horizon models with discounting optimization of functions of several expectations and so on several examples are discussed our results readily generalize to general root finding problems

the action of outer automorphisms on bundles of chiral blocks on the bundles of wzw chiral blocks over the moduli space of a punctured rational curve we construct isomorphisms that implement the action of outer automorphisms of the underlying affine lie algebra these bundle isomorphisms respect the knizhnik zamolodchikov connection and have finite order when all primary fields are fixed points the isomorphisms are endomorphisms in this case the bundle of chiral blocks is typically a reducible vector bundle a conjecture for the trace of such endomorphisms is presented the proposed relation generalizes the verlinde formula our results have applications to conformal field theories based on non simply connected groups and to the classification of boundary conditions in such theories

almost sure convergence of stochastic approximation algorithms with non additive noise abstract stochastic approximation algorithms with non additive noise are discussed in studying strong convergence of such algorithms traditionally one assumes that the iterates return to a bounded or compact set infinitely often or that the function under consideration grows with certain rate the usual projection algorithms require that the bounded projection region is known beforehand it is desirable to weaken these boundedness conditions by introducing randomly varying truncations chen and zhu 1986 achieved this for stochastic approximation algorithms with additive noise here we extend their result to a more general setting