moment maps and riemannian symmetric pairs we study hamiltonian actions of a compact lie group on a symplectic manifold in the presence of an involution on the group and an antisymplectic involution on the manifold the fixed point set of the involution on the manifold is a lagrangian submanifold we investigate its image under the moment map and conclude that the intersection with the weyl chamber is an easily described subpolytope of the kirwan polytope of special interest is the integral kahler case where much stronger results hold in particular we obtain convexity theorems for closures of orbits of the noncompact dual group in the sense of the theory of symmetric pairs in the abelian case these results were obtained earlier by duistermaat we derive explicit inequalities for polytopes associated with real flag varieties

domain dependence of solutions to compressible navier stokes equations the minimization of the drag functional for the stationary isentropic compressible navier stokes equations nse in three spatial dimensions is considered in order to establish the existence of an optimal shape the general result on compactness of families of generalized solutions to the nse is established within in the framework of the modern theory of nonlinear pdes p l lions mathematical topics in fluid dynamics vol 2 compressible models oxford university press clarendon press new york 1998 e feireisl dynamics of viscous compressible fluids oxford university press oxford uk 2004 the family of generalized solutions to the nse is constructed over a family of admissible domains mathcal u ad any admissible domain omega b setminus s contains an obstacle s e g a wing profile compactness properties of the family of admissible domains in the form of the condition mathcal h omega is imposed roughly speaking the condition mathcal h omega is satisfied provided that for any sequence of admissible domains omega n subset mathcal u ad there is a subsequence convergent both in hausdorff metrics and in the sense of kuratowski and mosco the analysis is performed for the adiabatic constant gamma 1 in the pressure law p rho rho gamma and it is based on the technique used in p i plotnikov and j sokolowski comm math phys 258 2005 pp 567 608 in the case of discretized nse the result is a generalization to the stationary equations with gamma 1 of the results obtained in e feireisl a h novotny and h petzeltova math methods appl sci 25 2002 pp 1045 1073 e feireisl appl math optim 47 2003 pp 59 78 for evolution equations within the range gamma 3 2 for the adiabatic ratio

construction of invariant whiskered tori by a parameterization method part ii quasi periodic and almost periodic breathers in coupled map lattices we construct quasi periodic and almost periodic solutions for coupled hamiltonian systems on an infinite lattice which is translation invariant the couplings can be long range provided that they decay moderately fast with respect to the distance for the solutions we construct most of the sites are moving in a neighborhood of a hyperbolic fixed point but there are oscillating sites clustered around a sequence of nodes the amplitude of these oscillations does not need to tend to zero in particular the almost periodic solutions do not decay at infinity we formulate an invariance equation solutions of this equation are embeddings of an invariant torus on which the motion is conjugate to a rotation we show that if there is an approximate solution of the invariance equation that satisfies some non degeneracy conditions there is a true solution close by the proof of this emph a posteriori theorem is based on a nash moser iteration which does not use transformation theory simpler versions of the scheme were developed in e fontich r de la llave y sire emph j differential equations bf 246 3136 2009 one technical tool important for our purposes is the use of weighted spaces that capture the idea that the maps under consideration are local interactions using these weighted spaces the estimates of iterative steps are similar to those in finite dimensional spaces in particular the estimates are independent of the number of nodes that get excited using these techniques given two breathers we can place them apart and obtain an approximate solution which leads to a true solution nearby by repeating the process infinitely often we can get solutions with infinitely many frequencies which do not tend to zero at infinity

what is the optimal shape of a pipe we consider an incompressible fluid in a three dimensional pipe following the navier stokes system with classical boundary conditions we are interested in the following question is there any optimal shape for the criterion energy dissipated by the fluid moreover is the cylinder the optimal shape we prove that there exists an optimal shape in a reasonable class of admissible domains but the cylinder is not optimal for that purpose we explicit the first order optimality condition thanks to adjoint state and we prove that it is impossible that the adjoint state be a solution of this over determined system when the domain is the cylinder at last we show some numerical simulations for that problem

some variational formulas for hausdorff dimension topological entropy and srb entropy for hyperbolic dynamical systems we derive several variational formulas for the topological entropy and srb entropy of axiom a flows on compact manifolds and for the hausdorff dimension of basic sets for axiom a diffeomorphisms on compact surfaces

convex functions on symmetric spaces side lengths of polygons and the stability inequalities for weighted configurations at infinity in a symmetric space of noncompact type x g k oriented geodesic segments correspond modulo isometries to vectors in the euclidean weyl chamber we can hence assign vector valued lengths to segments our main result is a system of homogeneous linear inequalities which we call the generalized triangle inequalities or stability inequalities describing the restrictions on the vector valued side lengths of oriented polygons it is based on the mod 2 schubert calculus in the real grassmannians g p for maximal parabolic subgroups p the side lengths of polygons in euclidean buildings are studied in the related paper klm2 applications of the geometric results in both papers to algebraic group theory are given in klm3

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

multiplicative lidskii s inequalities and optimal perturbations of frames abstract in this paper we study two design problems in frame theory on the one hand given a fixed finite frame f f j j i n for c d we compute those dual frames g of f that are optimal perturbations of the canonical dual frame for f under certain restrictions on the norms of the elements of g on the other hand we compute those v f v f j j i n for invertible operators v which are close to the identity that are optimal perturbations of f that is we compute the optimal perturbations of f among frames g g j j i n that have the same linear relations as f in both cases optimality is measured with respect to submajorization of the eigenvalues of the frame operators hence our optimal designs are minimizers of a family of convex potentials that include the frame potential and the mean squared error the key tool for these results is a multiplicative analogue of lidskii s inequality in terms of log majorization and a characterization of the case of equality

the high road to an exponential formula a conjectured exponential formula is proved using a recent result of klyachko linear algebra appl 319 2000 37 then the exponential formula is applied to confirm the equality of two types of spectral triples used to describe all homogeneous eigenvalue inequalities for sum of hermitian matrices and product of positive definite matrices

temperley lieb pfaffinants and schur q positivity conjectures we study pfaffian analogues of immanants which we call pfaffinants our main object is the tl pfaffinants which are analogues of rhoades and skandera s tl immanants we show that tl pfaffinants are positive when applied to planar networks and explain how to decompose products of complementary pfaffians in terms of tl pfaffinants we conjecture in addition that tl pfaffinants have positivity properties related to schur q functions

analyticity of the srb measure of a lattice of coupled anosov diffeomorphisms of the torus we consider the thermodynamic limit of a d dimensional lattice of hyperbolic dynamical systems on the 2 torus interacting via weak and nearest neighbor coupling we prove that the srb measure is analytic in the strength of the coupling the proof is based on symbolic dynamics techniques that allow us to map the srb measure into a gibbs measure for a spin system on a d 1 dimensional lattice this gibbs measure can be studied by an extension decimation of the usual cluster expansion techniques

least squares approximation by elements from matrix orbits achieved by gradient flows on compact lie groups let s a denote the orbit of a complex or real matrix a under a certain equivalence relation such as unitary similarity unitary equivalence unitary congruences etc efficient gradient flow algorithms are constructed to determine the best approximation of a given matrix a0 by the sum of matrices in s a1 s an in the sense of finding the euclidean least squares distance

geometric quantization and multiplicities of group representations the heisenberg uncertainty principle says that it is impossible to determine simultaneously the position and momentum of a quantum mechanical particle this can be rephrased as follows the smallest subsets of classical phase space in which the presence of a quantum mechanical particle can be detected are its lagrangian submanifolds for this reason it makes sense to regard the lagrangian submanifolds of phase space as being its true points see weinstein 17 now let g be a compact lie group and g x x x a hamiltonian action of g on x see w for definitions it is well known that the fixed points of this action form a symplectic submanifold of x see for instance guillemin and sternberg 5 however what can one say about the fixed points of g we will show that they are also the points of a symplectic manifold xc this manifold is the marsden weinstein reduction of x with respect to the zero orbit in g and will be described in sect 2 it was introduced in a completely different context fl om ours by marsden and weinstein 12 problems in classical mechanics can often be reduced to the study of hamiltonian systems on symplectic manifolds and problems in quantum mechanics to the study of linear operators on hilbert space this fact has inspired a number of efforts to quantize symplectic geometry by devising schemes for associating hilbert space to symplectic manifolds the no go theorems of groenwald and van hove impose some embarrassing limitations on all such schemes however it seems to be a useful idea heuristically to think of every symplectic manifold x a ic 1 as being symbiotically associated with a hilbert space xq a t in such a way that the classical observables on the first space correspond to quantum observables on the second space the heuristics further suggests that if g is a group of symmetries of x la i j it should also be a group of symmetries of xq a tum in this heuristic spirit we will state the main conjecture of this paper

singular values and invariant factors of matrix sums and products abstract some remarks are made concerning the range of individual singular values of sums and products of two complex square matrices when these are allowed to vary in their orbits under unitary equivalence an analogous question is considered for invariant factors of products of matrices over a principal ideal domain

the spectrum of a hermitian matrix sum abstract an announcement by lidskii b v lidskii functional anal appl 10 1982 76 77 russian 139 140 english claimed to establish the explicit description of the spectrum of a hermitian matrix sum in the form conjectured by horn a horn pacific j math 16 1962 225 241 but no supporting proof has been published this paper begins an analysis of the claim and is the first step towards bridging the distance between the elementary methods of horn s 1962 paper and the partial solution of the same problem using noncommutative harmonic analysis by dooley et al i dooley r repka n wildberger lin mult alg 36 1993 79 102 the methods of horn lidskii lead to combinatorial issues of independent interest

kazhdan lusztig immanants and products of matrix minors abstract using the kazhdan lusztig basis c w 1 w s n for the symmetric group algebra we obtain nonnegativity properties of certain polynomials in matrix minors in particular we show that the application of these polynomials to jacobi trudi matrices yields symmetric functions which are equal to nonnegative linear combinations of schur functions

random walks on symmetric spaces and inequalities for matrix spectra abstract using harmonic analysis on symmetric spaces we reduce the singular spectral problem for products of matrices to the recently solved spectral problem for sums of hermitian matrices this proves r c thompson s conjecture matrix spectral inequalities johns hopkins university press baltimore md 1988

uniqueness and exponential decay of correlations for some two dimensional spin lattice systems we consider a two dimensional lattice spin system which naturally arises in dynamical systems called coupled map lattice the configuration space of the spin system is a direct product of mixing subshifts of finite type the potential is defined on the set of all squares in z2 and decays exponentially with the linear size of the square via the polymer expansion technique we prove that for sufficiently high temperatures the limit gibbs distribution is unique and has an exponential decay of correlations

discrete gradient flows for shape optimization and applications we present a variational framework for shape optimization problems that establishes clear and explicit connections among the continuous formulation its full discretization and the resulting linear algebraic systems our approach hinges on the following essential features shape differential calculus a semi implicit time discretization and a finite element method for space discretization we use shape differential calculus to express variations of bulk and surface energies with respect to domain changes the semi implicit time discretization allows us to track the domain boundary without an explicit parametrization and has the flexibility to choose different descent directions by varying the scalar product used for the computation of normal velocity we propose a schur complement approach to solve the resulting linear systems efficiently we discuss applications of this framework to image segmentation optimal shape design for pde and surface diffusion along with the choice of suitable scalar products in each case we illustrate the method with several numerical experiments some developing pinch off and topological changes in finite time

horn inequalities for nonzero kronecker coefficients the kronecker coefficients and the littlewood richardson coefficients are nonnegative integers depending on three partitions by definition these coefficients are the multiplicities of the tensor product decomposition of two irreducible representations of symmetric groups resp linear groups by a classical littlewood murnaghan s result the kronecker coefficients extend the littlewood richardson ones the nonvanishing of a littlewood richardson coefficient implies linear inequalities on the triple of partitions called horn inequalities in this paper we extend the essential horn inequalities to the triples of partitions corresponding to a nonzero kronecker coefficient

on the distribution of the length of the longest increasing subsequence of random permutations let sn be the group of permutations of 1 2 n if 7r e sn we say that 7 i1 7f ik is an increasing subsequence in 7r if il i2 ik and 7r ii 7r i2 7r ik let 1n r be the length of the longest increasing subsequence for example if n 5 and 7r is the permutation 5 1 3 2 4 in one line notation thus 7r 1 5 7r 2 1 then the longest increasing subsequences are 1 2 4 and 1 3 4 and n 3 equip sn with uniform distribution

exact solutions for isotropic random flights in odd dimensions we give a method to solve the isotropic random flight in arbitrary dimension and show that when the dimension is odd it leads to a piecewise polynomial expression for the probability density the method is based on the fact that the projection of an isotropic function has an inverse and on the fact that the convolution commutes with the projection and the inverse projection in order to develop this method the abel transform and its inverse are extended by comparing with the expression for the isotropic random flight obtained using the convolution theorem some integrals of bessel functions can be expressed as piecewise polynomials the expressions for the five dimensional case for 2 3 and 4 steps are found explicitly although the main result derived from this method concerns odd dimensions it also yields an analytical result in the four dimensional isotropic random flight which is derived here the results of this article can also be considered as results on the convolution with itself of an isotropic probability density whose support is the surface of a sphere

shape sensitivity analysis for a microchannel cooling system abstract we analyze the theoretical framework of a shape optimization problem for a microchannel cooling system to this end a cost functional based on the tracking of absorbed energy by the cooler as well as some desired flow on a subdomain of the cooling system is introduced the flow and temperature of the coolant are modeled by a stokes system coupled to a convection diffusion equation we prove the well posedness of this model on a domain transformed by the speed method further we rigorously prove that the cost functional of our optimization problem is shape differentiable and calculate its shape derivative by means of a recent material derivative free adjoint approach

semi invariants of quivers and saturation for littlewood richardson coefficients q is defined in the space of all weights by one homogeneous linear equation and by a finite set of homogeneous linear inequalities in particular the set q is saturated i e if n q then also q these results when applied to a special quiver q tn n n and to a special dimension vector show that the gln module v appears in v v if and only if the partitions and satisfy an explicit set of inequalities this gives new proofs of the results of klyachko 7 3 and knutson and tao 8 the proof is based on another general result about semi invariants of quivers theorem 1 in the paper 10 schofield defined a semi invariant cw for each indecomposable representation w of q we show that the semi invariants of this type span each weight space in si q this seems to be a fundamental fact connecting semi invariants and modules in a direct way given this fact the results on sets of weights follow at once from the results in another paper of schofield 11

angular momenta of relative equilibrium motions and real moment map geometry chenciner and jimenez perez mosc math j 13 4 621 630 2013 showed that the range of the spectra of the angular momenta of all the rigid motions of a fixed central configuration in a general euclidean space form a convex polytope in this note we explain how this result follows from a general convexity theorem of o shea and sjamaar in real moment map geometry math ann 31 415 457 2000 finally we provide a representation theoretic description of the pushforward of the normalized measure under the real moment map for riemannian symmetric pairs

shape optimization with stokes constraints over the set of axisymmetric domains in this paper we are interested in the study of shape optimizations problems with stokes constraints within the class of axisymmetric domains represented by the graph of a function existence results with weak assumptions on the regularity of the graph are provided we strongly use these assumptions to get some topological properties we formulate the shape optimization problem using different constraints formulations uniform bound constraints on the function and its derivative and or volume global constraint writing the first order optimality conditions allows to provide quasi explicit solutions in some particular cases and to give some hints for the treatment of the generic problem furthermore we extend the negative result of 16 dealing with the non optimality of the cylinder

the sum of unitary similarity orbits containing only special operators abstract let b h be the algebra of bounded linear operator acting on a hilbert space h over the complex or real field characterization is given to a 1 a k b h such that for any unitary operators u 1 u k j 1 k u j a j u j is always in a special class s of operators such as normal operators self adjoint operators unitary operators as corollaries characterizations are given to a b h such that complex real or nonnegative linear combinations of operators in its unitary orbit u a u au u unitary always lie in s

the poisson geometry of su 1 1 we study the natural poisson structure on the group su 1 1 and related questions in particular we give an explicit description of the ginzburg weinstein isomorphism for the sets of admissible elements we also establish an analog of thompson s conjecture for this group

bethe algebra of homogeneous xxx heisenberg model has simple spectrum we show that the algebra of commuting hamiltonians of the homogeneous xxx heisenberg model has simple spectrum on the subspace of singular vectors of the tensor product of two dimensional mathfrak gl 2 modules as a byproduct we show that there exist exactly binom n l binom n l 1 two dimensional vector subspaces v subset mathbb c u with a basis f g in v such that deg f l deg g n l 1 and f u g u 1 f u 1 g u u 1 n

eigenvalues of hermitian matrices with positive sum of bounded rank we give a minimal list of inequalities characterizing the possible eigenvalues of a set of hermitian matrices with positive semidefinite sum of bounded rank this answers a question of barvinok

coincidences among skew schur functions new sufficient conditions and necessary conditions are developed for two skew diagrams to give rise to the same skew schur function the sufficient conditions come from a variety of new operations related to ribbons also known as border strips or rim hooks the necessary conditions relate to the extent of overlap among the rows or among the columns of the skew diagram

schur positivity and schur log concavity we prove okounkov s conjecture a conjecture of fomin fulton li poon and a special case of lascoux leclerc thibon s conjecture on schur positivity and give several more general statements using a recent result of rhoades and skandera we also give an intriguing log concavity property of schur functions

arrangements of equal minors in the positive grassmannian abstract we discuss arrangements of equal minors of totally positive matrices more precisely we investigate the structure of equalities and inequalities between the minors we show that arrangements of equal minors of largest value are in bijection with sorted sets which earlier appeared in the context of alcoved polytopes and grobner bases maximal arrangements of this form correspond to simplices of the alcoved triangulation of the hypersimplex and the number of such arrangements equals the eulerian number on the other hand we prove in many cases that arrangements of equal minors of smallest value are weakly separated sets weakly separated sets originally introduced by leclerc and zelevinsky are closely related to the positive grassmannian and the associated cluster algebra however we also construct examples of arrangements of smallest minors which are not weakly separated using chain reactions of mutations of plabic graphs

linear response function for coupled hyperbolic attractors the srb measures of a hyperbolic system are widely accepted as the measures that are physically relevant it has been shown by ruelle that they depend smoothly on the system furthermore ruelle showed by a separate argument that the first derivative i e the linear response function admits a geometric interpretation in this paper we consider thermodynamic limits of srb measures in lattices of coupled hyperbolic attractors in a previous paper using markov partitions and thermodynamic formalism we had established the smooth dependence of thermodynamic limits of srb measures here we establish that the linear response function admits a geometric interpretation the formula is analogous to the one found by ruelle for finite dimensional systems if one term is reinterpreted appropriately we show that the limiting derivative is the thermodynamic limit of the derivatives in finite volume we also obtain similar results for the derivatives of the entropy

dynamical systems on lattices with decaying interaction ii hyperbolic sets and their invariant manifolds abstract this is the second part of the work devoted to the study of maps with decay in lattices here we apply the general theory developed in fontich et al 2011 3 to the study of hyperbolic sets in particular we establish that any close enough perturbation with decay of an uncoupled lattice map with a hyperbolic set has also a hyperbolic set with dynamics on the hyperbolic set conjugated to the corresponding of the uncoupled map we also describe how the decay properties of the maps are inherited by the corresponding invariant manifolds

the recursive nature of cominuscule schubert calculus the necessary and sufficient horn inequalities which determine the non vanishing littlewood richardson coefficients in the cohomology of a grassmannian are recursive in that they are naturally indexed by non vanishing littlewood richardson coefficients on smaller grassmannians we show how non vanishing in the schubert calculus for cominuscule flag varieties is similarly recursive for these varieties the non vanishing of products of schubert classes is controlled by the non vanishing products on smaller cominuscule flag varieties in particular we show that the lists of schubert classes whose product is non zero naturally correspond to the integer points in the feasibility polytope which is defined by inequalities coming from non vanishing products of schubert classes on smaller cominuscule flag varieties while the grassmannian is cominuscule our necessary and sufficient inequalities are different than the classical horn inequalities

convexity properties of gradient maps abstract we consider the action of a real reductive group g on a kahler manifold z which is the restriction of a holomorphic action of a complex reductive group h we assume that the action of a maximal compact subgroup u of h is hamiltonian and that g is compatible with a cartan decomposition of h we have an associated gradient map p z p where g k p is the cartan decomposition of g for a g stable subset y of z we consider convexity properties of the intersection of p y with a closed weyl chamber in a maximal abelian subspace a of p our main result is a convexity theorem for real semi algebraic subsets y of z p v where v is a unitary representation of u

a local fluctuation theorem abstract a mechanism for the validity of a local version of the fluctuation theorem uniform in the system size is discussed for a reversible chain of weakly coupled anosov systems

decomposable representations and lagrangian submanifolds of moduli spaces associated to surface groups the importance of explicit examples of lagrangian submanifolds of moduli spaces is revealed by papers such as dostoglou and salamon ann of math 2 139 3 581 640 1994 and salamon proceedings of the international congress of mathematicians vol 1 2 zurich 1994 pp 526 536 birkhauser basel 1995 given a 3 manifold m with boundary m dostoglou and salamon use such examples to obtain a proof of the atiyah floer conjecture relating the symplectic floer homology of the representation space hom 1 m u u associated to an explicit pair of lagrangian submanifolds of this representation space and the instanton homology of the 3 manifold m in the present paper we construct a lagrangian submanifold of the space of representations mathcal m g l hom mathcal c pi g l u u of the fundamental group g lof a punctured riemann surface g l into an arbitrary compact connected lie group u this lagrangian submanifold is obtained as the fixed point set of an anti symplectic involution hat beta defined on mathcal m g l we show that the involution hat beta is induced by a form reversing involution defined on the quasi hamiltonian space u times u g times mathcal c 1 times cdots times mathcal c l the fact that hat beta has a non empty fixed point set is a consequence of the real convexity theorem for group valued momentum maps proved in schaffhauser a real convexity theorem for quasi hamiltonian actions submitted 25 p 2007 http arxiv org abs math 0705 0858 the notion of decomposable representation provides a geometric interpretation of the lagrangian submanifold thus obtained

on products of sln characters and support containment abstract let and be dominant weights of sl n satisfying let v denote the highest weight module corresponding to lam postnikov pylyavskyy conjectured a sufficient condition for v v to be contained in v v as sl n modules in this note we prove a weaker version of the conjecture namely we prove that under the conjectured conditions every irreducible sl n module which appears in the decomposition of v v does appear in the decomposition of v v

finite and infinite dimensional generalizations of klyachko s theorem abstract we describe the convex set of the eigenvalues of hermitian matrices which are majorized by a sum of m hermitian matrices with prescribed eigenvalues we extend our characterization to selfadjoint nonnegative definite compact operators on a separable hilbert space we give necessary and sufficient conditions on the eigenvalue sequence of a selfadjoint nonnegative compact operator of trace class to be a sum of m selfadjoint nonnegative compact operators of trace class with prescribed eigenvalue sequences

the limit of the product of the parameterized exponentials of two operators abstract given two self adjoint positive compact operators a b on a separable hilbert space we show that there exists a self adjoint positive compact operator c commuting with b such that lim t e bt 2 e at e bt 2 1 t e c 0

the horn conjecture for sums of compact selfadjoint operators we determine the possible nonzero eigenvalues of compact selfadjoint operators a b 1 b 2 dots b m with the property that a b 1 b 2 cdots b m when all these operators are positive the eigenvalues were known to be subject to certain inequalities which extend horn s inequalities from the finite dimensional case when m 2 we find the proper extension of the horn inequalities and show that they along with their reverse analogues provide a complete characterization our results also allow us to discuss the more general situation where only some of the eigenvalues of the operators a and b k are specified a special case is the requirement that b 1 b 2 cdots b m be positive of rank at most rho ge1

differentiation of srb states abstract let f be a diffeomorphism of a manifold m and f a generalized srb state for f if supp f is a hyperbolic compact set we show that the map is differentiable in a suitable functional setup and we compute the derivative when supp f is an attractor the derivative is given by where x is the vector field this formula can be extended to time dependent situations and also at least formally to nonuniformly hyperbolic situations the above results will find their use in the study of the onsager reciprocity relations and the fluctuation dissipation formula of nonequilibrium statistical mechanics

eigenvalues singular values and littlewood richardson coefficients we characterize the relationship between the singular values of a hermitian resp real symmetric complex symmetric matrix and the singular values of its off diagonal block we also characterize the eigenvalues of a hermitian or real symmetric matrix c a b in terms of the combined list of eigenvalues of a and b the answers are given by horn type linear inequalities the proofs depend on a new inequality among littlewood richardson coefficients

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

stein s method and characters of compact lie groups stein s method is used to study the trace of a random element from a compact lie group or symmetric space central limit theorems are proved using very little information character values on a single element and the decomposition of the square of the trace into irreducible components this is illustrated for lie groups of classical type and dyson s circular ensembles the approach in this paper will be useful for the study of higher dimensional characters where normal approximations need not hold

cartan decomposition of the moment map we investigate a class of actions of real lie groups on complex spaces using moment map techniques we establish the existence of a quotient and a version of luna s slice theorem as well as a version of the hilbert mumford criterion a global slice theorem is proved for proper actions we give new proofs of results of mostow on decompositions of groups and homogeneous spaces

dynamical systems on lattices with decaying interaction i a functional analysis framework abstract we consider weakly coupled map lattices with a decaying interaction that is we consider systems which consist of a phase space at every site such that the dynamics at a site is little affected by the dynamics at far away sites we develop a functional analysis framework which formulates quantitatively the decay of the interaction and is able to deal with lattices such that the sites are manifolds this framework is very well suited to study systematically invariant objects one obtains that the invariant objects are essentially local we use this framework to prove a stable manifold theorem and show that the manifolds are as smooth as the maps and have decay properties i e the derivatives of one of the coordinates of the manifold with respect to the coordinates at far away sites are small other applications of the framework are the study of the structural stability of maps with decay close to uncoupled possessing hyperbolic sets and the decay properties of the invariant manifolds of their hyperbolic sets in the companion paper by fontich et al 2011 10

minimax problems on grassmann manifolds sums of eigenvalues courbures principales et energies de hartree fock grassmanniens des domaines de forme quadratique cycles de schubert le plongement de plucker fonctions de morse quadratiques caracterisation minimax de sommes de valeurs propres

quivers long exact sequences and horn type inequalities we give necessary and sufficient inequalities for the existence of long exact sequences of m finite abelian p groups with fixed isomorphism types this problem is related to some generalized littlewood richardson coefficients that we define in this paper we also show how this problem is related to eigenvalues of hermitian matrices satisfying certain in equalities when m 3 we recover the horn type inequalities that solve the saturation conjecture for littlewood richardson coefficients and horn s conjecture

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

convexity properties of the moment mapping re examined abstract consider a hamiltonian action of a compact lie group on a compact symplectic manifold a theorem of kirwan s says that the image of the momentum mapping intersects the positive weyl chamber in a convex polytope i present a new proof of kirwan s theorem which gives explicit information on how the vertices of the polytope come about and on how the shape of the polytope near any point can be read off from infinitesimal data on the manifold it also applies to some interesting classes of noncompact or singular hamiltonian spaces such as cotangent bundles and complex affine varieties

infinite dimensional srb measures we review the basic steps leading to the construction of a sinai ruelle bowen srb measure for an infinite lattice of weakly coupled expanding circle maps and we show that this measure has exponential decay of space time correlations first using the perron frobenius operator one connects the dynamical system of coupled maps on a d dimensional lattice to an equilibrium statistical mechanical model on a lattice of dimension d 1 this lattice model is for weakly coupled maps in a high temperature phase and we use a general but very elementary method to prove exponential decay of correlations at high temperatures

equilibrium measures for coupled map lattices existence uniqueness and finite dimensional approximations we consider coupled map lattices of hyperbolic type i e chains of weakly interacting hyperbolic sets attractors over multi dimensional lattices we describe the thermodynamic formalism of the underlying spin lattice system and then prove existence uniqueness mixing properties and exponential decay of correlations of equilibrium measures for a class of holder continuous potential functions with a sufficiently small holder constant we also study finite dimensional approximations of equilibrium measures in terms of lattice systems approximations and lattice spin systems d approximations we apply our results to establish existence uniqueness and mixing property of srb measures as well as obtain the entropy formula

the canonical correlations of a 2 2 block matrix with given eigenvalues abstract let 1 k n 2 and a a 11 a 12 1pt a 21 a 22 be an n n positive definite matrix so that a11 is k k suppose that a has given eigenvalues 1 n 0 the singular values j a11 1 2a12a22 1 2 j 1 k are known as the canonical correlations of the partitioned matrix a and have been extensively studied with regard to the inefficiency of the ordinary least squares method in statistics the object of this paper is to provide proofs of some new inequalities for the canonical correlations in terms of 1 n

instability of an inverse problem for the stationary navier stokes equations this paper provides a theoretical study of the detection of an object immersed in a fluid when the fluid motion is governed by the stationary navier stokes equations with nonhomogeneous dirichlet boundary conditions to solve this inverse problem we make a boundary measurement on a part of the exterior boundary first we present an identifiability result we then use a shape optimization method in order to identify the obstacle we minimize a nonlinear least squares thus we prove the existence of the first order shape derivative of the state characterize it and deduce the gradient of the least squares functional finally we study the stability of this setting doing a shape sensitivity analysis of order two hence we prove the existence of the second order shape derivatives and we give the expression of the shape hessian at possible solutions of the original inverse problem then the compactness of the riesz operator corresponding to this shape hessian is shown and the ill posedness of the ident

convexity properties of the moment mapping ii be its associated momen t mapping see w for definitions the set x is the union of co adjoint orbits the main result of this paper is a description of the orbit structure of this set to describe this result we first note that the coadjoint orbits in q can be parametr ized as follows let t be a caf tan subalgebra of g and let b be a positive definite g invariant bilinear form on by means of b we get a g equivariant identification q let t be the subspace of q corresponding to t then every co adjoint orbit intersects i in a weyl group orbit so there is a one one correspondence between co adjoint orbits in q and weyl group orbits in t to get an even sharper parametrization let t be a weyl chamber in then every weyl group orbit intersects t in a single point so t is a modul i space for the co adjoint orbits two orbits in x are said to be of the same type if they are isomorphic as homogeneous g spaces since x is compact there are at most a finite number of different types of orbits see 1113 let h be a closed subgroup of t of codimension r and let cit be the set of all orbits in x of type g h

real loci of symplectic reductions let m be a compact connected symplectic manifold with a hamiltonian action of a compact n dimensional torus t suppose that m is equipped with an anti symplectic involution a compatible with the t action the real locus of m is the fixed point set m of duistermaat introduced real loci and extended several theorems of symplectic geometry to real loci in this paper we extend another classical result of symplectic geometry to real loci the kirwan surjectivity theorem in addition we compute the kernel of the real kirwan map these results are direct consequences of techniques introduced by tolman and weitsman in some examples these results allow us to show that a symplectic reduction m t has the same ordinary cohomology as its real locus m t red with degrees halved this extends duistermaat s original result on real loci to a case in which there is not a natural hamiltonian torus action

eigenvalues of majorized hermitian matrices and littlewood richardson coefficients abstract answering a question raised by s friedland we show that the possible eigenvalues of hermitian matrices or compact operators a b and c with c a b are given by the same inequalities as in klyachko s theorem for the case where c a b except that the equality corresponding to tr c tr a tr b is replaced by the inequality corresponding to tr c tr a tr b the possible types of finitely generated torsion modules a b and c over a discrete valuation ring such that there is an exact sequence b c a are characterized by the same inequalities

hamiltonian systems with a discrete symmetry abstract if m m is an antisymplectic involution of a symplectic manifold m then the fixed set of is a lagrangian submanifold l m moreover there exist cotangent bundle coordinates in a neighborhood of l in m such that in these coordinates maps a covector into its negative thus classical examples which have a discrete symmetry such as the restricted three body problems are locally like a reversible system

sinai ruelle bowen measures for lattice dynamical systems for weakly coupled expanding maps on the unit circle bricmont and kupiainen showed that the sinai ruelle bowen srb measure exists as a gibbs state via thermodynamic formalism we prove that this srb measure is indeed the unique equilibrium state for a holder continuous potential function on the infinite dimensional phase space for a more general class of lattice systems that are small perturbations of the uncoupled map lattice we present the variational principle the entropy formula and the formula for the potential function for the srb measures for coupled map lattices with nearest neighbor interactions we give an explicit formula of the potential function for the srb measure and consequently obtain the entropy in terms of coupling parameters

the convexity package for hamiltonian actions on conformal symplectic manifolds consider a hamiltonian action of a compact connected lie group on a conformal symplectic manifold we prove a convexity theorem for the moment map under the assumption that the action is of lee type which establishes an analog of kirwan s convexity theorem in conformal symplectic geometry

intersections of schubert varieties and eigenvalue inequalities in an arbitrary finite factor the intersection ring of a complex grassmann manifold is generated by schubert varieties and its structure is governed by the littlewood richardson rule given three schubert varieties s1 s2 s3 with intersection number equal to one we show how to construct an explicit element in their intersection this element is obtained generically as the result of a sequence of lattice operations on the spaces of the corresponding flags and is therefore well defined over an arbitrary field of scalars moreover this result also applies to appropriately defined analogues of schubert varieties in the grassmann manifolds associated with a finite von neumann algebra the arguments require the combinatorial structure of honeycombs particularly the structure of the rigid extremal honeycombs it is known that the eigenvalue distributions of self adjoint elements a b c with a b c 0 in the factor r are characterized by a system of inequalities analogous to the classical horn inequalities of linear algebra we prove that these inequalities are in fact true for elements of an arbitrary finite factor in particular if x y z are self adjoint elements of such a factor and x y z 0 then there exist self adjoint a b c r such that a b c 0 and a respectively b c has the same eigenvalue distribution as x respectively y z a complete matricial form of this result is known to imply an affirmative answer to an embedding question formulated by connes the critical point in the proof of this result is the production of elements in the intersection of three schubert varieties when the factor under consideration is the algebra of n n complex matrices our arguments provide new and elementary proofs of the horn inequalities which do not require knowledge of the structure of the cohomology of the grassmann manifolds

smooth dependence of thermodynamic limits of srb measures the thermodynamic limits of srb measures for uniformly hyperbolic sets are smoothly dependent on the map in an appropriate functional space

the symplectic and algebraic geometry of horn s problem one version of horn s problem asks for which does h h h 0 have solutions where h are hermitian matrices with spectra this turns out to be a moment map condition in hamiltonian geometry many of the results around horn s problem proven with great effort by hand are in fact simple consequences of the modern machinery of symplectic geometry and the subtler ones provable via the connection to geometric invariant theory we give an overview of this theory which was not available to horn including all definitions and how it can be used in linear algebra