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

a schur horn theorem for symplectic eigenvalues abstract let x and y be positive n vectors we show that there exists a 2 n 2 n positive definite real matrix whose symplectic spectrum is y and the symplectic spectrum of whose diagonal is x if and only if x is weakly supermajorised by y

on the soliton resolution for equivariant wave maps to the sphere we consider finite energy corotational wave maps with target manifold s2 we prove that for a sequence of times they decompose as a sum of decoupled harmonic maps in the light cone and a smooth wave map in the blowup case or a linear scattering term in the global case up to an error which tends to 0 in the energy space 2015 wiley periodicals inc

two bubble dynamics for threshold solutions to the wave maps equation we consider the energy critical wave maps equation mathbb r 1 2 rightarrow mathbb s 2 in the equivariant case with equivariance degree k ge 2 it is known that initial data of energy 8 pi k and topological degree zero leads to global solutions that scatter in both time directions we consider the threshold case of energy 8 pi k we prove that the solution is defined for all time and either scatters in both time directions or converges to a superposition of two harmonic maps in one time direction and scatters in the other time direction in the latter case we describe the asymptotic behavior of the scales of the two harmonic maps the proof combines the classical concentration compactness techniques of kenig merle with a modulation analysis of interactions of two harmonic maps in the absence of excess radiation

rough solutions for the wave maps equation we consider the wave maps equation with values into a riemannian manifold which is isometrically embedded in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i our main result asserts that the cauchy problem is globally well posed for initial data which is small in the critical sobolev spaces this extends and completes recent work of tao and other authors

scattering below critical energy for the radial 4d yang mills equation and for the 2d corotational wave map system given g and f gg we consider solutions to the following non linear wave equation left begin array l displaystyle u tt u rr frac 1 r u r frac f u r 2 u u t t 0 u 0 u 1 end array right under suitable assumptions on g this equation admits non constant stationary solutions we denote q one with least energy we characterize completely the behavior as time goes to of solutions u ut corresponding to data with energy less than or equal to the energy of q either it is q 0 up to scaling or it scatters in the energy space

renormalization and blow up for charge one equivariant critical wave maps we prove the existence of equivariant finite time blow up solutions for the wave map problem from 2 1 s 2 of the form u t r q lambda t r mathcal r t r where u is the polar angle on the sphere q r 2 arctan r is the ground state harmonic map t t 1 and mathcal r t r is a radiative error with local energy going to zero as t 0 the number nu frac 1 2 can be prescribed arbitrarily this is accomplished by first renormalizing the blow up profile followed by a perturbative analysis

the three dimensional ising model revisited numerically several basic universal amplitude ratios are studied afresh for three dimensional nearest neighbor ising models in revising earlier work modern estimates of the critical temperature and exponents are used in conjunction with biased inhomogeneous differential approximants to extrapolate the longest available series expansions to find the critical amplitudes c for the susceptibility 1 for the correlation length 1 a for the specific heat c t and b for the spontaneous magnetization m0 we find c c 4 95 0 15 1 1 1 960 0 01 a a 0 523 0 009 a c b 0 0581 0 0010 while a 1 3 verifies hyperuniversality to within 0 8 a method for calculating amplitude ratios which allows for corrections to scaling yields estimates for c c and 1 1 in excellent agreement with those derived from the individual amplitudes finally explicit formulae are given for the numerical evaluation of t 1 t c t and m0 t over the full temperature range from criticality to t 0 and corresponding plots and convenient near critical representations are also presented

symplectic surgery and the spinc dirac operator abstract letgbe a compact connected lie group and m a compact hamiltoniang space with moment mapj m g under the assumption that these data are pre quantizable one can construct an associated spinc dirac operator formula whose equivariant index yields a virtual representation ofg we prove a conjecture of guillemin and sternberg that if 0 is a regular value ofj the multiplicityn 0 of the trivial representation in the index space formula is equal to the index of the spinc dirac operator for the symplectic quotientm0 j 1 0 g this generalizes previous results for the case thatg tis abelian i e a torus

order parameter distribution function of finite o n symmetric systems in an external field abstract we study the effect of an external field h on the order parameter distribution function near the critical point of o n symmetric three dimensional 3d systems in a finite geometry the distribution function is calculated within the 4 field theory for a 3d cube with periodic boundary conditions by means of a new approach that appropriately deals with the goldstone modes below tc the result describes finite size effects near the critical point in the h t plane including the first order transition at the coexistence line at h 0 below tc theoretical predictions of the finite size scaling function are presented for the ising n 1 and xy n 2 models good agreement is found with recent monte carlo data for the distribution function of the magnetization of the 3d ising model at finite h above and below tc

on the formation of singularities in the critical o 3 sigma model we study the phenomena of energy concentration for the critical o 3 sigma model also known as the wave map flow from 2 1 minkowski space into the sphere s 2 we establish rigorously and constructively existence of a set of smooth initial data resulting in a dynamic finite time formation of singularities the construction and analysis are done in the context of the k equivariant symmetry reduction and we restrict to maps with homotopy class k 4 the concentration mechanism we uncover is essentially due to a resonant self focusing shrinking of a corresponding harmonic map we show that the phenomenon is generic e g in certain sobolev spaces in that it persists under small perturbations of initial data while the resulting blowup is bounded by a log modified self similar asymptotic

asymptotic stability of large energy harmonic maps under the wave map from 2d hyperbolic spaces to 2d hyperbolic spaces abstract in this paper we prove that the large energy harmonic maps from h 2 to h 2 are asymptotically stable under the wave map equation this work is the last paper in a series of works 46 45 44 which together prove the asymptotic stability for all admissible harmonic maps of arbitrary size without any symmetry assumptions

quantization of time like energy for wave maps into spheres in this article we consider large energy wave maps in dimension 2 1 as in the resolution of the threshold conjecture by sterbenz and tataru commun math phys 298 1 139 230 2010 commun math phys 298 1 231 264 2010 but more specifically into the unit euclidean sphere mathbb s n 1 subset mathbb r n with n geq2 and study further the dynamics of the sequence of wave maps that are obtained in sterbenz and tataru commun math phys 298 1 231 264 2010 at the final rescaling for a first finite or infinite time singularity we prove that on a suitably chosen sequence of time slices at this scaling there is a decomposition of the map up to an error with asymptotically vanishing energy into a decoupled sum of rescaled solitons concentrating in the interior of the light cone and a term having asymptotically vanishing energy dispersion norm concentrating on the null boundary and converging to a constant locally in the interior of the cone in the energy space similar and stronger results have been recently obtained in the equivariant setting by several authors cote commun pure appl math 68 11 1946 2004 2015 cote commun pure appl math 69 4 609 612 2016 cote am j math 137 1 139 207 2015 cote et al am j math 137 1 209 250 2015 krieger commun math phys 250 3 507 580 2004 where better control on the dispersive term concentrating on the null boundary of the cone is provided and in some cases the asymptotic decomposition is shown to hold for all time here however we do not impose any symmetry condition on the map itself and our strategy follows the one from bubbling analysis of harmonic maps into spheres in the supercritical regime due to lin and riviere ann math 149 2 785 829 1999 duke math j 111 177 193 2002 which we make work here in the hyperbolic context of sterbenz and tataru commun math phys 298 1 231 264 2010

characterization of large energy solutions of the equivariant wave map problem ii we consider 1 equivariant wave maps from bbb r 1 2 to bbb s 2 of finite energy we establish a classification of all degree one global solutions whose energies are less than three times the energy of the harmonic map q in particular for each global energy solution of topological degree one we show that the solution asymptotically decouples into a rescaled harmonic map plus a radiation term together with a companion article part i where we consider the case of finite time blow up this gives a characterization of all 1 equivariant degree 1 wave maps in the energy regime e q 3e q

convex bodies of states and maps we give a general solution to the question of when the convex hulls of orbits of quantum states on a finite dimensional hilbert space under unitary actions of a compact group have a non empty interior in the surrounding space of all density operators the same approach can be applied to study convex combinations of quantum channels the importance of both problems stems from the fact that usually only sets with non vanishing volumes in the embedding spaces of all states or channels are of practical importance for the group of local transformations on a bipartite system we characterize maximally entangled states by the properties of a convex hull of orbits through them we also compare two partial characteristics of convex bodies in terms of the largest balls and maximum volume ellipsoids contained in them and show that in general they do not coincide separable states mixed unitary channels and k entangled states are also considered as examples of our techniques

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

bohr sommerfeld orbits in the moduli space of flat connections and the verlinde dimension formula we show how the moduli space of flatsu 2 connections on a two manifold can be quantized in the real polarization of 15 using the methods of 6 the dimension of the quantization given by the number of integral fibres of the polarization matches the verlinde formula which is known to give the dimension of the quantization of this space in a kahler polarization

symmetry and bifurcations of momentum mappings the zero set of a momentum mapping is shown to have a singularity at each point with symmetry the zero set is diffeomorphic to the product of a manifold and the zero set of a homogeneous quadratic function the proof uses the kuranishi theory of deformations among the applications it is shown that the set of all solutions of the yang mills equations on a lorentz manifold has a singularity at any solution with symmetry in the sense of a pure gauge symmetry similarly the set of solutions of einstein s equations has a singularity at any solution that has spacelike killing fields provided the spacetime has a compact cauchy surface

renormalization and blow up for the critical yang mills problem we consider the yangs mills equations in 4 1 dimensions this is the energy critical case and we show that it admits a family of solutions which blow up in finite time they are obtained by the spherically symmetric ansatz in the so 4 gauge group and result by rescaling of the instanton solution the rescaling is done via a prescribed rate which in this case is a modification of the self similar rate by a power of vertical bar log t vertical bar the powers themselves take any value exceeding 3 2 and thus form a continuum of distinct rates leading to blow up the methods are related to the authors previous work on wave maps and the energy critical semi linear equation however in contrast to these equations the linearized yang mills operator around an instanton exhibits a zero energy eigenvalue rather than a resonance this turns out to have far reaching consequences amongst which are a completely different family of rates leading to blow up logarithmic rather than polynomial corrections to the self similar rate c 2009 elsevier inc all rights reserved

goldstone mode singularities in specific heats and non ordering susceptibilities of isotropic systems goldstone mode singularities in non ordering correlation functions of the o n symmetric phi 4 theory are investigated in 4 epsilon dimensions functions constructed from non invariant combinations of the transverse operator pi 2 and the longitudinal operators sigma 2 and sigma generally exhibit severe infrared divergences on the coexistence curve in low order perturbative calculations of for example the specific heat however these divergences cancel such cancellations are shown to be a general feature of o n symmetric correlation functions non leading singularities remain and lead to momentum dependence which can be expressed as a double power series in mod p mod epsilon and mod p mod 2 epsilon

frankel s theorem in the symplectic category we prove that if an n 1 dimensional torus acts symplectically on a 2n dimensional symplectic manifold then the action has a fixed point if and only if the action is hamiltonian one may regard it as a symplectic version of frankel s theorem which says that a kahler circle action has a fixed point if and only if it is hamiltonian the case of n 2 is the well known theorem by mcduff

on blowup of co rotational wave maps in odd space dimensions abstract we consider co rotational wave maps from the 1 d dimensional minkowski space into the d sphere for d 3 odd this is an energy supercritical model which is known to exhibit finite time blowup via self similar solutions based on a method developed by the second author and schorkhuber we prove the asymptotic nonlinear stability of the ground state self similar solution

global well posedness scattering and blow up for the energy critical focusing non linear wave equation we study the energy critical focusing non linear wave equation with data in the energy space in dimensions 3 4 and 5 we prove that for cauchy data of energy smaller than the one of the static solution w which gives the best constant in the sobolev embedding the following alternative holds if the initial data has smaller norm in the homogeneous sobolev space h1 than the one of w then we have global well posedness and scattering if the norm is larger than the one of w then we have break down in finite time

homogeneous quantization and multiplicities of group representations let b be a compact manifold a cone over b is a principal r bundle x with base b let a x a x be the mapping associated with the action of a r on x x is called a symplectic cone if it possesses a symplectic form such that a a a compact lie group g is said to act in a homogeneous fashion on x if it acts on x in such a way that both and the principal bundle structure are preserved it is known that to such an action one can associate in a fairly canonical way a representation of g on a hilbert space h see 3 in this paper we propose a symplectic recipe for the multiplicities with which h decomposes into g irreducibles and show that this recipe is correct generically

asymptotic decomposition for semilinear wave and equivariant wave map equations in this paper we give a unified proof to the soliton resolution conjecture along a sequence of times for the semilinear focusing energy critical wave equations in the radial case and two dimensional equivariant wave map equations including the four dimensional radial yang mills equation without using outer energy type inequalities such inequalities have played a crucial role in previous works with similar results roughly speaking we prove that along a sequence of times t n to t the maximal time of existence the solution decouples to a sum of rescaled solitons and a term vanishing in the energy space plus a free radiation term in the global case or a regular part in the finite time blow up case the main difficulty is that in general for instance for the radial four dimensional yang mills case and the radial six dimensional semilinear wave case we do not have a favorable outer energy inequality for the associated linear wave equations our main new input is the simultaneous use of two virial identities

convex analysis on cartan subspaces in 1937 von neumann 31 gave a famous characterization of unitarily invariant matrix norms that is norms f on cp q satisfying f uxv f x for all unitary matrices u and v and matrices x in cp q his result states that such norms are those functions of the form g where the map x cp q 7 x r has components the singular values 1 x 2 x p x of x assuming p q and g is a norm on rp invariant under sign changes and permutations of components furthermore he showed the respective dual norms satisfy

heckman kostant and steinberg formulas for symplectic manifolds abstract let m be a compact symplectic manifold on which a compact connected lie group k acts in a hamiltonian fashion in part i we derive a formula for j m which generalizes the heckman formula for co adjoint orbits here j is the moment map associated with the action of k on m and m the liouville measure on m in part ii we derive a quantum analogue of this formula which extends to the symplectic setting classical multiplicity formulas of kostant and steinberg

field theory of finite size effects for systems with a one component order parameter the field theoretic renormalization group approach is used to describe finite size effects near the critical point of the 4 model with a one component order parameter problems of previous perturbation approaches for t tc are discussed and an improved perturbation theory is employed that is applicable both above and below tc the susceptibility order parameter specific heat and a cumulant ratio are calculated for fixed d 4 in one loop order for the case of a cube with periodic boundary conditions finite size scaling functions are evaluated in three dimensions without using the e 4 d expansion quantitative agreement with monte carlo mc data of the three dimensional ising model is found in most cases additional mc data of larger systems would be desirable in order to test the detailed predictions of the finite size field theory more conclusively in the asymptotic region

12 24 and beyond abstract we generalize the well known 12 and 24 theorems for reflexive polytopes of dimension 2 and 3 to any smooth reflexive polytope our methods apply to a wider category of objects here called reflexive gkm graphs that are associated with certain monotone symplectic manifolds which do not necessarily admit a toric action as an application we provide bounds on the betti numbers for certain monotone hamiltonian spaces which depend on the minimal chern number of the manifold

energy partition for the linear radial wave equation we consider the radial free wave equation in all dimensions and derive asymptotic formulas for the space partition of the energy as time goes to infinity we show that the exterior energy estimate which duyckaerts et al obtained in odd dimensions duyckaerts et al j eur math soc 13 533 599 2011 j eur math soc 2013 fails in even dimensions positive results for restricted classes of data are obtained

noncommutative poisson structures on orbifolds author s halbout gilles tang xiang abstract in this paper we compute the gerstenhaber bracket on the hoch schild cohomology of c infty m rtimes g for a finite group g acting on a compact manifold m using this computation we obtain geometric descriptions for all noncommutative poisson structures on c infty m rtimes g when m is a symplectic manifold we also discuss examples of deformation quantizations of these noncommutative poisson structures

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

stokes matrices poisson lie groups and frobenius manifolds the purpose of this paper is to point out and then draw some consequences of the fact that the poisson lie group g dual to g gln c may be identified with a certain moduli space of meromorphic connections over the unit disc having an irregular singularity at the origin g will be fully described in sect 2 the key feature of this point of view is that there is a holomorphic map g g from the dual of the lie algebra to the group g for each choice of diagonal matrix a0 with distinct eigenvalues the irregular type this map is essentially the riemann hilbert map or de rham morphism for such connections we will call it the monodromy map it is generically a local analytic isomorphism the main result is

lie algebraic aspects of the finite nonperiodic toda flows in this paper we review various aspects of nonperiodic full and tridiagonal toda flow theory in particular we consider gradient structures in the dynamics and geometry of these systems and we compare and contrast a number of formulations of the nonperiodic toda equations we describe the structure of these systems on general semisimple lie algebras in the case of the full kostant asymmetric toda flow we describe some of the background behind its integrability and explain the role of noncommutative integrability in its qualitative behavior we describe the relationship between the asymmetric toda flows and the symmetric and indefinite toda flows and show how one may conjugate from the full kostant toda flows to the full symmetric toda flows via a poisson map we describe briefly related toda systems such as nonabelian toda and the peakon flows

the topology of reduces phase spaces of the motion of vortices on a sphere abstract the motion of point vortices in a perfect incompressible fluid on a two dimensional sphere can be represented by a hamiltonian flow which is invariant under the action of the special orthogonal group so 3 following the reduction method of marsden and weinstein this motion can be described by hamiltonian flows on reduced phase spaces some topological invariants of these reduced phase spaces are calculated for suitable vorticities the results obtained can be used to give lower bounds on numbers of relative equilibria of the original system

lie poisson structure on some poisson lie groups poisson lie groups appeared in the work of drinfel d see e g drl dr2 as classical objects corresponding to quantum groups going in the other direction we may say that a poisson lie group is a group of symmetries of a phase space that are allowed to twist in a certain sense the symplectic or poisson structure the poisson structure on the group controls this twisting in a precise way quantizing both the phase space and the symmetries one may obtain a quantum group acting on a quantum phase space in recent work lu and ratiu lr used so called standard poisson structures on a compact semisimple lie group k and on its poisson dual k in order to give a new proof of the nonlinear convexity theorem of kostant ko their method is analogous to the famous symplectic proof of the linear convexity theorem given by atiyah a and guillemin and stemnberg gs the nonlinear convexity theorem like the linear one follows from a very general result on convexity of the image of the momentum map a gs however in lr it is applied not to a coadjoint orbit in t but to a symplectic leaf in k the main result of this paper is that the standard poisson structure on the poisson dual k to a compact semisimple poisson lie group k is actually isomorphic to the linear one on t this theorem seems to be related to some facts in the theory of quantum groups namely for generic q the universal enveloping algebra u t and its quantum deformation uq t are isomorphic as algebras though not as coalgebras of course since the quantum version is not cocommutative in particular there is a bijective correspondence between their representations a direct connection between our work and its quantum analogues though is still to be found the present work supplies a positive answer to question 5 1 in lr and strongly depends on that paper to simplify reading we keep the notations of lr wherever possible on one hand but give all necessary definitions on the other our work is also related to du in which the nonlinear convexity theorem is reduced to the linear one by a deformation argument not unlike the one that we use in 5 the paper is organized as follows in 2 we define poisson lie groups discuss

equivariant wave maps in two space dimensions singularities of corotational wave maps from 1 2 dimensional minkowski space into a surface n of revolution after a suitable rescaling give rise to nonconstant corotational harmonic maps from 2 into in consequence for noncompact target surfaces of revolution the cauchy problem for wave maps is globally well posed 2003 wiley periodicals inc

on the variation in the cohomology of the symplectic form of the reduced phase space is called the momentum mapping of the hamiltonian t action given 1 1 the condition 1 2 just means that t acts along the fibers of j for the basic definitions and properties of non commutative hamiltonian group actions see am the results of this paper can easily be extended to hamiltonian actions of arbitrary compact connected lie groups by applying our results to the action of its maximal toms and using the equivariance of the momentum mapping for some more details see the remarks at the end of sect 2 we will assume throughout this paper that the momentum map is proper that is j i u is compact for each compact subset u of t now let r be a regular value of j that is t j tmm t is surjective for all meye j 1 4 then ye is a smooth submanifold of m compact because

high frequency approximation of solutions to critical nonlinear wave equations this work is devoted to the description of bounded energy sequences of solutions to the equation 1 u u 4 0 in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i up to remainder terms small in energy norm and in every strichartz norm the proof relies on scattering theory for 1 and on a structure theorem for bounded energy sequences of solutions to the linear wave equation in particular we infer the existence of an a priori estimate of strichartz norms of solutions to 1 in terms of their energy

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

reduction theory and the lagrange routh equations reduction theory for mechanical systems with symmetry has its roots in the classical works in mechanics of euler jacobi lagrange hamilton routh poincare and others the modern vision of mechanics includes besides the traditional mechanics of particles and rigid bodies field theories such as electromagnetism fluid mechanics plasma physics solid mechanics as well as quantum mechanics and relativistic theories including gravity symmetries in these theories vary from obvious translational and rotational symmetries to less obvious particle relabeling symmetries in fluids and plasmas to subtle symmetries underlying integrable systems reduction theory concerns the removal of symmetries and their associated conservation laws variational principles along with symplectic and poisson geometry provide fundamental tools for this endeavor reduction theory has been extremely useful in a wide variety of areas from a deeper understanding of many physical theories including new variational and poisson structures to stability theory integrable systems as well as geometric phases this paper surveys progress in selected topics in reduction theory especially those of the last few decades as well as presenting new results on non abelian routh reduction we develop the geometry of the associated lagrange routh equations in some detail the paper puts the new results in the general context of reduction theory and discusses some future directions

polar and coisotropic actions on kahler manifolds the main result of the paper is that a polar action on a compact irreducible homogeneous kahler manifold is coisotropic this is then used to give new examples of polar actions and to classify coisotropic and polar actions on quadrics

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

global regularity of wave maps ii small energy in two dimensions we show that wave maps from minkowski space 1 n to a sphere sm 1 are globally smooth if the initial data is smooth and has small norm in the critical sobolev space in all dimensions n 5 this generalizes the results in the prequel 40 of this paper which addressed the high dimensional case n 5 in particular in two dimensions we have global regularity whenever the energy is small and global regularity for large data is thus reduced to demonstrating non concentration of energy

an analytic proof of the geometric quantization conjecture of guillemin sternberg we present a direct analytic approach to the guillemin sternberg conjecture gs that geometric quantization commutes with symplectic reduction which was proved recently by meinrenken m1 m2 and vergne v1 v2 et al besides providing a new proof of this conjecture our methods also lead immediately to further extensions in various contexts

when is a pure state of three qubits determined by its single particle reduced density matrices using techniques from symplectic geometry we prove that a pure state of three qubits is up to local unitaries uniquely determined by its one particle reduced density matrices exactly when their ordered spectra belong to the boundary of the so called kirwan polytope otherwise the states with given reduced density matrices are parameterized up to local unitary equivalence by two real variables given inevitable experimental imprecision this means that already for three qubits a pure quantum state can never be reconstructed from single particle tomography we moreover show that the knowledge of the reduced density matrices is always sufficient if one is given the additional promise that the quantum state is not convertible to the greenberger horne zeilinger state by stochastic local operations and classical communication and discuss generalizations of our results to an arbitrary number of qubits

diversity of critical behavior within a universality class we study spatial anisotropy effects on the bulk and finite size critical behavior of the mathrm o n symmetric anisotropic ensuremath varphi 4 lattice model with periodic boundary conditions in a d dimensional hypercubic geometry above at and below t c the absence of two scale factor universality is discussed for the bulk order parameter correlation function the bulk scattering intensity and for several universal bulk amplitude relations the anisotropy parameters are observable by scattering experiments at t c for the confined system renormalization group theory within the minimal subtraction scheme at fixed dimension d for 2ldl4 is employed in contrast to the ensuremath epsilon 4 ensuremath d expansion the fixed d finite size approach keeps the exponential form of the order parameter distribution function unexpanded for the case of cubic symmetry and for n 1 our perturbation approach yields excellent agreement with the monte carlo mc data for the finite size amplitude of the free energy of the three dimensional ising model at t c by mon phys rev lett 54 2671 1985 the ensuremath epsilon expansion result is in less good agreement below t c a minimum of the scaling function of the excess free energy is found we predict a measurable dependence of this minimum on the anisotropy parameters the relative anisotropy effect on the free energy is predicted to be significantly larger than that on the binder cumulant our theory agrees quantitatively with the nonmonotonic dependence of the binder cumulant on the ferromagnetic next nearest neighbor nnn coupling of the two dimensional ising model found by mc simulations of selke and shchur j phys a 38 l739 2005 our theory also predicts a nonmonotonic dependence for small values of the antiferromagnetic nnn coupling and the existence of a lifshitz point at a larger value of this coupling the nonuniversal anisotropy effects in the finite size scaling regime are predicted to satisfy a kind of restricted universality the tails of the large l behavior at t ensuremath ne t c violate both finite size scaling and universality even for isotropic systems as they depend on the bare four point coupling of the ensuremath varphi 4 theory on the cutoff procedure and on subleading long range interactions

comparison between finite size field theory and monte carlo simulations for the three dimensional ising model abstract recent field theoretic predictions of finite size effects for various thermodynamic quantities near the critical point are compared with new accurate monte carlo mc data for the l l l ising model obtained from a cluster algorithm special purpose computer owing to the large size l 32 64 128 256 of the ising models these mc data permit to test the finite size field theory in the asymptotic scaling region above and below t c the results constitute the most accurate confirmation of finite size scaling in three dimensions obtained to date in a wide range of temperatures agreement with the predicted shape of the finite size scaling functions is found on a quantitative level

geometric proofs of some theorems of schur horn type abstract we describe elementary geometric proofs of the classical schur horn theorem for symmetric matrices and analogous results for skew symmetric matrices in the real and complex cases we compare this approach to the proofs obtained by making use of symplectic techniques as a further application we present a new proof of horn s characterization of diagonal entries of orthogonal matrices

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

parameterization by polytopes of intersections of orbits by conjugation abstract let s be an n n real symmetric matrix with spectral decomposition s q t q where q is an orthogonal matrix and is diagonal with simple spectrum 1 n also let o s and r s be the orbits by conjugation of s by respectively orthogonal matrices and upper triangular matrices with positive diagonal denote by f s the intersection o s r s we show that the map f f s r n taking s q t q to diag q q t is a smooth bijection onto its range p s the convex hull of some subset of the n points 1 2 n is apermutation we also find necessary and sufficient conditions for p s to have n vertices

critical behaviour of the 3d xy model a monte carlo study we present the results of a study of the three dimensional xy model on a simple cubic lattice using the single cluster updating algorithm combined with improved estimators we have measured the susceptibility and the correlation length for various couplings in the high temperature phase on lattices of size up to l 112 at the transition temperature we studied the fourth order cumulant and other cumulant like quantities on lattices of size up to l 64 from our numerical data we obtain for the critical coupling kc 0 45420 2 and for the static critical exponents v 1 976 6 and v 0 662 7