localization and diagonalization a review of functional integral techniques for low dimensional gauge theories and topological field theories localization techniques for functional integrals which have recently been used to perform calculations in and gain insight into the structure of certain topological field theories and low dimensional gauge theories are reviewed these are the functional integral counterparts of the mathai quillen formalism the duistermaat heckman theorem and the weyl integral formula respectively in each case the necessary mathematical background euler classes of vector bundles equivariant cohomology topology of lie groups is introduced and the finite dimensional integration formulae described then some applications to path integrals are discussed and an overview of the relevant literature is given the applications include supersymmetric quantum mechanics cohomological field theories phase space path integrals and two dimensional yang mills theory

computations of complex equivariant bordism rings we give explicit computations of the coefficients of homotopical complex equivariant cobordism theory mu g when g is abelian we present a set of generators which is complete for any abelian group we present a set of relations which is complete when g is cyclic and which we conjecture to be complete in general we proceed by first computing the localization of mu g obtained by inverting euler classes of representations we then define a family of operations which essentially divide by euler classes and use these operations to define our generating sets we give geometric applications of these computations to the study of equivariant genera circle actions on four manifolds and cobordism relations between lens spaces

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

geometric versus homotopy theoretic equivariant bordism by results of loffler and comezana the pontrjagin thom map from geometric g equivariant bordism to homotopy theoretic equivariant bordism is injective for compact abelian g if g s1 s1 we prove that the associated fixed point square is a pull back square thus confirming a recent conjecture of sinha 22 this is used in order to determine the image of the pontrjagin thom map for toralg

the relation between primal and dual boundary conditions for hyperbolic systems of equations abstract in this paper we study boundary conditions for linear hyperbolic systems of equations and the corresponding dual problem in particular we show that the primal and dual boundary conditions are related by a simple scaling relation it is also shown that the weak dual problem can be derived directly from the weak primal problem based on the continuous analysis we discretize and perform computations with a high order finite difference scheme on summation by parts form with weak boundary conditions it is shown that the results obtained in the continuous analysis lead directly to stability results for the primal and dual discrete problems numerical experiments corroborate the theoretical results

numerical approximation of incompressible flows with net flux defective boundary conditions by means of penalty techniques we consider incompressible flow problems with defective boundary conditions prescribing only the net flux on some inflow and outflow sections of the boundary as a paradigm for such problems we simply refer to stokes flow after a brief review of the problem and of its well posedness we discretize the corresponding variational formulation by means of finite elements and looking at the boundary conditions as constraints we exploit a penalty method to account for them we perform the analysis of the method in terms of consistency boundedness and stability of the discrete bilinear form and we show that the application of the penalty method does not affect the optimal convergence properties of the finite element discretization since the additional terms introduced to account for the defective boundary conditions are non local we also analyze the spectral properties of the equivalent algebraic formulation and we exploit the analysis to set up an efficient solution strategy in contrast to alternative discretization methods based on lagrange multipliers accounting for the constraints on the boundary the present scheme is particularly effective because it only mildly affects the structure and the computational cost of the numerical approximation indeed it does not require neither multipliers nor sub iterations or additional adjoint problems with respect to the reference problem at hand

adaptive finite element methods for optimal control of partial differential equations basic concept a new approach to error control and mesh adaptivity is described for the discretization of optimal control problems governed by elliptic partial differential equations the lagrangian formalism yields the first order necessary optimality condition in form of an indefinite boundary value problem which is approximated by an adaptive galerkin finite element method the mesh design in the resulting reduced models is controlled by residual based a posteriori error estimates these are derived by duality arguments employing the cost functional of the optimization problem for controlling the discretization error in this case the computed state and costate variables can be used as sensitivity factors multiplying the local cell residuals in the error estimators this results in a generic and simple algorithm for mesh adaptation within the optimization process this method is developed and tested for simple boundary control problems in semiconductor models

coupling requirements for multiphysics problems posed on two domains we consider two hyperbolic systems in first order form of different size posed on two domains our ambition is to derive general conditions for when the two systems can and cannot be coupled the a

a posteriori error estimates for control problems governed by stokes equations in this paper we derive a posteriori error estimates for the finite element approximation of distributed optimal control problems governed by the stokes equations we obtain a posteriori error estimators for both the state and the control approximation in the l2 norm and the h1 norm these estimates can be used to construct reliable adaptive finite element approximation for the control problems

flow rate boundary problems for an incompressible fluid in deformable domains formulations and solution methods in this paper we consider the numerical solution of the interaction of an incompressible fluid and an elastic structure in a truncated computational domain as well known in this case there is the problem of prescribing realistic boundary data on the artificial sections when only partial data are available this problem has been investigated extensively for the rigid case in this work we start considering the compliant case by focusing on the flow rate conditions for the fluid we propose three formulations of this problem different algorithms for its numerical solution and carry out several 2d numerical simulations with the aim of comparing the performances of the different algorithms

analysis and approximation of the cross model for quasi newtonian flows with defective boundary conditions a numerical method for a generalized newtonian fluid with flow rate boundary conditions is considered the defect boundary condition problem is formulated as a constrained optimal control problem where a flow balance is forced on the inflow and outflow boundaries using a neumann control the control problem is analyzed for an existence result and the lagrange multiplier rule a decoupling solution algorithm is presented and numerical experiments are provided to validate robustness of the algorithm

an effective fluid structure interaction formulation for vascular dynamics by generalized robin conditions in this work we focus on the modeling and numerical simulation of the fluid structure interaction mechanism in vascular dynamics we first propose a simple membrane model to describe the deformation of the arterial wall which is derived from the koiter shell equations and is applicable to an arbitrary geometry secondly we consider a reformulation of the fluid structure problem in which the newly derived membrane model thanks to its simplicity is embedded into the fluid equations and will appear as a generalized robin boundary condition the original problem is then reduced to the solution of subsequent fluid equations defined on a moving domain and may be achieved with a fluid solver only we also derive a stability estimate for the resulting numerical scheme finally we propose new outflow absorbing boundary conditions which are easy to implement and allow us to reduce significantly the spurious pressure wave reflections that typically appear in artificially truncated computational domains we present several numerical results showing the effectiveness of the proposed approaches

optimal control for quasi newtonian flows with defective boundary conditions we consider a generalized newtonian fluid with defective boundary conditions where only flow rates or mean pressures are prescribed on parts of boundary the defect boundary condition problem is formulated as an optimal control problem in which a neumann or dirichlet boundary control is used for matching given flow rates or mean pressures for the constrained optimization problem an optimality system is derived from which a solution of the problem is obtained computational algorithms are discussed and numerical results are also presented

bohr sommerfeld conditions for integrable systems with critical manifolds of focus focus type we present a detailed study in the semi classical regime h of microlocal properties of systems of two commuting h pseudo dierential operators ph ph such that the joint principal symbol p p p has a special kind of singularity called a focusfocus singularity typical examples include the quantum spherical pendulum or the quantum champagne bottle in the spirit of colin de verdiere and parisse we show that such systems have a universal behavior described by singular quantization conditions of bohr sommerfeld type these conditions are used to give a precise description of the joint spectrum of such systems including the phenomenon of quantum monodromy and dierent formulations of the counting function for the joint eigenvalues close to the singularity in which a logarithm of the semi classical constant h appears thanks to numerical computations done by ms child for the case of the champagne bottle we are able to accurately illustrate our statements

a new approach to numerical solution of defective boundary value problems in incompressible fluid dynamics we consider the incompressible navier stokes equations where on a part of the boundary flow rate and mean pressure boundary conditions are prescribed there are basically two strategies for solving these defective boundary problems the variational approach see j heywood r rannacher and s turek internat j numer methods fluids 22 1996 pp 325 352 and the augmented formulation see l formaggia j f gerbeau f nobile and a quarteroni siam j numer anal 40 2002 pp 376 401 and a veneziani and c vergara internat j numer methods fluids 47 2005 pp 803 816 however both of these approaches present some drawbacks for the flow rate problem the former resorts to nonstandard functional spaces which are quite difficult to discretize on the other hand for the mean pressure problem it yields exact solutions only in very special cases the latter is applicable only to the flow rate problem since for the mean pressure problem it provides unfeasible boundary conditions in this paper we propose a new strategy based on a reformulation of the problems at hand in terms of the minimization of an appropriate functional this approach allows us to treat the two kinds of problems flow rate and mean pressure successfully within the same framework which can be useful in view of a mixed problem where the two conditions are simultaneously prescribed on different artificial boundaries moreover it is more versatile being prone to be extended to other kinds of defective conditions we analyze the problems obtained with this approach and propose some algorithms for their numerical solution several numerical results are presented supporting the effectiveness of our approach

well posed and stable transmission problems we introduce the notion of a transmission problem to describe a general class of problems where different dynamics are coupled in time well posedness and stability is analysed for continuous and d

non hamiltonian actions with isolated fixed points we construct a non hamiltonian symplectic circle action on a closed connected six dimensional symplectic manifold with exactly 32 fixed points

stochastic equivariant cohomologies and cyclic cohomology we give two stochastic diffeologies on the free loop space which allow us to define stochastic equivariant cohomology theories in the chen souriau sense and to establish a link with cyclic cohomology with the second one we can establish a stochastic fixed point theorem

the fixed point theorem in equivariant cohomology in this paper we study the s equivariant de rham cohomology of infinite dimensional s manifolds our main example is the free loop space lx where x is a finite dimensional manifold with the circle acting by rotating loops we construct a new form of equivariant cohomology h which agrees with the usual periodic equivariant cohomology in finite dimensions and we prove a suitable analogue of the classical fixed point theorem which is valid for loop spaces lx this gives a cohomological framework for studying differential forms on loop spaces and we apply these methods to various questions which arise from the work of witten 16 atiyah 2 and bismut 5 in particular we show following atiyah in 2 that the a polynomial of x arises as an equivariant characteristic class in the theory h of the normal bundle to x considered as the space of constant loops in lx

numerical treatment of defective boundary conditions for the navier stokes equations we present a formulation for accommodating defective boundary conditions for the incompressible navier stokes equations where only averaged values are prescribed on measurable portions of the boundary in particular we consider the case where the flow rate is imposed on several domain sections this methodology has an interesting application in the numerical simulation of flow in blood vessels when only a reduced set of boundary data are generally available for the upstream and downstream sections

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

augmented lagrangian method for constraining the shape of velocity profiles at outlet boundaries for three dimensional finite element simulations of blood flow in three dimensional blood flow simulations of the cardiovascular system velocity and pressure fields in the computational domain are highly affected by outlet boundary conditions this fact has motivated the development of novel methods to couple three dimensional computational domains with one dimensional numerical models or alternatively with zero dimensional or one dimensional analytic models in all such methods described to date whether they are explicit or implicit the relationship between flow and pressure at the outlet boundary is enforced weakly this coupling does not include any constraints on the shape of the velocity profiles nor on the distribution of pressure at the interface as a result there remain some classes of problems that are at best difficult to solve and at worst intractable with current numerical methods for simulating blood flow these include problems with significant flow reversal during part of the cardiac cycle or geometric complexity in the proximity of the outlet of the computational domain we have implemented a novel method to resolve these challenging problems whereby an augmented lagrangian method is used to enforce constraints on the shape of the velocity profile at the interface between the upstream computational domain and the downstream analytic domain these constraints on the shape of the velocity profile are added to the coupled multidomain method in order to implicitly couple the computational domain with downstream analytic models in this study an axisymmetric profile is imposed after ensuring that each constrained outlet boundary is circular we demonstrate herein that including constraints on the shape of the velocity profile does not affect velocity and pressure fields except in the immediate vicinity of the constrained outlet boundaries furthermore this new method enables the solution of problems which diverged with an unconstrained method

an approximate method for solving incompressible navier stokes problems with flow rate conditions we consider the incompressible navier stokes problem with flow rate boundary conditions this problem has been investigated in j heywood r rannacher s turek artificial boundaries and flux and pressure conditions for the incompressible navier stokes equations int j numer methods fluids 22 1996 325 352 l formaggia j f gerbeau f nobile a quarteroni numerical treatment of defective boundary conditions for the navier stokes equation siam j numer anal 40 1 2002 376 401 a veneziani c vergara flow rate defective boundary conditions in haemodynamics simulations int j numer methods fluids 47 2005 803 816 in the latter two papers a feasible numerical formulation of this problem based on a lagrange multiplier approach has been investigated this approach has the drawback of high computational costs in this paper we propose an approximate formulation of this problem yielding a strong reduction of the computational costs the error analysis shows that the error introduced by this approximate formulation is confined in a small region of the boundary this is confirmed by the numerical simulations

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

on the construction of certain 6 dimensional symplectic manifolds with hamiltonian circle actions let m w be a connected compact 6 dimensional symplectic manifold equipped with a semi free hamiltonian s 1 action such that the fixed point set consists of isolated points or surfaces assume dim h 2 m 3 in an earlier paper we defined a certain invariant of such spaces which consists of fixed point data and twist type and we divided the possible values of these invariants into six types in this paper we construct such manifolds with these types as a consequence we have a precise list of the values of these invariants

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

localization formulas superconnections and the index theorem for families in this paper we give a new proof of the localization formulas of berline and vergne 9 and duistermaat and heckman 18 when interpreted in the framework of atiyah 2 the probabilistic heat equation proof of the index theorem given in our paper 12 appears as the rigorous infinite dimensional version of this new proof of the localization formulas in finite dimensions the results of quillen 25 on superconnections are briefly presented the heat equation proofs 15 of the index theorem for families are described it is shown that in this framework the superconnections formalism is the operator theoretic description of integration along the fiber in the loop space

bordism of semi free s1 actions we calculate geometric and homotopical bordism rings associated to semi free s1 actions on complex manifolds giving explicit generators for the geometric theory the classification of semi free actions with isolated fixed points up to cobordism complements similar results from symplectic geometry

localization and completion theorems for mu module spectra let g be a finite extension of a torus working with highly structured ring and module spectra let m be any module over mu examples include all of the standard homotopical mu modules such as the brown peterson and morava k theory spectra we shall prove localization and completion theorems for the computation of m bg and m bg the g spectrum mug that represents stabilized equivariant complex cobordism is an algebra over the equivariant sphere spectrum sg and there is an mug module mg whose underlying mu module is m this allows the use of topological analogues of constructions in commutative algebra the computation ofm bg andm bg is expressed in terms of spectral sequences whose respective e2 terms are computable in terms of local cohomology and local homology groups that are constructed from the coefficient ring mu and its module m the central feature of the proof is a new norm map in equivariant stable homotopy theory the construction of which involves the new concept of a global i functor with smash product