conditions for the equivalence of internal and external stability for distributed parameter systems conditions are given on a controlled observed distributed parameter system under which external stability also known as input output stability implies internal stability which is the exponential stability of the underlying semigroup generator it is shown that when the system satisfies a general definition of stabilizability and detectability external stability is equivalent to internal stability unlike the case previous definitions of stabilizability and detectability a system which satisfies these does not necessarily have a spectrum decomposition therefore in order to check internal stability it is often sufficient to check the boundedness of the transfer function in the right half complex plane this is illustrated with a simple example

minimum fill in inapproximability and almost tight lower bounds abstract given an n n sparse symmetric matrix with m nonzero entries performing gaussian elimination may turn some zeroes into nonzero values so called fill ins the minimum fill in problem asks whether it is possible to perform the elimination with at most k fill ins we exclude the existence of polynomial time approximation schemes for this problem assuming p np and the existence of 2 o n 1 time approximation schemes for any positive assuming the exponential time hypothesis we also give a 2 o k 1 2 n o 1 parameterized lower bound all these results come as corollaries of a new reduction from vertex cover to the minimum fill in problem which might be of its own interest all previous reductions for similar problems start from some kind of graph layout problems and hence have limited use in understanding their fine grained complexity

the semiclassical limit of the two dimensional quantum yang mills model the semiclassical limit of the quantum yang mills partition function on a compact oriented surface is related to the symplectic volume of the moduli space of flat connections by using an explicit expression for the symplectic form this gives an independent proof of some recent results of witten and forman

the semiclassical limit for su 2 and so 3 gauge theory on the torus we prove that forsu 2 andso 3 quantum gauge theory on a torus holonomy expectation values with respect to the yang mills measure open image in new window converge ast 0 to integrals with respect to a symplectic volume measure 0 on the moduli space of flat connections on the bundle these moduli spaces and the symplectic structures are described explicitly

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

on the existence of finite dimensional compensators for retarded and neutral systems in this paper we study a large class of neutral systems containing the class of retarded systems a necessary and sufficient condition is given for the existence of a finite dimensional compensator achieving exponential stability

invariant functions on lie groups and hamiltonian flows of surface group representations si est le groupe fondamental d une surface orientee fermee s et g est un groupe de lie satisfaisant des conditions tres generales alors l espace hom g g des classes de conjugaison de representation g a une structure symplectique naturelle on etudie la geometrie de cette structure symplectique a l aide d une famille naturelle de fonctions sur hom g g

the kauffman skein algebra of a surface at sqrt 1 we study the structure of the kauffman algebra of a surface with parameter equal to sqrt 1 we obtain an interpretation of this algebra as an algebra of parallel transport operators acting on sections of a line bundle over the moduli space of flat su 2 connections over the surface we analyse the asymptotics of traces of curve operators in tqft in non standard regimes where the root of unity parametrizing the tqft accumulates to a root of unity we interpret the case of sqrt 1 in terms of parallel transport operators

on the symplectic geometry of deformations of a hyperbolic surface let r be a riemann surface in this manuscript we consider a geometry on the moduli space x r for r which we regard as the space of equivalence classes of constant curvature metrics on the underlying smooth manifold of r classically the space of flat metrics for a torus is the locally symmetric space 0 2 sl 2 r sl 2 z we shall describe a symplectic geometry for the space of hyperbolic metrics on a surface of negative euler characteristic the teichmiiller space t r a covering of the moduli space is a complex kihler manifold a kuhler metric for t r defined in terms of the petersson product for automorphic forms was introduced by weil 1 the weil petersson metric is invariant under the covering transformations and so projects to the moduli space x r the metric provides a link between the function theory of r and the geometry of x r in the fenchel nielsen manuscript 8 a deformation based on an amalgamation construction for fuchsian groups is introduced the deformation is defined geometrically by cutting the surface along a simple closed geodesic rotating one side of the cut relative to the other and attaching the sides in their new position the hyperbolic metric in the complement of the cut extends to a hyperbolic metric on the new surface choose a free homotopy class a on the surface r then for each marked surface r realize a by the closed geodesic ar the fenchel nielsen deformations for the athen define a 1 parameter group of diffeomorphisms of t r whose infinitesimal generator by definition is the fenchel nielsen vector field ta in 21 the fenchel nielsen deformation was described in terms of quasiconformal mappings and an investigation of the vector fields t was begun the fenchel nielsen vector fields were found to be related to the geodesic length functions 1 introduced by fricke klein to provide coordinates for t r

reducing rank of the adjacency matrix by graph modification the main topic of this article is to study a class of graph modification problems a typical graph modification problem takes as input a graph g a positive integer k and the objective is to add delete k vertices edges so that the resulting graph belongs to a particular family mathcal f of graphs in general the family mathcal f is defined by forbidden subgraph minor characterization in this paper rather than taking a structural route to define mathcal f we take algebraic route more formally given a fixed positive integer r we define mathcal f r as the family of graphs where for each g in mathcal f r the rank of the adjacency matrix of g is at most r using the family mathcal f r we initiate algorithmic study both in classical and parameterized complexity of following graph modification problems r rank vertex deletion r rank edge deletion and r rank editing these problems generalize the classical vertex cover problem and a variant of the d cluster editing problem we first show that all the three problems are np complete then we show that these problems are fixed parameter tractable fpt by designing an algorithm with running time 2 mathcal o k log r n mathcal o 1 for r rank vertex deletion and an algorithm for r rank edge deletion and r rank editing running in time 2 mathcal o f r sqrt k log k n mathcal o 1 we complement our fpt result by designing polynomial kernels for these problems

the transmission zeros of systems with delays in this paper we study the properties of the zeros of a system with general delays in the input state and control variables the main result of this paper shows that the zeros of the transfer function are always zeros of the system matrix with of at least the same multiplicity moreover when the system is canonical the zeros of the system matrix are zeros of the transfer function with the same multiplicity

the semiclassical limit for gauge theory on s 2 it is shown that the yang mills measurez h 1 e s h d whereh 0 describing gauge fields on the two sphere converges to a probability measure on the moduli space of yang mills connections ons2 ash 0

cobordism for hamiltonian loop group actions and flat connections on the punctured two sphere we develop a theory of symplectic cobordism and a duistermaat heckman principle for hamiltonian loop group actions as an application we construct a symplectic cobordism between moduli spaces of flat connections on the three holed sphere and disjoint unions of toric varieties this cobordism yields formulas for the mixed pontrjagin numbers of the moduli spaces equivalent to witten s formulas in the case of symplectic volumes

internal and external stability and robust stability condition for a class of infinite dimensional systems abstract in the current study of robust stability of infinite dimensional systems internal exponential stability is not necessarily guaranteed this paper introduces a new class of impulse responses called r in which the usual notion of l2 input output stability guarantees not only external but also internal exponential stability the result is applied to derive a closed loop stability condition and a version of the small gain theorem with internal exponential stability this leads to a robust stability condition that also assures internal stability an application to repetitive control systems is shown to illustrate the results

intersection cohomology of quotients of nonsingular varieties let m p be a nonsingular projective variety acted on by a connected complex reductive group g kc via a homomorphism g gl n 1 which is the complexification of a homomorphism k u n 1 geometric invariant theory git gives us a recipe to form a quotient m m g of the set of semistable points and many interesting spaces in algebraic geometry are constructed in this manner mfk94 but often this quotient is singular and hence intersection cohomology with middle perversity is an important topological invariant the purpose of this paper is to present a way to compute the middle perversity intersection cohomology of the singular quotients the choice of an embedding m p provides us with a moment map for the k action and the morse stratification of m with respect to its norm square is kequivariantly perfect with the unique open dense stratum m kir84 hence one can compute the betti numbers for the equivariant cohomology h k m as well as the cup product at least in principle see also kir92 when g acts locally freely on m we get an orbifold m g m g and h k m h m g ih m g

quadratic vertex kernel for split vertex deletion abstract a graph is called a split graph if its vertex set can be partitioned into a clique and an independent set split graphs have rich mathematical structure and interesting algorithmic properties making it one of the most well studied special graph classes in the split vertex deletion problem given a graph and a positive integer k the objective is to test whether there exists a subset of at most k vertices whose deletion results in a split graph in this paper we design a kernel for this problem with o k 2 vertices improving upon the previous cubic bound known also by giving a simple reduction from the vertex cover problem we establish that split vertex deletion does not admit a kernel with o k 2 edges for any 0 unless np conp poly

ergodic theory on moduli spaces let m be a compact surface with x m 0 and let g be a compact lie group whose levi factor is a product of groups locally isomorphic to su 2 for example su 2 itself then the mapping class group rm of m acts on the moduli space x m of flat g bundles over m possibly twisted by a fixed central element of g when m is closed then fm preserves a symplectic structure on x m which has finite total volume on m more generally the subspace of x m corresponding to flat bundles with fixed behavior over am carries a rm invariant symplectic structure the main result is that rm acts ergodically on x m with respect to the measure induced by the symplectic structure

large induced subgraphs via triangulations and cmso we obtain an algorithmic metatheorem for the following optimization problem let varphi be a counting monadic second order logic cmso formula and t geq 0 be an integer for a given graph g v e the task is to maximize x subject to the following there is a set f subseteq v such that x subseteq f the subgraph g f induced by f is of treewidth at most t and the structure g f x models varphi i e g f x models varphi we give an algorithm solving this optimization problem on any n vertex graph g in time cal o pi g cdot n t 4 cdot f t varphi where pi g is the set of all potential maximal cliques in g and f is a function of t and varphi only pipelined with the known bounds on the number of potential maximal cliques in different graph classes there are a plethora of algorithmic consequences extending and subsuming many known results on polynomial time algorithms for graph classes we also show that all potential maximal cliques of g can be enumerated in time cal o 1 7347 n this implies the existence of an exact exponential algorithm of running time cal o 1 7347 n for many np hard problems related to finding maximum induced subgraphs with different properties

triangulated riemann surfaces with boundary and the weil petersson poisson structure given a hyperbolic surface with geodesic boundary s the lengths of a maximal system of disjoint simple geodesic arcs on s that start and end at s perpendicularly are coordinates on the teichmuller space t s we express the weil petersson poisson structure of t s in this system of coordinates and we prove that it limits pointwise to the piecewise linear poisson structure defined by kontsevich on the arc complex of s at the same time we obtain a formula for the first order variation of the distance between two closed geodesics under fenchel nielsen deformation

edge deletion problems edge deletion problems ask for a minimum set of edges whose deletion makes a graph have a certain property when this property can be characterized by a finite set of forbidden induced subgraphs the problem can be solved in fixed parameter time by a naive bounded search tree algorithm sometimes deleting an edge to break an erstwhile forbidden induced subgraph might introduce new ones which may involve the neighbors of the original forbidden induced subgraph therefore in considering possible ways to break a forbidden induced subgraph one naturally takes its neighborhood into consideration this observation easily yields more efficient branching rules but a naive implementation will require too many tedious case analyses here we take advantage of modular decomposition which allows us to focus on far simpler quotient graphs instead of the original graphs they together yield simple improved algorithms for the edge deletion problems to chain graphs and trivially perfect graphs

sewing yang mills measures and moduli spaces over compact surfaces consider a closed surface obtained by pasting two surfaces 1and 2along some of their boundary components a formula is derived expressing quantum yang mills functional integrals for in terms of the corresponding integrals for the surfaces i using this a formula is derived which expresses the symplectic volume measure on the moduli space of flatsu 2 connections over a closed riemann surface in terms of the corresponding volume measures for the surfaces 1and 2 when the latter have one boundary component each

characteristic classes and representations of discrete subgroups of lie groups a volume invariant is used to characterize those representations of a countable group into a connected semisimple lie group g which are injective and whose image is a discrete cocompact subgroup of g let it be a discrete cocompact subgroup of g and consider the analytic variety hom 7r g consisting of homomorphisms 0 tt g denote by k a maximal compact subgroup of g and x k g the associated symmetric space let m be the orbit space x tt for convenience we shall henceforth assume that n is torsionfree by selberg s lemma 12 this may be accomplished by replacing tt by a subgroup of finite index this insures that m is a compact smooth manifold having tt as its fundamental group the case when n has torsion follows from the torsionfree case with minor modifications but these modifications need not concern us here to every representation 0 g hom 7r g we associate a foliated bundle e over m with fibre x and structure group g see e g 6 if co is a closed ginvariant differential fc form on x then we may spread co over the fibres of e copies of x to obtain a closed fc form co on e we define co 0 fj f where is any section of e moreover co 0 is independent of the choice of section for example taking co to be the g invariant volume form on x we obtain a real number u 0 which depends on 0 when x is even dimensional the chern gauss bonnet theorem implies that u 0 may be described as an euler number i e the self intersection number of any section which is a topological invariant of e when x is odd dimensional this volume invariant is related to a more recent kind of topological invariant based on bounded cohomology and due to gromov 3 and is constant on the connected components of hom 7r g the volume invariant satisfies an inequality

group systems groupoids and moduli spaces of parabolic bundles moduli spaces of homomorphisms or more generally twisted homomorphisms from fundamental groups of surfaces to compact connected lie groups were connected with geometry through their identification with moduli spaces of holomorphic vector bundles see 29 atiyah and bott 2 initiated a new approach to the study of these moduli spaces by identifying them with moduli spaces of projectively fiat constant central curvature connections on principal bundles over riemann surfaces which they analyzed by methods of gauge theory in particular they showed that an invariant inner product on the lie algebra of the lie group in question induces a natural symplectic structure on a certain smooth open stratum although this moduli space is a finite dimensional object generally a stratified space which is locally semialgebraic 19 but sometimes a manifold its symplectic structure on the stratum just mentioned was obtained by applying the method of symplectic reduction to the action of an infinite dimensional group the group of gauge transformations on an infinite dimensional symplectic manifold the space of all connections on a principal bundle

quantum yang mills on a riemann surface we obtain the quantum expectations of gauge invariant functions of the connection on ag su n product bundle over a riemann surface of genusg we show that the spacea g m of connections modulo those gauge transformations which are the identity at one point is itself a principal bundle with affine linear fiber the base space path2gg consists of 2g tuples of paths ing subject to a relation on their endpoint values quantum expectations are iterated path integrals over first the fiber then over path2gg each with respect to the push forward toa g m of the measuree s a da here s a denotes the yang mills action ona we exhibit a global section ofa g m to define a choice of origin in each fiber relative to which the measure on the fiber is gaussian the induced measure on path2gg is the product of wiener measures on the component paths conditioned to preserve the endopoint relation conformal transformations of the metric onm act by reparametrizing these paths we explicitly compute the partition function in the general case and the expectations of functions of certain products of wilson loops in the caseg 1

the hyperbolic moduli space of flat connections and the isomorphism of symplectic multiplicity spaces the hyperbolic moduli space of flat connections and the isomorphism of symplectic multiplicity spaces

degeneration of riemann surfaces and intermediate polarization of the moduli space of flat connections we construct a family of polarizations of the moduli space of flatsu n connections on a closed 2 manifold of genusg 2 these are generalizations of various polarizations known until now that is our family of polarizations includes weitsman s real polarizations in the case ofn 2 17 as well as the kahler polarizations which are well known since 2 and 18 our construction is based on an original formulation of degeneration of riemann surfaces the relation between our polarizations and the complex structures of the moduli spaces of parabolic bundles are also studied

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

subexponential parameterized algorithm for minimum fill in the minimum fill in problem is used to decide if a graph can be triangulated by adding at most k edges in 1994 kaplan shamir and tarjan showed that the problem is solvable in time mathcal o 2 mathcal o k k 2 nm on graphs with n vertices and m edges and thus is fixed parameter tractable here we give the first subexponential parameterized algorithm solving minimum fill in in time mathcal o 2 mathcal o sqrt k log k k 2 nm this substantially lowers the complexity of the problem techniques developed for minimum fill in can be used to obtain subexponential parameterized algorithms for several related problems including minimum chain completion chordal graph sandwich and triangulating colored graph

a note on stability and stabilizability of neutral systems this note presents frequency domain characterizations of exponential stability and stabilizability of neutral systems based on transfer function matrices and the existence of nice solutions of certain bezout equations it turns out that the existence of h sup spl infin solutions is not sufficient for exponential stabilizability but that they have to satisfy an additional growth assumption as well whilst the proofs of the authors results are based on an abstract infinite dimensional representation of the neutral system they emphasize that the results are expressed in terms of the original parameters of the neutral equation and do not require a reformulation of the system in an abstract state space form the sufficiency parts of the results hold even when the delay operator acting on the derivative contains a singular part

quantum yang mills on the two sphere we obtain the quantum expectations of gauge invariant functions of the connection on a principalg su n bundle overs 2 we show that the spacea g m of connections modulo gauge transformations which are the identity at one point is itself a principal bundle over g based loops in the symmetry group the fiber ina g m is an affine linear space quantum expectations are iterated path integrals first over this fiber then over g each with respect to the push forward toa g m of the measure s s a da s a denotes the yang mills action ona there is a global section ofa g m on which the first integral is a gaussian the resulting measure on g is the conditional wiener measure we explicitly compute the expectations of a special class of wilson loops

intersection theory on moduli spaces of holomorphic bundles of arbitrary rank on a riemann surface let n and d be coprime positive integers and define m n d to be the moduli space of semi stable holomorphic vector bundles of rank n degree d and fixed determinant on a compact riemann surface e this moduli space is a compact kifhler manifold which has been studied from many different points of view for more than three decades see for instance narasimhan and seshadri 41 the subject of this article is the characterization of the intersection pairings in the cohomology ring1 h m n d a set of generators of this ring was described by atiyah and bott in their seminal 1982 paper 2 on the yangmills equations on riemann surfaces where in addition inductive formulas for the betti numbers of m n d obtained earlier using number theoretic methods 13 25 were rederived by poincare duality knowledge of the intersection pairings between products of these generators or equivalently knowledge of the evaluation on the fundamental class of products of the generators completely determines the structure of the cohomology ring in 1991 donaldson 15 and thaddeus 47 gave formulas for the intersection pairings between products of these generators in h m 2 1 in terms of bernoulli numbers then using physical methods witten 50 found formulas for generating functions from which could be extracted the intersection pairings between products of these generators in h m n d for general rank n these generalized his rigorously proved formulas 49 for the symplectic volume of m n d for instance the symplectic volume of m 2 1 is given by

lie group valued moment maps we develop a theory of quasi hamiltonian g spaces for which the moment map takes values in the group g itself rather than in the dual of the lie algebra the theory includes counterparts of hamiltonian reductions the guillemin sternberg symplectic cro

intersection numbers of polygon spaces we study the intersection ring of the space m 1 m of polygons in r 3 we find homology cycles dual to generators of this ring and prove a recursion relation in m the number of edges for their intersection numbers this result is an analog of the recursion relation appearing in the work of witten and kontsevich on moduli spaces of punctured curves and in the work of weitsman on moduli spaces of flat connections on two manifolds of genus g with m marked points based on this recursion formula we obtain an explicit expression for the computation of the intersection numbers of polygon spaces and use it in several examples among others we study the special case of equilateral polygon spaces where all i s are the same and compare our results with the expressions for these particular spaces that have been determined by kamiyama and tezuka finally we relate our explicit formula for the intersection numbers with the generating function for intersection pairings of the moduli space of flat connections of yoshida as well as with equivalent expressions for polygon spaces obtained by takakura and konno through different techniques

the symplectic nature of fundamental groups of surfaces soit 7b g la categorie des groupes de lie g avec une forme bilineaire symetrique non singuliere b sur l algebre de lie soit le groupe fondamental d une surface orientee close de genre 1 et soit 7b s la categorie des out espaces symplectiques alors la correspondance g b hom g g b definit un foncteur 7b g 7b s ou b est la structure symplectique sur hom g g definie par b

around polygons in ℝ3 and s3 we survey certain moduli spaces in low dimensions and some of the geometric structures that they carry and then construct identifications among all of these spaces in particular we identify the moduli spaces of polygons in 3 and s 3 the moduli space of restricted representations of the fundamental group of a punctured 2 sphere the moduli space of flat connections on a punctured sphere the moduli space of parabolic bundles on a sphere the moduli space of weighted points on 1 and the symplectic quotient of so 3 acting diagonally on s 2 n all of these spaces depend on parameters and some the above identifications require the parameters to be small one consequence of this work is that these spaces are all biholomorphic with respect to the most natural complex structures they can each be given

an explicit description of the symplectic structure of moduli spaces of flat connections the moduli space of flat connections on a principal g bundle over a compact oriented surface of genus g 1 is considered herein using the holonomies around noncontractible loops the moduli space is described as a quotient of a submanifold of g2g an explicit expression is obtained for the symplectic form on the smooth part of moduli space and several properties of this form are established

the fenchel nielsen deformation the uniformization theorem provides that a riemann surface s of negative euler characteristic has a metric of constant curvature 1 a hyperbolic structure can be understood in terms of its deformations unfortunately the variation of the hyperbolic metric arising from a deformation is not determined by local data if z is a conformal coordinate for s and a ii h ii 00 1 a beltrami differential then dw dz p dj defines a new conformal coordinate and structure si the tensor p may vanish on the open set 0 c s and yet the hyperbolic metrics of s and si do not coincide on this phenomenon presents a s erious difficulty in the deformation theory of hyperbolic structures the fenchel nielsen twist deformation does not share this defect the deformation is defined by cutting the surface along a simple closed geodesic rotating one side of the cut relative to the other and attaching the sides in their new position a geodesic intersecting the cut is deformed to a broken geodesic the hyperbolic structure in the complement of the cut extends naturally to a hyperbolic structure on the new surface the deformation is concentrated at the cut the basic idea for the deformation appears in the work of fricke klein dehn and fenchel nielsen 71 8 in the fenchel nielsen manuscript the deformation was considered extensively coordinates for teichmiller space are defined in terms of the deformation the proper definition is given by a fuchsian group construction calculations are made in terms of unimodular matrices this approach is often cumbersome our basic objective is to give a description of the deformation in terms of quasiconformal mappings the bers embedding of teichmiiller space is central to the analytic theory of deformations using our characterization we calculate the first variation of the bers embedding for the fenchel nielsen deformation the result is a poincare series 6 which already appears in the work of petersson 16

gauge theory on asymptotically periodic 4 manifolds il existe une famille non denombrable de classes de diffeomorphismes des 4 varietes orientees homeomorphes a r 4

small volume limits of 2 d yang mills by examining the lattice gauge approximation we show that the small volume limit of the 2 dimensional yang mills functional integral is the natural symplectic measure on the moduli space of flat connections

floer cohomology and pencils of quadrics there is a classical relationship in algebraic geometry between a hyperelliptic curve and an associated pencil of quadric hypersurfaces we investigate symplectic aspects of this relationship with a view to applications in low dimensional topology

left and right centers in quasi poisson geometry of moduli spaces abstract we introduce left central and right central functions and left and right leaves in quasi poisson geometry generalizing central or casimir functions and symplectic leaves from poisson geometry they lead to a new type of quasi poisson reduction which is both simpler and more general than known quasi hamiltonian reductions we study these notions in detail for moduli spaces of flat connections on surfaces where the quasi poisson structure is given by an intersection pairing on homology

toric structures on the moduli space of flat connections on a riemann surface volumes and the moment map abstract in earlier papers we constructed a hamiltonian torus action on an open dense set in the moduli space of flat su 2 connections on a compact riemann surface where the dimension of the torus is half the dimension of the moduli space this torus action shows that this set can be viewed symplectically as a noncompact toric variety the number of integral points of the moment map for the torus action turns out to be identical to the verlinde dimension d g k as an application we furnish a new proof of the relation between the large k limit of d g k and the volume of the moduli space from our point of view this relation follows from the equality between the symplectic volume of a toric variety and the euclidean volume of the image of the moment map similar considerations are shown to give rise to the volumes of moduli spaces of parabolic bundles on a riemann surface knowledge of these volumes has been shown to allow a proof of the verlinde formula for the dimension of the space of holomorphic sections of line bundles on this space

an area preserving action of the modular group on cubic surfaces and the painleve vi equation we construct an area preserving action of the modular group on a general 4 parameter family of affine cubic surfaces we present a geometrical background behind this construction that is a natural symplectic structure on a moduli space of rank two linear monodromy representations over the 2 dimensional sphere with four punctures and a natural symplectic action upon it of the braid group on three strings studying this action as a discrete dynamical system will be important in discussing the monodromy of the painleve vi equation

half density volumes of representation spaces of some 3 manifolds and their application in this paper we compute half density volumes of the irreducible su 2 representation spaces for seifert fibred manifolds and graph manifolds these representation spaces are not discrete so that the half density derived from the reidemeister torsion is used as a measure as an application of our result we get the exact value of the jeffrey weitsman witten invariant of a seifert fibred manifold with non discrete irreducible su 2 representation space

a symplectic structure for connections on surfaces with boundary for compact surfaces with one boundary component and semisimple gauge groups we construct a closed gauge invariant 2 form on the space of flat connections whose boundary holonomy lies in a fixed conjugacy class this form descends to the moduli space under the action of the full gauge group and provides an explicit description of a symplectic structure for this moduli space

group cohomology construction of the cohomology of moduli spaces of flat connections on 2 manifolds we use group cohomology and the de rham complex on simplicial manifolds to give explicit differential forms representing generators of the cohomology rings of moduli spaces of representations of fundamental groups of 2 manifolds these generators are constructed using the de rham representatives for the cohomology of classifying spaces bk where k is a compact lie group such representatives universal characteristic classes were found by bott and shulman thus our representatives for the generators of the cohomology of moduli spaces are given explicitly in terms of the maurer cartan form this work solves a problem posed by weinstein who gave a corresponding construction following karshon and goldman of the symplectic forms on these moduli spaces we also give a corresponding construction of equivariant differential forms on the extended moduli space x which is a finite dimensional symplectic space equipped with a hamiltonian action of k for which the symplectic reduced space is the moduli space of representations of the 2 manifold fundamental group in k

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 transfer matrix of a neutral system characterizations of exponential stability in input output terms abstract this paper investigates the stability of linear autonomous multivariable neutral systems from an input output viewpoint several frequency domain and input output characterizations for exponential stability of neutral systems are given we provide two examples which illustrate that the behaviour of neutral systems may be quite different from that of retarded systems moreover we give necessary and sufficient conditions for the transfer functions of a neutral system to belong to certain algebras of meromorphic functions introduced in this paper

topological sectors and measures on moduli space in quantum yang mills on a riemann surface previous path integral treatments of yang mills on a riemann surface automatically sum over principal fiber bundles of all possible topological types in computing quantum expectations this paper extends the path integral formulation to treat separately each topological sector the formulation is sufficiently explicit to calculate wilson line expectations exactly further it suggests two new measures on the moduli space of flat connections one of which proves to agree with the small volume limit of the yang mills measure

symplectic manifolds and isomonodromic deformations abstract we study moduli spaces of meromorphic connections with arbitrary order poles over riemann surfaces together with the corresponding spaces of monodromy data involving stokes matrices natural symplectic structures are found and described both explicitly and from an infinite dimensional viewpoint generalising the atiyah bott approach this enables us to give an intrinsic symplectic description of the isomonodromic deformation equations of jimbo miwa and ueno thereby putting the existing results for the six painleve equations and schlesinger s equations into a uniform framework

yang mills on surfaces with boundary quantum theory and symplectic limit the quantum field measure for gauge fields over a compact surface with boundary with holonomy around the boundary components specified is constructed loop expectation values for general loop configurations are computed for a compact oriented surface with one boundary component let be the moduli space of flat connections with boundary holonomy lying in a conjugacy class in the gauge group g we prove that a certain natural closed 2 form on introduced in an earlier work by c king and the author is a symplectic structure on the generic stratum of for generic we then prove that the quantum yang mills measure with the boundary holonomy constrained to lie in converges in a natural sense to the corresponding symplectic volume measure in the classical limit we conclude with a detailed treatment of the case and determine the symplectic volume of this moduli space

stratified symplectic spaces and reduction let m w be a hamiltonian g space with proper momentum map j m g it is well known that if zero is a regular value of j and g acts freely on the level set j 0 then the reduced space mo j 0 g is a symplectic manifold we show that if the regularity assumptions are dropped the space mo is a union of symplectic manifolds i e it is a stratified symplectic space arms et al 2 proved that mo possesses a natural poisson bracket using their result we study hamiltonian dynamics on the reduced space in particular we show that hamiltonian flows are strata preserving and give a recipe for lifting a reduced hamiltonian flow to the level set j 0 finally we give a detailed description of the stratification of mo and prove the existence of a connected

on the variation of the poisson structures of certain moduli spaces let pi be the fundamental group of a compact connected orientable topological surface with finitely many boundary circles our aim is to construct poisson algebras of continuous functions on certain open subspaces cal r pi g vec w of the space of representations of pi in a lie group g whose lie algebra is supposed to be endowed with a nondegenerate invariant symmetric bilinear form when g is compact and connected the spaces cal r pi g vec w may be taken to be dense in the space of all representations the spaces cal r pi g vec w contain spaces of representations rm rep pi g vec c where the values of those generators of the fundamental group which correspond to the boundary circles are constrained to lie in fixed conjugacy classes and the representation spaces rm rep pi g vec c being endowed with universal poisson algebras constructed elsewhere earlier the restrictions from the spaces cal r pi g vec w to any of the spaces rm rep pi g vec c are poisson maps thus the poisson algebra on each cal r pi g vec w gives a description of the variation of the universal poisson structures on the spaces rm rep pi g vec c as the chosen conjugacy classes move across g we also give twisted versions of these results

the moduli space of flat connections on oriented surfaces with boundary abstract a 2 form is constructed on the space of connections on a principal bundle over an oriented surface with boundary this induces a symplectic structure for the moduli space of flat connections with boundary holonomies lying in prescribed conjugacy classes the yang mills quantum field measure is described for this situation this measure converges to the normalized symplectic volume measure in the classical limit

the yang mills measure for s2 abstract the yang mills measure for gauge fields over the two sphere is constructed using a conditioned white noise process stochastic differential equations are used to study parallel translation under this measure expectation values of a broad class of wilson loop configurations are computed explicitly and it is shown that these values are invariant under area preserving diffeomorphisms of the sphere