the cohomology rings of moduli spaces of bundles over riemann surfaces the cohomology of the moduli space ji n d of semistable bundles of coprime rank n and degree d over a fixed riemann surface m of genus g 2 has been much studied over the last two decades and a great deal is known about it it is known that the cohomology has no torsion 2 the betti numbers have been computed using various different methods 2 5 11 and a set of generators for the cohomology ring is known 2 the purpose of this article is to provide a method of deciding when an expression in these generators is a relation and in particular when the rank n is two to prove two conjectures about the relations between the generators due to mumford and to newstead and ramanan newstead and ramanan conjectured that certain relations hold among the generators whereas mumford conjectured that a certain set of relations is a complete set in fact the conjecture of newstead and ramanan has two parts the first of which was proved by gieseker in 1982 4 the second part of their conjecture which is the part examined in this article was given an independent proof recently by thaddeus 19 who has a very different method of obtaining information about the relations in the cohomology ring the computation of the betti numbers of the moduli spaces i n d in 2 stimulated a general method of computing betti numbers of quotients in the sense of mumford s geometric invariant theory 14 of nonsingular complex projective varieties by complex reductive group actions see 8 9 10 similarly the ideas applied in this article to finding generators and relations for the cohomology rings of the moduli spaces i n d also lead to an analogous method of obtaining generators and relations for the rational cohomology rings of geometric invariant theoretic quotients the first section provides background material on the cohomology of the moduli spaces i n d in the second section mumford s conjecture is proved when the rank n is two the method could in principle be used to decide whether the conjecture is true for rank n 2 but appears to become significantly more complicated in the general case the third section gives a method for deciding when an expression in the generators is a relation and the fourth

fundamental groups of character varieties surfaces and tori we compute the fundamental group of moduli spaces of lie group valued representations of surface and torus groups

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

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

hilbert schemes separated variables and d branes we explain sklyanin s separation of variables in geometrical terms and construct it for hitchin and mukai integrable systems we construct hilbert schemes of points on t for c c or elliptic curve and on c 2 and show that their complex deformations are integrable systems of calogero sutherland moser type we present the hyperkahler quotient constructions for hilbert schemes of points on cotangent bundles to the higher genus curves utilizing the results of hurtubise kronheimer and nakajima finally we discuss the connections to physics of d branes and string duality

gromov invariants for holomorphic maps from riemann surfaces to grassmannians two compactifications of the space of holomorphic maps of fixed degree from a compact riemann surface to a grassmannian are studied it is shown that the uhlenbeck compactification has the structure of a projective variety and is dominated by the algebraic compactification coming from the grothendieck quot scheme the latter may be embedded into the moduli space of solutions to a generalized version of the vortex equations studied by bradlow this gives an effective way of computing certain intersection numbers known as gromov invariants on the space of holomorphic maps into grassmannians we carry out these computations in the case where the riemann surface has genus one department of mathematics university of utah salt lake city utah 84112 e mail address bertram math utah edu department of mathematics brown university providence rhode island 02912 e mail address daskal gauss math brown edu department of mathematics university of california irvine california 92717 e mail address raw math uci edu license or copyright restrictions may apply to redistribution see http www ams org journal terms of use

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

the yang mills heat flow on the moduli space of framed bundles on a surface we study the analog of the yang mills heat flow on the moduli space of framed bundles on a cut surface existence and convergence of the heat flow give a stratification of morse type invariant under the action of the loop group we use the stratification to prove versions of kahler quantization commutes with reduction and kirwan surjectivity

universal moduli spaces of surfaces with flat bundles and cobordism theory for a compact connected lie group g we study the moduli of pairs e w here is a genus g riemann surface and e is a flat g bundle varying both the riemann surface and the flat bundle leads to a moduli space m g parametrizing families riemann surfaces with flat g bundles we show that there is a stable range in which the homology of m g is independent of g the stable range depends on the genus of the surface we then identify the homology of this moduli space in the stable range in terms of the homology of an explicit infinite loop space rationally the stable cohomology of this moduli space is generated by the mumford morita miller classes and the ring of characteristic classes of principal g bundles h bg equivalently our theorem calculates the homology of the moduli space of semi stable holomorphic bundles on riemann surfaces we then identify the homotopy type of the category of one manifolds and surface cobordisms each equipped with a flat g bundle our methods combine the classical techniques of atiyah and bott with the new techniques coming out of madsen and weiss s proof of mumford s conjecture on the stable cohomology of the moduli space of riemann surfaces

the limit of the hermitian yang mills flow on reflexive sheaves abstract in this paper we study the asymptotic behavior of the hermitian yang mills flow on a reflexive sheaf we prove that the limiting reflexive sheaf is isomorphic to the double dual of the graded sheaf associated to the harder narasimhan seshadri filtration this answers a question by bando and siu

monte carlo renormalization group study of crosslinked polymer chains on fractals we study the problem of two crosslinked polymer chains in a good solvent modelled by two mutually crossing self avoiding walks situated on fractals that belong to the sierpinski gasket sg family whose members are labelled by an integer b by applying the monte carlo renormalization group mcrg method we calculate the critical exponent y associated with the number of crossings of the two self avoiding walk paths for a sequence of sg fractals with for the problem under study we find that our mcrg approach provides results that are virtually rigorous that is results with exceptionally small deviations at most 0 07 from the available exact renormalization group results we discuss our set of mcrg data for y as a function of the fractal parameter b and compare its behaviour with the finite size scaling predictions

partial desingularisations of quotients of nonsingular varieties and their betti numbers when a reductive group g acts linearly on a nonsingular complex projective variety x one can define a projective quotient variety x g using mumford s geometric invariant theory if the condition that every semistable point be properly stable is satisfied this quotient is the ordinary topological quotient of an open subset xss of the variety by the group in k a formula is obtained for the rational cohomology of x g under this condition the formula involves the rational cohomology of x and various linear sections of x together with the rational cohomology of the classifying spaces of g and certain reductive subgroups of g in many interesting examples the condition required in k is not satisfied thus the question arises as to what information we can obtain in general the quotient x g can now have serious singularities in contrast to the good case considered in k where the only singularities are those caused by finite isotropy groups it will be shown in this paper that there is a systematic way of blowing up x along a sequence of nonsingular subvarieties to obtain a variety x with a linear action of g such that every semistable point of x is stable the only assumption that we have to make is that there exists at least one stable point of x then x g is almost a resolution of singularities of x1 g in the sense that the most serious singularities have been resolved moreover there is a formula for the rational cohomology of x g again involving the rational cohomology of x and certain linear sections of x together with the rational cohomology of the classifying spaces of g and some reductive subgroups of g see theorem 8 14 for convenience we shall assume throughout that g is connected however the construction of x works in general and it is straightforward to modify the cohomological formulas to apply to the general case the construction of x g can also be modified to apply in some cases when x has no stable points the layout of the paper is as follows section 1 is a review of the basic facts of geometric invariant theory which will be needed and section 2 describes the relationship of geometric invariant theory with symplectic geometry and the moment map in section 3 semistability and stability in a blow up of x along a nonsingular c invariant subvariety are related to semistability and stability in x

rational intersection cohomology of quotient varieties given a linear action of a complex reductive group g on a complex projective variety x one can define a projective quotient variety x g using mumford s geometric invariant theory 18 from the construction of this quotient it is not immediately obvious how its topology is related to the topology of x and the group action however when x is nonsingular it is possible at least if one technical condition is satisfied to compute the intersection betti numbers of x g in terms of the rational cohomology of certain nonsingular linear sections z of x and the classifying spaces of certain reductive subgroups h of g see 15 the technical condition is that there should exist at least one point of x which is properly stable for the action of g this is satisfied in most interesting examples such as those coming from moduli problems the sections of x and subgroups of g which occur have a simple description in terms of the representation of g which induces the linear action on x it thus seems reasonable to ask what happens when x is singular the aim of this paper is to show that the procedure for computing the intersection betti numbers of x g can be generalised to the singular case see 2 25 and 2 28 however now one needs to know not only the intersection cohomology of the various sections z of x but also how they are positioned relative to the singularities of x in the special case when they meet the singularities transversely then one has exactly the same formulas as when x is nonsingular except that ordinary cohomology is replaced by intersection cohomology throughout the situation which it is most important to understand is that in which every semistable point is stable and this case is considered in detail in w 2 after the necessary preliminaries have been covered in w 1 some brief remarks are made about the more general case when one only assumes that there exist some stable points a simple example is worked out 2 27 which shows that the formulas can also be used in reverse to compute the intersection betti numbers of x from knowledge of the intersection betti numbers of the quotient x g even simply from the knowledge that these vanish in dimensions greater than dim x g

the self duality equations on a riemann surface in this paper we shall study a special class of solutions of the self dual yang mills equations the original self duality equations which arose in mathematical physics were defined on euclidean 4 space the physically relevant solutions were the ones with finite action the so called instantons the same equations may be dimensionally reduced to euclidean 3 space by imposing invariance under translation in one direction these equations also have physical relevance the solutions which have finite action in three dimensions are the magnetic monopoles if we take the reduction process one step further and consider solutions which are invariant under two translations we obtain a set of equations in the plane here however there is no clear physical meaning and indeed attempts to find finite action solutions have failed nevertheless these are the equations we shall consider despite the lack of interesting solutions in r2 the equations have the important property conformal invariance which allows them to be defined on manifolds modelled on r2 by conformal maps namely riemann surfaces we shall consider here solutions of the self duality equations defined on a compact riemann surface there are in fact solutions as we shall show and the moduli space of all solutions turns out to be a manifold with an extremely rich geometric structure which will be the focus of our study it brings together in a harmonious way the subjects of riemannian geometry topology algebraic geometry and symplectic geometry illuminating all these facets of the same object accounts for the length of this paper the self duality equations are equations from gauge theory geometrically they are defined in terms of connections on principal bundles while the group of the principal bundle may be chosen arbitrarily for the equations to make sense we restrict attention here to the simplest case of su 2 or so 3 there are two reasons for this the first and most obvious is that it simplifies calculations and avoids the use of inductive processes which are inherent in the consideration of a general lie group of higher rank the second reason is that solutions for su 2 have an intimate relationship with the internal structure of the riemann surface as a consequence of results we shall prove about solutions to the self duality equations we learn something about the moduli space of complex structures on the surface itself namely teichmuller space

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

adsorption of ideal polymers on an infinitely ramified fractal we study ideal polymer chains interacting attractively with the borders of the lacunas of an infinitely ramified fractal the sierpinski carpet ideal chains are simulated on finite stages of construction of this fractal at various temperatures the mean square displacement and the mean number of adsorbed monomers of n step chains are estimated in these lattices and extrapolations to the fractal limit infinite lattice consider the exact forms of finite size corrections as previously predicted by the series expansion method in the noninteracting case a finite fraction of the monomers is adsorbed and this fraction increases as the temperature decreases however there is evidence that the critical exponent v which governs the growth of the chains varies with the temperature in a nonmonotonic way at high temperatures v increases with decreasing temperature and thus the chains are more stretched than in the noninteracting case at an intermediate temperature v starts to decrease and is still positive at very low temperatures when the chains grow along the borders of several lacunas occasionally crossing the bulk between them

intersection homology ii in 19 20 we introduced topological invariants ih x called intersection homology groups for the study of singular spaces x these groups depend on the choice of a perversity p a perversity is a function from 2 3 to the non negative integers such that both c and c 2 c are positive and increasing functions of c 2 1 the group ihr is defined for spaces x called pseudomanifolds a pseudomanifold of dimension n is a space that admits a stratification x x n x n z x n 3 x 1 x 0

gauge theories on four dimensional riemannian manifolds this paper develops the riemannian geometry of classical gauge theories yang mills fields coupled with scalar and spinor fields on compact four dimensional manifolds some important properties of these fields are derived from elliptic theory regularity an energy gap theorem the manifold structure of the configuration space and a bound for the supremum of the field in terms of the energy it is then shown that finite energy solutions of the coupled field equations cannot have isolated singularities this extends a theorem of k uhlenbeck

analytic cycles bott chern forms and singular sets for the yang mills flow on kahler manifolds abstract it is shown that the singular set for the yang mills flow on unstable holomorphic vector bundles over compact kahler manifolds is completely determined by the harder narasimhan seshadri filtration of the initial holomorphic bundle we assign a multiplicity to irreducible top dimensional components of the singular set of a holomorphic bundle with a filtration by saturated subsheaves we derive a singular bott chern formula relating the second chern form of a smooth metric on the bundle to the chern current of an admissible metric on the associated graded sheaf this is used to show that the multiplicities of the top dimensional bubbling locus defined via the yang mills density agree with the corresponding multiplicities for the harder narasimhan seshadri filtration the set theoretic equality of singular sets is a consequence

on the blow up set of the yang mills flow on kahler surfaces the yang mills flow on a kahler surface with holomorphic initial data converges smoothly away from a singular set determined by the harder narasimhan seshadri filtration of the initial holomorphic bundle

calderon projector for the hessian of the perturbed chern simons function on a 3 manifold with boundary the existence and continuity for the calderon projector of the perturbed odd signature operator on a 3 manifold is established as an application we give a new proof of a result of taubes relating the modulo 2 spectral flow of a family of operators on a homology 3 sphere with the difference in local intersection numbers of the character varieties coming from a heegard decomposition

the intersection cohomology of the satake compactification of mathcal a _g for g le 4 we completely determine the intersection cohomology of the satake compactifications mathcal a 2 mathrm sat mathcal a 3 mathrm sat and mathcal a 4 mathrm sat except for ih 10 mathcal a 4 mathrm sat we also determine all the ingredients appearing in the decomposition theorem applied to the map from a toroidal compactification to the satake compactification in these genera as a byproduct we obtain in addition several results about the intersection cohomology of the link bundles involved

intersection homology and torus actions in the bialynicki birula decomposition of x see 15 6 10 13 the same formula holds for appropriate integers my when c is replaced by a torus t c r in 9 it is shown that the formula 0 1 is valid even when x is singular provided that the bialynicki birula decomposition is good the aim of this paper is to generalize 0 1 to the case when x is singular in a different way which involves replacing ordinary homology by intersection homology with respect to the middle perversity however only rational coefficients are considered when x is nonsingular its intersection homology and ordinary homology coincide but when x is singular its intersection homology behaves better in many respects than its ordinary cohomology it is shown that when x is singular just as when x is nonsingular the rational intersection homology of x is determined by the action of t on an arbitrarily small neighborhood of the fixed point set xt as might be expected the formula 0 1 does not carry over directly when intersection homology replaces ordinary homology the terms hi 2m f z appearing in the right hand side are replaced by hypercohomology groups of certain complexes of sheaves over the f which depend upon how the f meet the singularities of x see they y orem 2 3 1 of this paper contains a review of a proof of 0 1 for rational coefficients which uses equivariant morse theory in 2 it is shown how this proof can be extended to apply to singular varieties when intersection homology is used the

hyperkahler metrics and supersymmetry we describe two constructions of hyperkahler manifolds one based on a legendre transform and one on a sympletic quotient these constructions arose in the context of supersymmetric nonlinear models but can be described entirely geometrically in this general setting we attempt to clarify the relation between supersymmetry and aspects of modern differential geometry along the way reviewing many basic and well known ideas in the hope of making them accessible to a new audience

connections with l p bounds on curvature we show by means of the implicit function theorem that coulomb gauges exist for fields over a ball inr n when the integrall n 2 field norm is sufficiently small we then are able to prove a weak compactness theorem for fields on compact manifolds withl p integral norms bounded p n 2

the reverse yang mills higgs flow in a neighbourhood of a critical point the main result of this paper is a construction of solutions to the reverse yang mills higgs flow converging in the c topology to a critical point the construction uses only the complex gauge group action which leads to an algebraic classification of the isomorphis m classes of points in the unstable set of a critical point in terms of a filtration of the underlying higgs bundle analysing the compatibility of this filtration with the hard er narasimhan seshadri double filtration gives an algebraic criterion for two critical points to be connect ed by a flow line as an application we can use this to construct hecke modifications of higgs bundles via the yan g mills higgs flow when the higgs field is zero corresponding to the yang mills flow this criterion has a geometric interpretation in terms of secant varieties of the projectivisation of the underlying bundle inside the unstable manifold of a critical point which gives a precise description of broken and unbroken flow lines connec ting two critical points for non zero higgs field at generic critical points the analogous interpretation in volves the secant varieties of the spectral curve of the higgs bundle

equivariant intersection cohomology of semi stable points the main result of the paper is that the equivariant intersection cohomology of the semi stable points on a complex projective variety for the action of a complex reductive group may be determined from the equivariant intersection cohomology of the semi stable points for the action of a maximal torus it extends the work of brion who considered the smooth case using equivariant cohomology equivariant intersection cohomology is a theory due to brylinski and the second author as an application a surprising relation between the intersection cohomology of chow hypersurfaces is established in the last section of the paper in this paper we show that the equivariant intersection cohomology of the semi stable points on a complex projective variety for the action of a complex reductive group may be determined from the equivariant intersection cohomology of the semi stable points for the action of a maximal torus this extends the work of brion see br who considered the smooth case using equivariant cohomology the equivariant intersection cohomology we employ is due to brylinski and the

monopoles and geodesics using the holomorphic geometry of the space of straight lines in euclidean 3 space it is shown that every static monopole of chargek may be constructed canonically from an algebraic curve by means of the atiyah ward ansatz a k

twisted harmonic maps and the self duality equations si m est une surface de riemann compacte on utilise des resultats de la theorie des applications harmoniques pour trouver une solution aux equations d auto dualite associees a une representation irreductible p 1 m psl 2 c

continuous and first order polymer adsorption on hierarchical fractal walls polymer adsorption on fractally rough walls of varying dimensionality is studied by renormalization group methods on hierarchical lattices exact results are obtained for deterministic walls the adsorption transition is found continuous for low dimension dw of the adsorbing wall and the corresponding crossover exponent monotonically increases with dw eventually overcoming previously conjectured bounds for dw exceeding a threshold value dw becomes one and the transition changes to first order dw dsaw the fractal dimension of the polymer in the bulk an accurate numerical approach to the same problem with random walls gives evidence of the same scenario

the moduli space of stable vector bundles over a real algebraic curve we study the spaces of stable real and quaternionic vector bundles on a real algebraic curve the basic relationship is established with unitary representations of an extension of mathbb z 2 by the fundamental group by comparison with the space of real or quaternionic connections some of the basic topological invariants of these spaces are calculated