isomonodromic deformations of connections with singularities of parahoric formal type in previous work the authors have developed a geometric theory of fundamental strata to study connections on the projective line with irregular singularities of parahoric formal type in this paper the moduli space of connections that contain regular fundamental strata with fixed combinatorics at each singular point is constructed as a smooth poisson reduction the authors then explicitly compute the isomonodromy equations as an integrable system this result generalizes work of jimbo miwa and ueno to connections whose singularities have parahoric formal type

middle convolution and harnad duality dettweiler reiter s additive description of katz s middle convolution can be interpreted in terms of the harnad duality using this interpretation together with mumford s geometric invariant theory we generalize the additive middle convolution to have a multi parameter and act on systems of linear ordinary differential equations with irregular singularities we show that the generalized operation holds basic properties of the original one and additionally show that katz s algorithm involving our generalization works well in generic case

on the τ function of the painleve equations abstract concerning the hamiltonian structure associated with painleve equations we define the function related to a hamiltonian and show that the correlation function given in the study of the two dimensional ising model is a function for the third painleve equation the list of the hamiltonians which are polynomials in the two canonical variables is given and we prove that the function related to each of them is holomorphic

geometrizing infinite dimensional locally compact groups we study groups having invariant metrics of curvature bounded below in the sense of alexandrov such groups are a generalization of lie groups with invariant riemannian metrics but form a much larger class we prove that every locally compact arcwise connected first countable group has such a metric these groups may not be even infinite dimensional manifolds we show a number of relationships between the algebraic and geometric structures of groups equipped with such metrics many results do not require local compactness

confluent kz equations for mathfrak sl _n with poincare rank 2 at infinity we construct confluent kz equations with poincar rank 2 at infinity for the case of and the integral representation for the solutions hamiltonians of these confluent kz equations are derived from suitable quantization of dlog constructed in the theory of monodromy preserving deformation mpd in jimbo et al 1981 physica d 2 306 52 our confluent kz equations can be viewed as a quantization of mpd with poincar rank 2 at infinity

isomonodromic deformations of a rational differential system and reconstruction with the topological recursion the sl2 case in this paper we show that it is always possible to deform a differential equation x x l x x with l x sl2 c x by introducing a small formal parameter in such a way that it satisfies the topological type properties of bergere borot and eynard annales henri poincare 16 12 2713 2782 2015 this is obtained by including the former differential equation in an isomonodromic system and using some homogeneity conditions to introduce the topological recursion is then proved to provide a formal series expansion of the corresponding tau function whose coefficients can thus be expressed in terms of intersections of tautological classes in the deligne mumford compactification of the moduli space of surfaces we present a few examples including any fuchsian system of sl2 c x as well as some elements of painleve hierarchies

on the o 1 t convergence rate of ye yuan s modified alternating direction method of multipliers the alternating direction method of multipliers admm is known to be a classic and efficient method for constrained optimization problem with two blocks of variables and its empirical efficiency has been well illustrated in various fields specially for improving its speed performance ye and yuan suggested to do an additional extension with an optimal step size on the variables after each iteration of the primary admm indeed the numerical experiments indicate that this modified admm improves the speed performance of the admm by around 40 without changing the algorithmic framework much recently the o 1 t convergence rate of the primary admm is established inspired by its idea in this paper we show that this improved admm also has o 1 t convergence rate the reason that larger c yields better speed performance is also investigated and explained

gravitational instantons with faster than quadratic curvature decay iii this is our third paper in a series on the gravitational instantons in this paper we classify alg and alh gravitational instantons in alg case we extend tian yau hein s construction slightly and show that it is the only alg gravitational instanton in alh case we prove a torelli type theorem

isospectral hamiltonian flows in finite and infinite dimensions the approach to isospectral hamiltonian flow introduced in part i is further developed to include integration of flows with singular spectral curves the flow on finite dimensional ad invariant poisson submanifolds of the dual widetilde gl r of the positive part of the loop algebra widetilde gl r is obtained through a generalization of the standard method of linearization on the jacobi variety of the invariant spectral curves these curves are embedded in the total space of a line bundlet p1 c allowing an explicit analysis of singularities arising from the structure of the image of a moment map tilde j r m n r times m n r to widetilde gl r from the space of rank r deformations of a fixedn n matrixa it is shown how the linear flow of line bundles e t to tilde s over a suitably desingularized curve tilde s may be used to determine both the flow of matricial polynomialsl and the hamiltonian flow in the spacemn r mn r in terms of functions the resulting flows are proved to be completely integrable the reductions to subalgebras developed in part i are shown to correspond to invariance of the spectral curves and line bundles e t to tilde s under certain linear or anti linear involutions the integration of two examples from part i is given to illustrate the method the rosochatius system and the cnls coupled non linear schrodinger equation

from klein to painleve via fourier laplace and jimbo we describe a method for constructing explicit algebraic solutions to the sixth painleve equation generalising that of dubrovin and mazzocco there are basically two steps first we explain how to construct finite braid group orbits of triples of elements of rm sl 2 rm c out of triples of generators of three dimensional complex reflection groups this involves the fourier laplace transform for certain irregular connections then we adapt a result of jimbo to produce the painleve vi solutions in particular this solves a riemann hilbert problem explicitly each step is illustrated using the complex reflection group associated to klein s simple group of order 168 this leads to a new algebraic solution with seven branches we also prove that unlike the algebraic solutions of dubrovin and mazzocco and hitchin this solution is not equivalent to any solution coming from a finite subgroup of rm sl 2 rm c the results of this paper also yield a simple proof of a recent theorem of inaba iwasaki and saito on the action of okamoto s affine rm d 4 symmetry group as well as the correct connection formulae for generic painleve vi equations

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

stokes phenomenon gelfand zeitlin systems and relative ginzburg weinstein linearization abstract in 2007 alekseev meinrenken proved that there exists a ginzburg weinstein diffeomorphism from the dual lie algebra u n to the dual poisson lie group u n compatible with the gelfand zeitlin integrable systems in this paper we explicitly construct such diffeomorphisms via stokes phenomenon and boalch s dual exponential maps then we introduce a relative version of the ginzburg weinstein linearization motivated by irregular riemann hilbert correspondence and generalize the results of enriquez etingof marshall to this relative setting in particular we prove the connection matrix for a certain irregular riemann hilbert problem satisfies a relative gauge transformation equation of the alekseev meinrenken dynamical r matrices this gauge equation is then derived as the semiclassical limit of the relative drinfeld twist equation

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

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

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

the dependence on the monodromy data of the isomonodromic tau function the isomonodromic tau function defined by jimbo miwa ueno vanishes on the malgrange s divisor of generalized monodromy data for which a vector bundle is nontrivial or which is the same a certain riemann hilbert problem has no solution in their original work jimbo miwa ueno provided an algebraic construction of its derivatives with respect to isomonodromic times however the dependence on the generalized monodromy data i e monodromy representation and stokes parameters was not derived we fill the gap by providing a simpler and more general description in which all the parameters of the problem monodromy changing and monodromy preserving are dealt with at the same level we thus provide variational formulae for the isomonodromic tau function with respect to the generalized monodromy data the construction applies more generally given any sufficiently well behaved family of riemann hilbert problems rhp where the jump matrices depend arbitrarily on deformation parameters we can construct a one form not necessarily closed on the deformation space malgrange s differential defined off malgrange s divisor we then introduce the notion of discrete schlesinger transformation it means that we allow the solution of the rhp to have poles or zeros at prescribed point s even if is not closed its difference evaluated along the original solution and the transformed one is shown to be the logarithmic differential on the deformation space of a function as a function of the position of the points of the schlesinger transformation it yields a natural generalization of the sato formula for the baker akhiezer vector even in the absence of a tau function and it realizes the solution of the rhp as such ba vector some exemplifications in the setting of the painleve ii equation and finite toplitz hankel determinants are provided

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

moduli spaces of meromorphic connections and quiver varieties abstract we describe the moduli spaces of meromorphic connections on trivial holomorphic vector bundles over the riemann sphere with at most one unramified irregular singularity and arbitrary number of simple poles as nakajima s quiver varieties this result enables us to solve partially the additive irregular deligne simpson problem

quasi hamiltonian geometry of meromorphic connections for each connected complex reductive group g we find a family of new examples of complex quasi hamiltonian g spaces with g valued moment maps these spaces arise naturally as moduli spaces of suitably framed meromorphic connections on principal g bundles over a disk and they generalise the conjugacy class example of alekseev malkin and meinrenken 3 which appears in the simple pole case using the fusion product in the theory this gives a finite dimensional construction of the natural symplectic structures on the spaces of monodromy stokes data of meromorphic connections over arbitrary genus riemann surfaces together with a new proof of the symplectic nature of isomonodromic deformations of such connections

flat parabolic vector bundles on elliptic curves we describe the moduli space of logarithmic rank 2 connections on elliptic curves with 2 poles

geometry and braiding of stokes data fission and wild character varieties a family of new algebraic poisson varieties will be constructed generalising the complex character varieties of riemann surfaces then the well known poisson mapping class group actions on the character varieties will be generalised

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

the stokes groupoids we construct and describe a family of groupoids over complex curves which serve as the universal domains of definition for solutions to linear ordinary differential equations with singularities as a consequence we obtain a direct functorial method for resumming formal solutions to such equations

remarks on the confluent kz equation for mathfrak sl _2 and quantum painleve equations for the lie algebra we give an explicit form of an irregular singular version of the kz equation in the simplest non trivial cases these kz equations reproduce the quantum painlev equations possessing affine weyl group symmetry found previously by one of the authors