We solve the metrisability problem for the six Painlevé equations, and more generally for all 2nd order ODEs with Painlevé property, and determine for which of these equations their integral curves are geodesics of a (pseudo) Riemannian metric on a surface.
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The first, second and fourth Painlevé equations are studied by means of dynamical systems theory and three dimensional weighted projective spaces $\C P^3(p,q,r,s)$ with suitable weights determined by the Newton diagrams of the equations or the versal deformations of vector fields. Singular normal forms of t…
Characterizes Bonnet surfaces using analytic conditions.
problem Characterizing Bonnet surfaces by geometric conditions.
method Derives Bonnet surfaces using two analytic conditions: mean curvature reduction to an ODE and the Painlevé property.
result Bonnet surfaces characterized by analytic conditions.
In this paper, we will give a complete geometric background for the geometry of Painlevé and Garnier equations. By geometric invariant theory, we will construct a smooth coarse moduli space $M_n^{\balpha}(\bt, \blambda, L) $ of stable parabolic connection on $\BP^1$ with logarithmic poles at $D(\bt) = t_1 + ... + …
A symmetry reduction of the Painlevé IV hierarchy to the Flaschka-Newell Painlevé II hierarchymath-ph
Symmetry reduction of Painlevé IV to Flaschka-Newell Painlevé II
problem Isomonodromic deformation problem associated with rank-two meromorphic connections
method Symmetry
result Induced isomonodromic dynamics coincides with Flaschka-Newell Painlevé II hierarchy
New findings on Malgrange-Galois groupoid for Painlevé VI equation parameters.
problem Understanding transformations preserving specific forms for Painlevé VI equation.
method Computed Malgrange-Galois groupoid for Painlevé VI family with all parameters.
result Solutions of Painlevé VI do not satisfy new partial differential equations.
We classify the SU(2)-invariant anti-self-dual metrics with a signature (+,+,-,-). The metrics are specified by a solution of Painleve VI, V, III or II. Moreover we show the geometric meaning of the metrics specified by each type of Painlevé functions.
Inverse Problem for semisimple Frobenius Manifolds, Monodromy Data and the Painleve' VI Equationmath.CV
We study critical behaviour and connection problem for a Painleve' 6 equation. We construct solutions of WDVV eqs. using the isomonodromic deformation method and the Painleve' equations. We find algebraic solutions of WDVV and Gromov-Witten invariants of projective space.
Study Galois groupoids of discret Painlevé equations.
problem Computing Galois groupoids for discret Painlevé equations.
method Using semi-continuity theorem for Galois groupoid in confluence of difference to differential equations.
result Computed Galois groupoids for discret Painlevé equations.
Nous montrons que les équations du repère mobile des surfaces de Bonnet conduisent à une paire de Lax matricielle isomonodromique d'ordre deux pour la sixième équation de Painlevé. We show that the moving frame equations of Bonnet surfaces can be extrapolated to a second order, isomonodromic matrix Lax pair of the sixt…
We prove generic fibre of Painlevé moduli spaces are Weinstein handlebodies.
problem Understanding the generic fibre of Painlevé moduli spaces.
method Attach Weinstein handles to Stokes Legendrian.
result Generic fibre of Painlevé moduli spaces are Weinstein handlebodies.
Null Kähler metrics are characterized by Painlevé I or II ODEs.
problem Characterizing null-Kähler metrics in four dimensions.
method Cohomogeneity-one anti-self-dual null-Kähler metrics, twistor methods, Painlevé I and II ODEs.
result Cohomogeneity-one anti-self-dual null-Kähler metrics are generically characterized by solutions to Painlevé I or Painlevé II ODEs.
Solve Painleve VI to relate instanton bundles.
problem Relate instanton bundles to Painleve VI solutions.
method Generalize Hitchin's logarithmic connection to vector bundles with SL2 action.
result Identify Okamoto transformations as creation operators.
Geometry of the Ising persistence problem and the universal Bonnet-Manin Painlevé VI distributionmath-ph
New geometric insights reveal the persistence distribution in spin systems.
problem Determining the full persistence probability distribution in non-Markovian stochastic processes.
method Exact Fredholm Pfaffian structure and Painlevé VI system analysis.
result Recovery of the universal persistence exponent and its geometric interpretation.
Paper connects Painlevé VI equation to irregular systems, solving monodromy data.
problem Solving monodromy data for irregular systems related to Painlevé VI.
method Expressed Frobenius integrability in terms of PVI, computed monodromy data for coalescing eigenvalues.
result Computed monodromy data for transcendentals holomorphic at critical points of PVI.
In this paper we \emph{explicitly} compute the transformation that maps the generic second order differential equation to the Painlevé first equation (resp. the Painlevé second equation ). This change of coordinates, which is function of and its partial derivativ…
Proves resurgent nature of a series solution to deformed Painlevé I equation.
problem Analyzing the resurgent nature of a series solution to the deformed Painlevé I equation.
method Proves resurgent nature through formal -power series solution and Borel summability.
result Borel transform defines a global multivalued holomorphic function on a Fermat quintic surface.
Painleve Field Theorymath-ph
We propose multidimensional versions of the Painlevé VI equation and its degenerations. These field theories are related to the isomonodromy problems of flat holomorphic infinite rank bundles over elliptic curves and take the form of non-autonomous Hamiltonian equations. The modular parameter of curves plays the role o…
Explicit solutions to the Riemann-Hilbert problem will be found realising some irreducible non-rigid local systems. The relation to isomonodromy and the sixth Painleve equation will be described. Keywords: Riemann-Hilbert problem, Painleve equations, algebraic solutions, Heun equations, tetrahedral/octahedral group, tr…
We investigate the duality between local (complex analytic) projective structures on surfaces and two dimensional (complex analytic) neighborhoods of rational curves having self-intersection +1. We study the analytic classification, existence of normal forms, pencil/fibration decomposition, infinitesimal symmetries. We…
Study Galois groupoids of vector fields, proving lower semicontinuity.
problem Computing Galois groupoids for general parameter values of Painlevé equations.
method Prove lower semicontinuity of Galois groupoids of vector fields.
result Results can compute Galois groupoids for general parameter values of Painlevé equations.
Study of gauge theory blowups and Painlevé VI identity.
problem Understanding gauge theory blowups and their relation to Painlevé VI.
method Analyzing four-dimensional supersymmetric gauge theory with defects and proposing identities.
result Formula relating tau-function to conformal blocks of Liouville theory.
The abstract discusses convergence properties of Lipschitz functions and sets defined by equations.
problem Convergence of Lipschitz functions and sets defined by equations.
method Painlevé-Kuratowski convergence applied to Lipschitz functions and sets defined by equations.
result Generalizations and reverses of classical theorems on convergence of functions and sets.
This is an review on the point classification of second order ODE's by Ruslan Sharipov. His works were published in 1997-1998 at the Electronic Archive at LANL and undeservedly forgotten. Last chapter is an application of this classification to the investigation of Painleve equations.
We carry out a Painlevé analysis of the systems of differential equations corresponding to the steady and the expanding, rotationally symmetric, gradient Ricci solitons on . For the steady case, dimensions of the form are singled out, with dimensions 2, 5, and 10 being particularly distinguished…
Study of dynamics on cubic surfaces and their connection to Painlevé 6 Equation.
problem Understanding the dynamics of automorphism groups on cubic surfaces.
method Analyzing holomorphic automorphisms and character varieties.
result Several open questions about the dynamics of automorphism groups.
It was observed by Tod and later by Dunajski and Tod that the Boyer-Finley (BF) and the dispersionless Kadomtsev-Petviashvili (dKP) equations possess solutions whose level surfaces are central quadrics in the space of independent variables (the so-called central quadric ansatz). It was demonstrated that generic solutio…
Generalizes isomonodromic-isospectral correspondence for twisted connections.
problem Extending isomonodromic-isospectral correspondence to twisted cases.
method Construction of isospectral approach for Painlevé I hierarchy, two maps linking isomonodromic and isospectral Hamiltonians, and apparent singularities to isospectral coordinates.
result Established a correspondence between isomonodromic and isospectral systems for twisted connections.
The quadric ansatz for the -dispersionless KP equation, and supersymmetric Einstein-Weyl spacesnlin.SI
The quadric ansatz solves dKP equations in arbitrary dimensions, leading to Einstein-Weyl structures.
problem Characterizing solutions of the dispersionless KP equation in arbitrary dimensions.
method Quadric ansatz for the dKP equation, constructing Einstein-Weyl spaces.
result Explicit new family of Einstein-Weyl spaces constructed and characterized.
We give a gauge invariant characterisation of the elliptic affine sphere equation and the closely related Tzitzéica equation as reductions of real forms of $SL(3, \C)$ anti--self--dual Yang--Mills equations by two translations, or equivalently as a special case of the Hitchin equation. We use the Loftin--Yau--Zaslow co…
Elizabethan vorticeshep-th
Vortices on conical surfaces embedded in hyperbolic space.
problem Embedding Abelian vortices on conical surfaces in hyperbolic space.
method Analyzing elliptic sinh-Gordon and Tzitzeica equations, asymptotic analysis of Painleve III ODE.
result Radial solutions can be globally embedded in hyperbolic space.
Einstein--Weyl spaces and dispersionless Kadomtsev--Petviashvili equation from Painlevé I and IInlin.SI
We present two constructions of new solutions to the dispersionless KP (dKP) equation arising from the first two Painlevé transcendents. The first construction is a hodograph transformation based on Einstein--Weyl geometry, the generalised Nahm's equation and the isomonodromy problem. The second construction, motivated…
Study symplectic structures in moduli spaces of meromorphic connections.
problem Understanding symplectic structures in moduli spaces of meromorphic connections.
method Explicit and infinite dimensional description of symplectic structures.
result Intrinsic symplectic description of isomonodromic deformation equations.
We carry out a Painlevé analysis to find the cases where the cohomogeneity one steady Ricci soliton equation can be integrable. We concentrate on two classes of solitons: warped products and complex line bundles over a Fano Kähler Einstein base. For warped products, the analysis singles out the case with one factor whe…
Summary of main work 1999-2012
We provide a Lax pair for the surfaces of Voss and Guichard, and we show that such particular surfaces considered by Gambier are characterized by a third Painlevé function.
The first half of the thesis concerns Abelian vortices and Yang-Mills (YM) theory. It is proved that the 5 types of vortices recently proposed by Manton are symmetry reductions of (A)SDYM equations with suitable gauge groups and symmetry groups acting as isometries in a 4-manifold. As a consequence, the twistor integra…
Study of meromorphic connections and their spectral duals in .
problem Exploring -deformed meromorphic connections and their spectral duals.
method Using apparent singularities and their dual partners as Darboux coordinates, the Hamiltonian evolutions and reductions are derived.
result Spectral duality extends to Hamiltonian evolutions, tau-functions, and Hermitian matrix models on both sides.
Study dynamics of automorphisms on cubic surfaces and their connection to Painlevé 6.
problem Dynamics of holomorphic automorphisms on cubic surfaces and their relation to Painlevé 6.
method Defined Julia and Fatou sets, studied locally discrete and non-discrete dynamics, and proved existence of non-empty Fatou and Julia sets.
result Existence of non-empty Fatou and Julia sets for the group action.
We study a family of fermionic extensions of the Camassa-Holm equation. Within this family we identify three interesting classes: (a) equations, which are inherently hamiltonian, describing geodesic flow with respect to an H^1 metric on the group of superconformal transformations in two dimensions, (b) equations which …
The tt* structure of the quantum cohomology of CP^1 from the viewpoint of differential geometrymath.DG
The quantum cohomology of CP^1 provides a distinguished solution of the third Painleve equation. S. Cecotti and C. Vafa discovered this from a physical viewpoint. We derive it from a differential geometric viewpoint, using the theory of harmonic maps and in particular the generalized Weierstrass representation (DPW rep…
The study extends removability results for quasiregular curves in Euclidean spaces.
problem Removability of singularities in quasiregular curves.
method Extending a fundamental inequality for volume forms to calibrations and proving a Caccioppoli inequality for quasiregular curves.
result Every non-constant quasiregular curve has infinite energy.
We study the anti-self-dual equation for non-diagonal SU(2)-invariant metrics and give an equivalent ninth-order system. This system reduce to a sixth-order system if the metric is in the conformal class of scalar-flat-Kaehler metric.
This is an extensive (published) survey on CR geometry, whose major themes are: formal analytic reflection principle; generic properties of Systems of (CR) vector fields; pairs of foliations and conjugate reflection identities; Sussmann's orbit theorem; local and global aspects of holomorphic extension of CR functions;…
We prove that conformally parametrized surfaces in Euclidean space $\Rcubec$ of curvature admit a symmetry reduction of their Gauss-Codazzi equations whose general solution is expressed with the sixth Painlevé function. Moreover, it is shown that the two known solutions of this type (Bonnet 1867, Bobenko, Eitner an…
We determine the set of Busemann points of an arbitrary finite-dimensional normed space. These are the points of the horofunction boundary that are the limits of "almost-geodesics". We prove that all points in the horofunction boundary are Busemann points if and only if the set of extreme sets of the dual unit ball is …
We show that the local equivalence problem for second-order ordinary differential equations under point transformations is completely characterized by differential invariants of order at most 10 and that this upper bound is sharp. We also show that, modulo Cartan duality and point transformations, the Painlevé-I equati…
Geometric formalism views optimization algorithms as discrete connections, revealing their algebraic curvature and flatness properties.
problem Understanding and optimizing the behavior of iterative optimization algorithms.
method Introducing a geometric and operator-theoretic formalism where optimization algorithms are encoded by coupled channels (drift and diffusion) whose algebraic curvature measures the deviation from ideal reversibility.
result Flat connections correspond to methods whose updates commute up to higher order, achieving minimal numerical dissipation and preserving stability.