arXiv research

Symmetry reduction of Painlevé IV to Flaschka-Newell Painlevé II

New findings on Malgrange-Galois groupoid for Painlevé VI equation parameters.

Study of meromorphic connections and their spectral duals in $\mathfrak{gl}_3(\mathbb{C})$.

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.

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.

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.

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.

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 $(p,q,r,s)$ determined by the Newton diagrams of the equations or the versal deformations of vector fields. Singular normal forms of t…

Null Kähler metrics are characterized by Painlevé I or II ODEs.

Solve Painleve VI to relate instanton bundles.

New geometric insights reveal the persistence distribution in spin systems.

In this paper we \emph{explicitly} compute the transformation that maps the generic second order differential equation $y''= f(x, y, y')$ to the Painlevé first equation $y''=6y^2+x$ (resp. the Painlevé second equation ${y''=2 y^{3}+yx+ α}$). This change of coordinates, which is function of $f$ and its partial derivativ…

Proves resurgent nature of a series solution to deformed Painlevé I equation.

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.

Study of gauge theory blowups and Painlevé VI identity.

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 $\mathbb{R}^n$. For the steady case, dimensions of the form $n=k^2+1$ are singled out, with dimensions 2, 5, and 10 being particularly distinguished…

Characterizes Bonnet surfaces using analytic conditions.

Study of dynamics on cubic surfaces and their connection to Painlevé 6 Equation.

In this paper, we will give a complete geometric background for the geometry of Painlevé $VI$ 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 + ... + …

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.

The quadric ansatz solves dKP equations in arbitrary dimensions, leading to Einstein-Weyl structures.

In this paper we study $F$-manifolds equipped with multiple flat connections (and multiple $F$-products), that are required to be compatible in a suitable sense. In the semisimple case we show that a necessary condition for the existence of such multiple flat connections can be expressed in terms of the integrability o…

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…

Vortices on conical surfaces embedded in hyperbolic space.

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.

Paper connects Painlevé VI equation to irregular systems, solving monodromy data.

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…

BGM-IV uses AI to estimate causal effects in complex data.

The abstract discusses convergence properties of Lipschitz functions and sets defined by equations.

Paper discovers valid IVs from data without domain knowledge.

Study dynamics of automorphisms on cubic surfaces and their connection to Painlevé 6.

A new method learns IV representation from data to estimate causal effects.

DML-IV improves IV regression for learning decision policies by reducing bias.

Ivy combines weak IV candidates to estimate causal effects robustly.

Flow IV uses IVs to infer counterfactuals in complex models.

The study extends removability results for quasiregular curves in Euclidean spaces.

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.