Related papers: The Indefinite Self-Dual Metrics and Painleve Equa…

Continuously symmetric solutions of the Adler-Bobenko-Suris class of discrete integrable equations are presented. Initially defined by their invariance under the action of both of the extended three point generalized symmetries admitted by…

We study polynomials that are orthogonal with respect to a varying quartic weight \exp(-N(x^2/2+tx^4/4)) for t<0, where the orthogonality takes place on certain contours in the complex plane. Inspired by developments in 2D quantum gravity,…

Using the twistor correspondence, we give a classification of toric anti-self-dual Einstein metrics: each such metric is essentially determined by an odd holomorphic function. This explains how the Einstein metrics fit into the…

We classify all left-invariant pseudo-Riemannian Einstein metrics on $\mathrm{SL}(2,\mathbb{R})\times \mathrm{SL}(2,\mathbb{R})$ that are bi-invariant under a one-parameter subgroup. We find that there are precisely two such metrics up to…

A class of self-dual and geodesically complete spacetimes with maximally superintegrable geodesic flows is constructed by applying the Eisenhart lift to mechanics in pseudo-Euclidean spacetime of signature (1,1). It is characterized by the…

The Painlev\'e--Kovalevskaya test is applied to find three matrix versions of the Painlev\'e II equation. All these equations are interpreted as group-invariant reductions of integrable matrix evolution equations, which makes it possible to…

We show that Pleba\'nski's equation for self-dual metrics is equivalent to a pair of equations describing canonical transformations in 2-dimensional phase spaces. Examples of linearizations of these equations are given.

We study the Witten--Reshetikhin--Turaev SU(2) invariant for the Seifert manifold $S^3/\Gamma$ where $\Gamma$ is a finite subgroup of SU(2). We show that the WRT invariants can be written in terms of the Eichler integral of the modular…

Unstable separatrix solutions for the first and second Painlev\'e transcendents are studied both numerically and analytically. For a fixed initial condition, say $y(0)=0$, there is a discrete set of initial slopes $y'(0)=b_n$ that give rise…

The set E of Levi-Civita connections of left-invariant pseudo-Riemannian Einstein metrics on a given semisimple Lie group always includes D, the Levi-Civita connection of the Killing form. For the groups SU(l,j) (or SL(n,R), or SL(n,C) or,…

We construct a 2-parameter family of new triaxial $SU(2)$-invariant complete negative Einstein metrics on the complex line bundle $\mathcal{O}(-4)$ over $\mathbb{C}P^1$. The metrics are conformally compact and neither K\"ahler nor…

Let \(\mu\) be a finite Borel measure on \((-\pi,\pi)\). Consider the one-dimensional Poisson equation \(-u''=\mu\), where equality holds in the sense of distributions, with Dirichlet boundary conditions \(u(\pm\pi)=0\). In this paper, we…

Let $\mu$ and $\nu$ be two probability measures on $\R^d$, where $\mu(\d x)= \e^{-V(x)}\d x$ for some $V\in C^1(\R^d)$. Explicit sufficient conditions on $V$ and $\nu$ are presented such that $\mu*\nu$ satisfies the log-Sobolev, Poincar\'e…

We find a four-parameter family of coupled Painlev\'e VI systems in dimension four with affine Weyl group symmetry of type $A_7^{(2)}$. This is the first example which gave higher-order Painlev\'e equations of type $A_{2l+5}^{(2)}$. We then…

Starting from the standard form of the five discrete Painlev\'e equations we show how one can obtain (through appropriate limits) a host of new equations which are also the discrete analogues of the continuous Painlev\'e equations. A…

We consider some measure-theoretic properties of functions belonging to a Sobolev-type class on metric measure spaces that admit a Poincar\'e inequality and are equipped with a doubling measure. The properties we have selected to study are…

Painleve's transcendental differential equation P_{VI} may be expressed as the consistency condition for a pair of linear differential equations with 2 by 2 matrix coefficients with rational entries. By a construction due to Tracy and…

We discuss symmetries of Hamiltonians of I$_{38}$ and I$_{49}$ equations that appear on Ince's list of fifty second-order differential equations with Painlev\'e property. This study is informed by structure of Weyl symmetries of Painlev\'e…

We prove that given two metrics $g_{+}$ and $g_{-}$ with curvature $\kappa <-1$ on a closed, oriented surface $S$ of genus $\tau\geq 2$, there exists an $AdS$ manifold $N$ with smooth, space-like, strictly convex boundary such that the…

The primary aim of this thesis is to investigate metrics which are induced by a differential form and arise as a critical point of Hitchin's variational principle. Firstly, we investigate metrics associated with the structure group PSU(3)…