String and dilaton equations for counting lattice points in the moduli space of curves (2009)
We apply the techniques of Eynard and Orantin to the plane curve xy-y^2=1 to enumerate covers of the two-sphere branched over three points. This produces new recursion relations - string and dilaton...
Magnetic monopoles on manifolds with boundary (2008)
Kapustin and Witten associate a Hecke modification of a holomorphic bundle over a Riemann surface to a singular monopole on a Riemannian surface times an interval satisfying prescribed boundary...
Counting lattice points in the moduli space of curves (2008)
We show how to define and count lattice points in the moduli space $\modm_{g,n}$ of genus g curves with n labeled points. This produces a polynomial with coefficients that include the Euler...
REAL INSTANTONS, DIRAC OPERATORS AND QUATERNIONIC CLASSIFYING SPACES (2007)
Abstract. Let M(k;SO(n)) be the moduli space of based gauge equivalence classes of SO(n) instantons on principal SO(n) bundles over S 4 with first Pontryagin class p 1 = 2k. In this paper, we use a...
Walter D. Neumann, Paul Norbury
Abstract. We describe the trivial summand for monodromy around a fibre of a polynomial map C n! C generalising and clarifying work of Artal Bartolo, Cassou-Nogu`es and Dimca, who proved similar...
Walter D. Neumann, Paul Norbury
! C a "coordinate " if there is a g such that (f; g) : C
Periodic instantons and the loop group (2007)
We construct a large class of periodic instantons. Conjecturally we produce all periodic instantons. This confirms a conjecture of Garland and Murray that relates periodic instantons to orbits of the...
ASYMPTOTIC VALUES OF HYPERBOLIC MONOPOLES (2007)
Abstract. We show that many hyperbolic monopoles can be distinguished from each other via their asymptotic values in contrast to the case of Euclidean monopoles. 1.
POLYNOMIALS WITH RATIONAL FIBRES (2007)
Abstract. We classify rational polynomials in two variables of simple type whose generic fibres do not form an isotrivial family, thus completing the classification of all rational polynomials in two
Nontrivial Rational Polynomials In Two Variables Have Reducible Fibres (2007)
Walter D. Neumann, Paul Norbury
oof. This theorem is implicit in [7]. Suppose f is rational. As in [7], [6], [9], etc., we consider a nonsingular compactification Y = C 2 [ E of C 2 such that f extends to a holomorphic map f : Y !...
Unfolding Polynomial Maps At Infinity (2007)
Walter D. Neumann, Paul Norbury
This paper describes some progress. We will restrict mostly to the case that f has only isolated singularities. We show that the data then needed are local monodromy maps obtained by transporting a...
Spectral curves and the mass of hyperbolic monopoles (2007)
Norbury, Paul, Romao, Nuno Miguel
The original publication can be found at www.springerlink.com
Spectral curves and the mass of hyperbolic monopoles (2007)
Norbury, Paul, Romao, Nuno Miguel
The original publication can be found at www.springerlink.com
Lengths of geodesics on non-orientable hyperbolic surfaces (2006)
We give an identity involving sums of functions of lengths of simple closed geodesics, known as a McShane identity, on any non-orientable hyperbolic surface with boundary which generalises...
Stable reduction and topological invariants of complex polynomials (2006)
A topological invariant of a polynomial map $p:X\to B$ from a complex surface containing a curve $C\subset X$ to a one-dimensional base is given by a rational second homology class in the...
Weil-Petersson volumes and cone surfaces (2006)
The moduli spaces of hyperbolic surfaces of genus g with n geodesic boundary components are naturally symplectic manifolds. Mirzakhani proved that their volumes are polynomials in the lengths of the...
Spectral curves and the mass of hyperbolic monopoles (2005)
The moduli spaces of hyperbolic monopoles are naturally fibred by the monopole mass, and this leads to a nontrivial mass dependence of the holomorphic data (spectral curves, rational maps,...
Cohen, Ralph L., Norbury, Paul
In this paper we define and study the moduli space of metric-graph-flows in a manifold M. This is a space of smooth maps from a finite graph to M, which, when restricted to each edge, is a gradient...
Closed geodesics on incomplete surfaces (2003)
Norbury, Paul, Rubinstein, J. Hyam
We consider the problem of finding embedded closed geodesics on the two-sphere with an incomplete metric defined outside a point. Various techniques including curve shortening methods are used.
Minimal Spheres of Arbitrarily High Morse Index (2003)
Hass, Joel, Norbury , Paul, Rubinstein , J. Hyam
We construct a smooth Riemannian metric on any 3-manifold with the property that there are genus zero embedded minimal surfaces of arbitrarily high Morse index.
Minimal spheres of arbitrarily high Morse index (2002)
Hass, Joel, Norbury, Paul, Rubinstein, J. Hyam
We construct a smooth Riemannian metric on any 3-manifold with the property that there are genus zero embedded minimal surfaces of arbitrarily high Morse index.
Hyperbolic monopoles and holomorphic spheres (2001)
Murray, Michael K., Norbury, Paul, Singer, Michael A.
We associate to an SU(2) hyperbolic monopole a holomorphic sphere embedded in projective space and use this to uncover various features of the monopole.
The Orevkov invariant of an affine plane curve (2001)
Neumann, Walter D., Norbury, Paul
We show that although the fundamental group of the complement of an algebraic affine plane curve is not easy to compute, it possesses a more accessible quotient, which we call the Orevkov invariant.
A proof of Atiyah's conjecture on configurations, of four points in Euclidean three-space (2001)
Eastwood, Michael, Norbury, Paul
From any configuration of finitely many points in Euclidean three-space, Atiyah constructed a determinant and conjectured that it was always non-zero. Atiyah and Sutcliffe (hep-th/0105179) amass a...
Boundary algebras of hyperbolic monopoles (2001)
We prove the conjecture that a monopole in three-dimensional anti-de Sitter space can be completely determined by its ``holographic'' image on the conformal boundary two-sphere.
Rational polynomials of simple type (2000)
Neumann, Walter D., Norbury, Paul
We classify two-variable polynomials which are rational of simple type. These are precisely the two-variable polynomials with trivial homological monodromy.
Asymptotic values of hyperbolic monopoles (1999)
We show that many hyperbolic monopoles can be distinguished from each other via their asymptotic values in contrast to the case of Euclidean monopoles.
Unfolding polynomial maps at infinity (1999)
Neumann, Walter D., Norbury, Paul
We give a topological model for a polynomial map from $\C^n$ to $\C$ in the neighborhood of a fiber with isolated singularities. This is motivated out of the ``unfolding of links'' described earlier...
Periodic instantons and the loop group (1999)
We construct a large class of periodic instantons. Conjecturally we produce all periodic instantons. This confirms a conjecture of Garland and Murray that relates periodic instantons to orbits of the...
Nontrivial rational polynomials in two variables have reducible fibres (1998)
Neumann, Walter D., Norbury, Paul
It has been proved several times in the literature that a polynomial map from $C^2$ to $C$ with irreducible rational fibers cannot be a component of a counterexample to the Jacobian Conjecture. This...
Compactification of hyperbolic monopoles (1997)
We prove that the space of SU(2) hyperbolic monopoles based at the centre of hyperbolic space is homeomorphic to the space of (unbased) rational maps of the two-sphere. The homeomorphism extends to a...