Past Geometry and Integrability

18 June 2015
17:00
Prof Nalini Joshi
Abstract

Abstract: The simplest solutions of integrable systems are special functions that have been known since the time of Newton, Gauss and Euler. These functions satisfy not only differential equations as functions of their independent variable but also difference equations as functions of their parameter(s).  We show how the inverse scattering transform method, which was invented to solve the Korteweg-de Vries equation, can be extended to its discrete version.

S.Butler and N.Joshi, An inverse scattering transform for the lattice potential KdV equation, Inverse Problems 26 (2010) 115012 (28pp)

  • Geometry and Integrability
12 June 2012
10:30
Tim Adamo
Abstract
Abstract: We'll try to learn something about Nekrasov's conjecture/theorem, which relates an instanton-counting partition function to the Seiberg-Witten prepotential of N=2 SYM theory on R^4. This will entail a review of some salient aspects of N=2 SYM theories, Witten's description of Donaldson invariants in terms of correlation functions in those theories, and the physical and mathematical definition of Nekrasov's partition function. Depending on time, I might talk about computational techniques for the partition function, methods of proof for Nekrasov's conjecture, or the partition function's role in the AGT conjectures.
  • Geometry and Integrability
13 September 2011
12:00
Allessandro Torielli
Abstract

We review the representation theory of the integrable model underlying the AdS_5/CFT_4 correspondence. We will discuss short and long multiplets, and their impact on the issue of the universal R-matrix. We will give special emphasis to the role of the so-called 'secret symmetry', which completes the Yangian symmetry of the system to a yet to be understood new type of quantum group.

  • Geometry and Integrability
2 May 2008
12:00
Dr Galina Filipuk
Abstract
Any nonlinear equation of the form y''=\sum_{n=0}^N a_n(z)y^n has a (generally branched) solution with leading order behaviour proportional to (z-z_0)^{-2/(N-1)} about a point z_0, where the coefficients a_n are analytic at z_0 and a_N(z_0)\ne 0. Jointly with R.G. Halburd we consider the subclass of equations for which each possible leading order term of this form corresponds to a one-parameter family of solutions represented near z_0 by a Laurent series in fractional powers of z-z_0. For this class of equations we show that the only movable singularities that can be reached by analytic continuation along finite-length curves are of the algebraic type just described. This work generalizes previous results of S. Shimomura. The only other possible kind of movable singularity that might occur is an accumulation point of algebraic singularities that can be reached by analytic continuation along infinitely long paths ending at a finite point in the complex plane. This behaviour cannot occur for constant coefficient equations in the class considered. However, an example of R. A. Smith shows that such singularities do occur in solutions of a simple autonomous second-order differential equation outside the class we consider here.
  • Geometry and Integrability
25 April 2008
12:00
Dr Maciej Dunajski
Abstract
Cover a plane with curves, one curve through each point in each direction. How can you tell whether these curves are the geodesics of some metric? This problem gives rise to a certain closed system of partial differential equations and hence to obstructions to finding such a metric. It has been an open problem for at least 80 years. Surprisingly it is harder in two dimensions than in higher dimensions. I shall present a solution obtained jointly with Robert Bryant and Mike Eastwood.
  • Geometry and Integrability