# University of Leeds Workshop on Algebro-Geometric Aspects of Integrable Systems, 13th/14th May, 2011

## Available Abstracts

### Abstract

I will report on results obtained with Hans Duistermaat for the completeness and connectedness of asymptotic behaviours of solutions of the first Painlev\'e equation ${d}^2y/{d}\! x^2=6\, y^2+ x$, in the limit $x\to\infty$, $x\in\C$. We prove that the complex limit set of solutions is non-empty, compact and invariant under the flow of the limiting autonomous Hamiltonian system, that the infinity set of the vector field is a repellor for the dynamics and obtain new proofs for solutions near the equilibrium points of the autonomous flow. The results rely on a realization of Okamoto's space, i.e., the space of initial values compactified and regularized by embedding in $\C\Proj 2$ through an explicit construction of nine blow-ups.

### Abstract

Michael Somos noticed that certain quadratic (or bilinear) recurrence relations surprisingly yielded sequences of integers. An explanation for this was provided by the further observation that these recurrences generated Laurent polynomials in the initial data. This was one of the first examples of the Laurent phenomenon, which more recently emerged as a general feature of Fomin and Zelevinsky's cluster algebras.

Somos sequences also arise in other contexts: elliptic divisibility sequences in number theory, solvable models of statistical mechanics, and discrete integrable systems (QRT maps and reductions of discrete Hirota equations), to name but a few. In this talk we consider the family of sixth-order Somos recurrences (Somos 6), and explain how it admits a general analytic solution in terms of genus two Kleinian sigma functions. The Lax pair for Somos 6, and associated Poisson structures, will also be briefly described, together with various open problems.

### Abstract

We consider various 2D lattice equations, and their integrability from the point of view of 3D consistency, Lax pairs and B\"acklund transformations. We bring to the light integrable lattice equations over black and white lattices, as well as variants of the functional Yang-Baxter equation.

### Abstract

The modern approach to integrability proceeds via a Riemann surface, the spectral curve, and solutions of the integrable system may be built from theta (and allied) functions on the curve. To implement this algebro-geometric construction requires a good understanding of the curve, for example its homology and period matrix. Computational algebraic geometry facilitates this. In some cases, including the construction of magnetic monopoles, physical symmetries are inherited by the spectral curve and there may be consequent simplifications of both the function theory and the data needed to construct solutions. We shall look at this interplay of ideas and report on several new tools and results.

### Abstract

We study equivariant properties of the gap sequences on plane algebraic curves and some implications for the structure of integrable systems of PDE associated with such curves, in particular identities for P-functions on Jacobians of hyperelliptic curves.

### Abstract

I'll survey some results (both old and new) related to the geometry of moduli spaces of irregular connections on curves. If time permits this will include: 1) new nonlinear geometric braid group actions, 2) new complete hyperkahler manifolds (including some gravitational instantons) [in work with O. Biquard], and 3) new ways to glue Riemann surfaces together to obtain (symplectic) generalisations of the complex character varieties of surface groups.

TBA