Timetable & Abstracts

13:00 - 13:10:  Welcome

13:10 - 14:00:  Mikhail Bershtein (Edinburgh) 

Title: Quantum spectral problems and isomonodromic deformations 

Abstract: We study the spectral properties of a class of quantum mechanical operators by using the knowledge about monodromies of 2 × 2 linear systems (Riemann-Hilbert correspondence). The main examples for the talk will be Mathieu operator and Lamé operator. The spectrum of such operators is given in terms of isomonodromic tau functions. By using the Kyiv formula for them, we obtain the spectrum in terms of self-dual Nekrasov functions. Through blowup relations, we also find Nekrasov-Shatashvili type of quantizations.

Based on joint work with Alba Grassi and Pavlo Gavrylenko.

14:00 - 14:50:  Clare Dunning (Kent) 

Title: Some rational solutions of the fifth Painlevé equation and related combinatorics.

Abstract: We consider a family of rational solutions of the fifth Painlevé equation that are expressed in terms of Wronskians of associated Laguerre polynomials.  We discuss various properties, and non-uniqueness of the solutions in special cases. We also show that several aspects and properties of the Wronksian Laguerre polynomials are related to properties of the partitions that label the associated Laguerre polynomials.  This is joint work with Peter Clarkson (Kent) based on DOI: 10.1111/sapm.12649. 

14:50 - 15:20:  Coffee Break  

15:20 - 16:10:  Thomas Bothner (Bristol)

Title: From Ginibre to Painlevé

Abstract: Random matrix eigenvalue spacings tend to show up in problems not directly related to random matrices: for instance, bumper to bumper distances of parked cars in a number of roads in central London are well represented by the eigenvalue bulk spacing distribution of a suitable Hermitian matrix model. In this talk we will first survey several occurrences of these Hermitian spacing distributions and afterwards try to generalise them to non-Hermitian models. As it turns out, the theory of integrable systems, especially Painlevé special function theory, plays a crucial role in this field. Based on arXiv:2212.00525, joint work with Alex Little (Lyon).

16:10 - 17:00:  Thomas Kecker (Portsmouth)

Title: Geometric approach for quasi-Painlevé Hamiltonian systems

Abstract: We present some new Hamiltonian systems of quasi-Painlevé type and their Okamoto's spaces of initial conditions. The geometric approach was introduced originally for the identification problem of Painlevé equations, comparing the irreducible components of the inaccessible divisors introduced in the blow-ups to obtain the space of initial conditions. Using this method, we find bi-rational coordinate changes between some of the systems we introduce, giving rise to a global symplectic structure for these systems. This scheme thus allows us to identify (quasi-)Painlevé Hamiltonian systems up to bi-rational symplectic maps, performed here for systems with solutions having movable singularities that are either square-root type algebraic poles or ordinary poles.