I currently welcome applications for a funded PhD on the projects related to the topics listed below.

**Details on scholarships and on how to apply can be found here:**

**PROJECTS**

**A. Rational solutions to Painlevé equations**

Investigation of rational solutions to Painlevé Equations using a variety of known and new analytical methods in combination with machine learning techniques.

Painlevé equations were said to be the most important nonlinear ordinary differential equations. The six Painlevé equations play a key role in a range of physical problems, in particular arising as universal models in reductions of the nonlinear Schrödinger equation, the Korteweg-de-Vries equation, the Boussinesq equation.

Their solutions appeared as special solutions of partial differential equations such as nonlinear wave equations as well as in a number of problems from neutron scattering theory, statistical mechanics, fibre optics, transportation problems, combinatorics, random matrices, quantum gravity, quantum field theory, number theory, combinatorics, electrochemistry, plasma physics, general relativity, among others.

For certain choices of the parameters, Painlevé equations have rational solutions. Several results are know and many questions are yet to unfold.

This project aims to explore further and deepen the knowledge on these rational solutions and beyond. The idea is to bring up a judicious use of numerical and analytical methods combined with machine learning techniques to investigate dynamical properties of the zeros of the polynomials involved on those rational solutions.

There is an extensive list of references on Painlevé equations. Here are a few (with further references therein):

[1] Robert J. Buckingham, Peter D. Miller (2022). On the algebraic solutions of the Painleve-III (D7) equation, arXiv:2202.04217.

[2] Peter A Clarkson (2019). Open Problems for Painlevé Equations, SIGMA 15 (2019), 006, 20 pages. https://www.emis.de/journals/SIGMA/2019/006/sigma19-006.pdf

[3] P. Clarkson, A.F. Loureiro and W. Van Assche (2016). Unique positive solution for an alternative discrete Painlevé I equation, Journal of Difference Equations and Applications. Taylor & Francis, pp. 656-675.

[4] Walter Van Assche (2017). Orthogonal Polynomials and Painlev\’e Equations (Australian Mathematical Society Lecture Series). Cambridge: Cambridge University Press.

[5] Walter Van Assche (2022). Orthogonal polynomials, Toda lattices and Painlevé equations, arXiv:2202.11017.

**B. Multiple orthogonal polynomials with applications to statistics and random matrix theory**

Multiple orthogonal polynomials are an extension to standard orthogonal polynomials. They consist of a sequence of polynomials of a single variable and on a multi-index, whose orthogonality measures are spread across a vector of measures. The number of measures and the dimension of the multi-index coincide. This multiple orthogonal systems offers new and efficient and more far reaching approaches to problems than standard orthogonality. The sequence satisfies finite dimensional recurrence relations, mimicking recurrence relations satisfied by standard orthogonal polynomials. Their applicability arises in many fronts such as probability, random matrix theory, statistical mechanics and in physics. The idea is to investigate a certain class that has strong links to Bayesian statistics and equally appearing in random matrix related problems. The study of such polynomials will have an immediate impact on statistical and stochastic methods, random matrix theory and potential applications to linear and nonlinear differential equations. The techniques involved will be prominently based on a combination of analytical and algebraic methods.

Some references (including the references therein) on multiple orthogonal polynomials:

[1] M.E.H. Ismail. Classical and Quantum Orthogonal Polynomials in One Variable. Cam- bridge University Press, 2005. (Chapter 23)

[2] H. Lima and A. F. Loureiro (2022), Multiple orthogonal polynomials with respect to Gauss’ hypergeometric function, Studies in Applied Mathematics. Wiley, pp. 154-185. doi: 10.1111/sapm.12437.

[3] Loureiro, A. F. and Van Assche, W. (2019) “Three-fold symmetric Hahn-classical multiple orthogonal polynomials”, Analysis and Applications. World Scientific Publishing, pp. 271-332. doi: 10.1142/S0219530519500106.