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Pieter Roffelsen

Room: 
A-709
Phone Number: 
+39 040 3787 286

Personal Info

From: 
Netherlands
Email: 
proffels (at) sissa.it
Essential Bio: 

I am a postdoctoral research fellow in the geometry and mathematical physics group at SISSA. In 2017, I obtained my PhD in mathematics at the University of Sydney under the supervision of Nalini Joshi.

I have been working on the global analysis of linear and nonlinear difference equations as well as Painlevé equations and anharmonic oscillators:


  • As of late, I have been developing a Riemann-Hilbert approach to (nonlinear) q-difference equations. In my first paper in this line of research (arXiv 1911.05854), in collaboration with Nalini Joshi, we constructed a Riemann-Hilbert representation for the general solution of the q-Painlevé equation qPIV in the non-resonant parameter case. Furthermore we constructed an explicit algebraic surface which is the moduli space of qPIV.
    In my PhD thesis, I made effective a q-analog of the isomonodromic deformation method to solve the nonlinear connection problem of a q-Painlevé equation. The q-Painlevé equation in question can be considered an eight parameter generalisation of the celebrated sixth Painlevé equation for which I derived a (conjecturally) complete list of all asymptotic behaviours of its solutions near its critical points, the origin and infinity.
  • In collaboration with Davide Masoero, I have been studying singularity distributions of Painlevé functions through the development of a scheme which associates them in a one-to-one fashion with certain anharmonic oscillators of (confluent) Heun type. In our first paper (doi 10.3842/SIGMA.2010.095/arXiv 1707.05222) we showed that singularities of Hermite-type PIV rationals are described by means of an inverse monodromy problem concerning quantum anharmonic oscillators of degree two and classified them by means of the monodromy representation of a class of meromorphic functions introduced by Nevanlinna. Furthermore, we computed the asymptotic distribution of the singularities as one of the two parameters grows large: the roots condensate on certain explicit curves in the complex plane and for each curve the real parts of the roots are distributed in accordance with Wigner's semicircle law.
    In our second paper (arXiv 1907.08552) we succeeded in computing the asymptotic distribution of the singularities as both parameters grow large: after rescaling the roots condensate in a compact region in the complex plane, organising themselves along the vertices of a deformed quadrilateral lattice within this compact region. See also my talk in the mathematical physics seminar at the University of Lisbon in which I present several of the main results of our papers.

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