My research program is concerned with analytic and probabilistic questions in mathematical physics and I place particular emphasis on topics in random matrix theory which display intimate connections to mathematical statistical mechanics and the field of integrable differential equations. The application of asymptotics methods, special functions, probability theory, orthogonal polynomials and potential theory is central to this work. Current areas of interest include (see below for PhD projects in those areas):

1) Random matrix theory and the theory of random processes: In a nutshell, my work in this field is concerned with the

• analysis of gap, distribution and correlation functions in invariant random matrix models and thinned versions thereof
• description of extreme values in non-Hermitian random matrix models
• identification of universality classes in Hermitian one- or multi-matrix models
• spectral analysis of integrable integral operators
• development of Hamiltonian approaches to the analysis of gap asymptotics

2) Exactly solvable lattice models in statistical mechanics: I have derived results for

• the six-vertex model with domain wall boundary conditions: computation of the free energy and subleading terms for the partition function and analysis of phase transitions
• the 2D Ising model: elementary derivation of the scaling function constant in the short distance expansion of the tau-function associated with 2-point functions

3) Integrable differential equations: Most of my work in this field is concerned with Painleve special functions, focusing on the

• unified asymptotic description of certain real-valued Painleve transcendents
• introduction of Schur/orthogonal polynomial methods to the analysis of rational Painleve functions
• development of nonlinear steepest descent techniques for singular Painleve transcendents
• total Painleve integral evaluations

Recent preprints as well as published work can be found on arXiv and MathSciNet as well as ORCID. Feel free to contact me if you are interested in one of the PhD projects below, I am happy to discuss specifics, prerequisites and learning outcomes.

Current and future work My current efforts lie at the forefront of research in mathematical statistical mechanics of highly correlated systems with focus on two major themes: exactly solvable lattice models and random matrices. The long-term goal is to unveil ground-breaking original connections between those themes and resolve a series of long-standing conjectures about the system’s underlying analytic and asymptotic behaviors. Here are two concrete PhD projects in this area

A) From non-Hermitan to Hermitian random matrices 

B) Topological expansions for (near) random matrix models