Possible Research Projects (February 2011):
A lattice ribbon model of supercoiling in DNA: A simple lattice ribbon model will be adapted to model twisting of super-coiled DNA. Within the lattice model one can successively simulate ever more complex situations, carefully studying the interplay of generic parameters such as stiffness and intrinsic twist of the DNA, and its response to external forcing and twisting.Simulation of the HP-model of protein folding: The HP-model, which has rightfully been called the "Ising model of protein folding," serves as a benchmark model for testing the performance of algorithms attempting to compute the density of states of proteins close to their folded ground state. This model provides an ideal testing ground for the comparison of new Monte-Carlo algorithms with established ones. The melting transition of a polymeric crystal: There has been a long-lasting controversy about the order of the melting transition of polymeric crystals. In two dimensions, there are conflicting statements in the literature, and details may very well depend on the precise lattice model studied. There is therefore continuing need for high-quality simulations in this area. Exact solution of two partially directed walk models in a slit: Lattice models of one or many partially directed self-avoiding walks (PDSAW) in a slit interacting with the walls will be studied by deriving exact solutions to their combinatorial generating function using the so-called kernel method for solving combinatorial functional equations. This can be done by studying more and more elaborate problems, and comparing and contrasting work on Dyck and Motzkin paths with work on PDSAW. Directed vesicle models in restricted geometries: Inclusion of area weights into directed walk models generally leads to q-deformed special functions, and it is planned to study the q-deformation of kernel method for the simplest cases for which the solution is well-understood, but that have not yet been studied with the proposed new approach. Starting with the model of Dyck paths enumerated with respect to length and enclosed area, we will gradually develop expertise for tackling more complicated models. Scaling functions for vesicle adsorption models: As yet unpublished case studies of concrete vesicle models, along with sketches of the asymptotics, can be given directly to a PhD student to work out details and learn the techniques needed for uniform asymptotic expansions near coalescing saddle points. This project is fairly straight-forward conceptually, but technically quite demanding.
Scientific Leadership Profile (February 2011):
My main research is concerned with problems from enumerative combinatorics, usually in the guise of lattice models in statistical mechanics. Here, my activities have ranged from rigorous combinatorial and asymptotic work to the development of Monte Carlo algorithms.A brief selection from my past research highlights is given by (a) a seminal paper on intermittent maps, (b) a uniform sampling algorithm based on stochastic growth, (c) the Takeuchi-Prellberg constant occurring in iterative functional equations, (d) work in uniform asymptotic analysis characterised by experts as a tour-de-force calculation, and (e) a record numerical estimate of a self-avoiding walk exponent. I am strongly embedded within a wide international research network. Examples of current active research collaborations include work on numbertheoretic spin chains with Peter Kleban from the University of Maine (where I spent a sabbatical term in 2008 as Visiting Professor of Diversity), an analysis of functional equations arising in the enumeration of lattice walks with Andrew Rechnitzer from the University of British Columbia and Buks van Rensburg from York University, and asymptotic analysis of physically motivated lattice models of polymers with Aleks Owczarek from the University of Melbourne. Other recent work includes enumerative combinatorics of quiver structures in cluster algebras with Martin Rubey from the University of Hannover. Having a background in both physics and mathematics allows me to draw from a larger range of collaborators, both internally and externally. In addition to having done work with several eminent combinatorialists, I have also successfully collaborated with experimental physical chemists and condensed matter physicists. Furthermore, I am equally at home at statistical physics workshops and combinatorics conferences: as early as 2002 I presented a plenary talk at FPSAC, a conference on formal power series and algebraic combinatorics. Indeed, I have frequently been invited to give talks at international conferences and summer schools in both mathematics and physics. In 2008 I gave a lecture series on techniques in enumerative combinatorics at a summer school at the Schr\"odinger Institute in Austria, and I am invited to give a lecture series on stochastic enumeration algorithms at a 2011 summer school on the simulation of extreme events. My international interests are not limited to conferences. I have received very positive referee feedback (a rare occurrence in Germany) on my final report for my last large grant from the German Science Foundation (DFG) on the development of stochastic enumeration algorithms. Part of this grant enabled me to employ a post-doctoral researcher which culminated in very successful results. As special recognition I have also been bestowed the title of Professor (Ausserplanmaessiger Professor) by Clausthal University of Technology. A large part of my research over the past six years has been supported by the ARC Centre of Excellence on the Mathematics and Statistics of Complex Systems (MASCOS) in Australia, funding for which is currently coming to an end. Besides Australia, Germany, the US and the UK, I have also held research positions in Norway and Israel. Because each of these countries has slightly different university systems I have accumulated considerable experience in teaching ranging from first-year to post-graduate level, and in supervising student's honours and postgraduate work. In 2006 I developed and taught a new calculus course which incorporates web-based learning software; in this, I was supported by an e-learning fellowship grant. I have also supervised a post-graduate student on the topic of non-Markovian random walks on random graphs, and am regularly asked to act as PhD examiner. I am active in key roles of departmental administration, such as postgraduate exam board chair or senior tutor, and I have been elected to serve on the Senate of Queen Mary.