# Current Research

I am interested in solvable lattice models, an area of mathematical physics and statistical mechanics which offers exciting research possibilities in pure as well as applied mathematics. The study of solvable lattice models uses a variety of techniques, ranging from algebraic concepts such as the Yang-Baxter equation, Hecke algebras and quantum groups to analytic methods such as complex analysis and elliptic curves. Due to this wide variety of methods, the study of solvable lattice models often produces unexpected links between different areas of research.

Solvable lattice models provide useful frameworks for modeling real world phenomena. Examples of solvable lattice models that are widely used in applications are quantum spin chains and ladders as models for metals and superconductivity, exclusion processes as models for traffic and fluid flow, more general stochastic processes to model random phenomena and random tilings as models for quasicrystals.

# Integrable stochastic processes

I have a longstanding interest in (quantum) stochastic particle models such as exclusion processes. Many interesting quantities for these processes can be computed using integrable models. These include non-trivial matrix product stationary states and universal current distribution functions such as the Gaussian and the Tracy-Widom distributions.

# Multivariable polynomials

I also study connections between enumerative combinatorics & statistical mechanics on the one hand, and (symmetric) polynomials and representation theory on the other. Schur and Macdonald polynomials are important classes of polynomials and we have developed a new framework to study and generalise such polynomials using solvable models and matrix product methods.

For example, we have developed new explicit expressions for classes of multi-variable polynomials $$f_\mu(x_1,\ldots,x_n)$$ indexed by compositions $\mu$, such as Macdonald polynomials, in the following matrix product form:
$f_\mu(x_1,\ldots,x_n) := \rho\big( A_{\mu_1}(x_1) \cdots A_{\mu_n}(x_n) \big),$
where \$$$\rho$$ is a linear form and the linear operators $$A_i(x)$$ obey the Zamolodchikov–Faddeev (ZF) algebra
$\check{R}(x/y)\cdot\big[\mathbb{A}(x)\otimes \mathbb{A}(y)\big] = \big[\mathbb{A}(y)\otimes \mathbb{A}(x)\big],$
where $$\check{R}(x)$$ is the R-matrix of an appropriate Hopf algebra or quantum group.

Traffic modelling

I was involved in a traffic modelling project in collaboration with researchers at Monash and VicRoads. Check out CEASAR, our traffic network simulator developed in the ARC Centre of Excellence for Mathematical and Statistical Frontiers (ACEMS).