Mathematics and Theoretical Physics
The Mathematics and Theoretical Physics section of the School of Physics Astronomy and Mathematics at the University of Hertfordshire is engaged in research in areas of pure and applied mathematics, as well as in theoretical physics.
Our research informs the teaching provision of the School, and provides links to collaborations in other areas of science and industry.
Algebra
Dr Catarina Carvalho studies the numerous connections between algebra and theoretical computer science. Her work has made it clear that the exploration and understanding of both areas is a productive endeavour for each of them. We are particularly interested in the mathematics of constraint satisfaction problems (CSPs), and the applications of universal algebra and combinatorial techniques to the study of the complexity of CSPs.
Quantum integrability
Dr Charles Young and Dr Benoit Vicedo have interest in the area of Quantum Integral Systems. The theory of quantum groups lies at the intersection of theoretical physics and pure mathematics.
From the physical point of view, quantum groups arise as the hidden symmetries of quantum integrable systems, while from a mathematical perspective they provide deformations of Lie groups and Lie algebras. In particular we are interested in quantum affine algebras and in non-ultralocal integrable systems.
Quantum affine algebras have a remarkably rich representation theory which is only partially understood, and that has deep connections to diverse areas of mathematics including algebraic geometry and combinatorics. Similarly, non-ultralocal integrable systems still lack a consistent quantum treatment since its first appearance over three decades ago. The most recent example of such an integrable system appears in the celebrated AdS/CFT conjecture.
We are particularly interested in the formulation of such integrable systems within the theory of quantum groups or its possible generalisations.
Mathematical modelling
Our interests encompass a wide spectrum of areas where we explore connections between mathematics and other branches of science. Dr Jesus Rogel-Salazar has interests in mathematical modelling using ordinary and partial differential equations, numerical analysis and scientific computation in applications that range from laser theory and nonlinear optics to quantum atom optics and cold atoms as well as organic semiconductors such as OLEDs and solar cells.
We are also interested in narrowing the gap between state-of-the-art numerical solution procedures in applied mathematics and the current practice in the physical sciences. It is very encouraging that some of the techniques used have relevant applications in areas other areas, such as microfluidics, econometrics and computational finance.
Theoretical optics
Dr Ole Steuernagel carries out research on theoretical optics ranging from properties of light to fundamental quantum mechanics. Recently, his work on ‘Wigner flow’ has established the quantum analog of classical particle flow along phase portrait lines. This is surprising because we know that Heisenberg's uncertainty principle precludes the existence of phase portrait lines since it stipulates that sharply defined trajectories do not exist. So, phase portraits do not exist but the associated flow does.
His work reveals hidden features of quantum dynamics and extra complexity. Being constrained by conserved flow winding numbers, it also introduces topological order into quantum dynamics.
Applied mathematics
We are also interested in applications of global optimisation and numerical solution of boundary-value problems with applications to practical challenges in areas as diverse as aircraft routing, particle identification and portfolio selection with threshold constraints.
Dr Stephen Kane and Prof Alan Davies have focused on using Laplace transform methods for the solution of difussion-type problems, in particular non-linear applications such as coupled thermal/electromagnetic and phase change problems. We have had an interest in parallel computing for nearly twenty years using a variety of architectures, the most recent of which comprises a cluster of PCs. Laplace transform formulations are ideally suited to such environments.