We believe that initiating new research projects at the interface between different disciplines within mathematics, and encouraging mathematical research motivated from various applications outside mathematics, will be vital to the further development of mathematics itself. Furthermore, we are convinced that cooperation with a mathematical centre like the CMA can lead to substantial progress in many important research areas in engineering, medicine, physical sciences, geosciences, biology, and so on. This belief is the main motivation for our strong commitment to interdisciplinary projects. During the first three years of the CMA we have built several research activities which we will characterize as interdisciplinary. In addition to the six research areas described below, the CMA-research on algebraic methods in computational geometry, which can be seen as an activity at the interface between algebraic and computational geometry, should also been seen as an interdisciplinary project. The same is true for the activity on the use of PDE methods, in particular level-set methods, for image processing. However, both these activities are already described above in the geometry section. Also, Snorre Christiansen's EURYI project on numerical methods for geometric wave equations is an activity in the intersection between PDEs and physics, but this research is described in the PDE section.
PDEs and stochastic analysis
The collaboration between the PDE-group and the stochasticians will be continued into the next five
year period. We have focused on the analysis of different types of linear and non-linear partial differential equations appearing in financial applications like option pricing and portfolio optimization, involving developing a viscosity theory for integro-PDEs. We plan to extend this collaboration to consider problems involving infinite dimensional analysis, in particular developing an infinite dimensional viscosity theory applicable to relevant financial problems. Interesting areas include the interest-rate market and electricity market, where infinite dimensional dynamical models find a natural context (see Stochastic Analysis above). Particular attention will be paid to jump processes and integro-PDEs, and work in this field has already been started by one of the Ph. D. students. Furthermore, we want to continue the activity in numerical analysis of partial (integro-)differential equations. There is also a focus on numerical methods for stochastic partial differential equations, an activity which also has links to physics (see below).
PDEs and quantum mechanics
Computational quantum mechanics will play a major role in growing fields like nano-technology and quantum computation. Deep mathematical analysis and sophisticated algorithms are required for this development. Research on the interface between PDEs and computational quantum mechanics will therefore continue to be a central research area for CMA. In particular, we plan to investigate further the role of compatible spatial discretization and symplectic time integrators for quantum mechanical systems and to relate these concepts to central questions on invariance properties such as gauge invariance for discrete schemes. Applications to rotating Bose-Einstein condesates of atoms are planned, with a particular emphasis on the treatment of vortices. The stability of double quantized vortices forms also an interesting research area for applications of partial differential equations to nonlinear systems of equations.
Furthermore, systems of great current experimental and industrial interest are so-called quantum dots, electrons confined to small almost two-dimensional regions. There is considerable experimental and theoretical activity on manipulation and control of such quantum mechanical systems. Nowadays it is possible to construct quantum circuits based on two dimensional systems (Nature Physics,Vol 1 (2005) pp. 177 – 183.) These are extremely promising candidates for building quantum computers. This needs to be accompanied by a theoretical understanding of the dynamics of these systems. The development of stable numerical schemes is crucial to this. In this connection, we have worked on finite element methods with time development of low-dimensional quantum mechanical systems such as quantum dots. The systems we will study first are two-particle systems in (4+1) dimensions and (6+1) dimensions and their time evolution under the influence of a time-dependent and spatially varying electromagnetic field.
PDEs and astrophysics
The modelling of the outer solar atmosphere is a challenging PDE problem based on an MHD system, including radiation and thermal effects. By now, CMA has developed a numerical code for MHD, using a finite volume approach. We will continue to develop and extend this code. In connection with this we aim to utilize results from both the study of boundary values for systems of equations, and from the study of conservation laws with discontinuous flux functions. Presently our code is first order, but we plan to use an ENO/WENO approach to extend it to higher order. We anticipate that this will lead to several research problems at the interface between PDEs and astrophysics, and that we will make a substantial improvement, both with regard to accuracy and efficiency, to existing codes.
Stochastic analysis and physics
Markov Chain Monte Carlo (MCMC) methods play an important role in the activities of both the Quantum Mechanics group and the Cosmology group at the Centre. The Cosmology group has recently achieved remarkable results in the research based on Gibbs sampling, while the research in quantum theory performed in the Physics group relies on advanced MCMC methods. To strengthen further the bounds between these two groups, and also create more activity on the analysis and development of MCMC methods in general, the Centre plans to hire a postdoctoral researcher during the next five year period. MCMC methods are also important in estimating the parameters of stochastic volatility and related processes appearing in mathematical finance, so such a postdoctorate position will pave the way for further collaboration between physics, cosmology, and stochastic analysis (extending the already started activity mentioned above).
PDEs and computational geometry
Many problems in computational geometry can be characterized as computational differential geometry. However, many well-known algorithms are derived and analyzed in a discrete setting, without reference to continuous objects and PDEs. In this project ideas from continuous differential geometry and PDEs will be systematically introduced in an attempt to improve the algorithms used in computational geometry today. As a starting point we are investigating methods for dealing with poorly parametrized surfaces, which frequently arise in CAD models. Many methods are based on sampling the given surface with a mesh, and then minimizing some discrete measure of metric distortion in order to create a new, improved mesh – so-called re-meshing. Instead, we are formulating measures of distortion directly on the smooth surface and following classical finite element methods to minimize these. We will also develop a framework for comparing continuous methods with the existing discrete ones.