Planning for the future of applied mathematics needs to take into account the challenges facing our nation and humanity. Do we as a society have the mathematical skills to understand the climate change debate, to follow modern biological technology and participate in industrial and business development? The role of mathematics in these discussions would seem clear: To provide a framework with which we can model novel scenarios, draw conclusions, reason with data and deal with uncertainties.
An important trend in the quantitative disciplines has been an ever increasing application of computational techniques. This trend is supported by the 1000 fold increase of the performance of top-end computers every decade. With the emergence of smart-phones, reasonably powerful computers (much more than just calculators) are now available. The processors of smart phones are powerful enough to do the computational assignments of a good introductory computational science course. In order to use the new computational resources to best effect it would seem important to understand the mathematics behind computation.
New challenges emerge which on one side originate from the complexity and extreme parallelism of new supercomputers. One challenge is that the main costs of computing nowadays is not the purchase but the energy costs. Of course energy is also a major driver for new phone technology. Another challenge of highly parallel computers is that at any time it is very likely that one of the components will break. Both challenges require new algorithms and with it new computational mathematics.
At the same time modern computational mathematics is increasingly making use of new pure mathematics. Traditionally, computational mathematics is based on calculus (Taylor series) and linear algebra. Well-known are the connections between representation theory and discrete Fourier transforms. New computational techniques use wavelets and compressed sensing. Hilbert and Banach spaces are important tools to understand the numerical solution of PDEs. Conversely, numerical algorithms are some times important components of existence proofs – because they are constructive. Some of the most exciting recent work links algebraic, differential and fractal geometry to computation. Finally, convex analysis is a major contributor to optimisation. It would appear that a
combination of computation and pure mathematics would be an ideal growth area – which indeed it is world-wide but just not in Australia (yet?).
When considering the profile of mathematical departments in Australia one notices that computational mathematics does not play a big role at all. Even some of the our best universities to not have any computational mathematicians. This is in stark contrast to what is found in the US, Britain, Germany and Singapore for example where computational mathematics is frequently a strong component both in terms of staff numbers and in the curriculum.
I think what we need in the next 10 years is less of isolated pure and applied mathematics departments but teams which include both pure and applied mathematicians working jointly on some computational problems. As mentioned in another contribution to the Decadal plan such groups can be found overseas and are the breeding ground for the new applied mathematics. Still, interdisciplinary teams joining researchers from different disciplines is important. But I think that for the development of mathematics the “inter-subdisciplinary” teams would have a much stronger impact. Mathematicians who have learned as postdocs to work in different parts of mathematics are clearly a big asset to small mathematics departments as they can teach different areas well. This inter-subdisciplinarity can of course be found in pure mathematics throughout the world. One also finds such cooperations between branches of applied areas like statistics and computation. Given the importance of computation for mathematics (which is clearly documented in the US Decadal plan) I think that we should make an effort to bridge pure and applied (here I suggest mostly computational) mathematics. Fostering such trends could clearly be done through supporting research proposals which build on such groups. I would hope that some advice in this direction from the Academy to funding agencies would be seen as helpful.