Pre-Restructuring Thoughts

This is my first post in 2019 and I will write about the institute's future and niche area.

It is perhaps an open secret that the institute will be undergoing restructuring process along with the rest of the university. One of the task given to us is to carve out niche areas perhaps unique to the institute itself, avoiding redundancies with say the Mathematics Department in the Faculty. Redundancy in itself is not necessarily bad and in fact is key to many technologies with critical components. Whether this is the case for our mathematical research perhaps is not clear. I certainly want to think that the institute has helped in many ways mathematical research in the university. Perhaps finding a unique identity to the institute is something that one should aim for after many years.

Each of us has one's own research passion and it is quite easy to be blinded into thinking that one's own research is more important or more fundamental or more difficult than another. To be able to objectively compare between fields, it is necessary to be actively involved in the research areas that one wants to compare. To master one particular research area already requires incredible feat; to be able to do more would probably take a few lifetimes. Comparing different fields is not our goal; more importantly is to ensure that such different fields synergize with each other. One hopes to have different groups working together for some common goals, which I hope will be realised in the future version of the institute. Another desirable thing to have is to get the right mix of birds and frogs (alluding to Dyson's categorization of researchers in his article in Notices of the AMS). Birds bring diverse ideas together and frogs do intricate details within one area. I would like to imagine myself as a bird, not a high-flying one but one that is close to the ground. I get very fascinated by mathematical topics that find their ways to physics in rather surprising ways. Each weekend I will normally scan the arXiv to see what new ideas are brewing; those that caught my eye are link-dumped at Equatorial Frequencies. Most of these will be left unread of course. Such scanning has helped me get some form of bird-eye's view of progress made in mathematical sciences. I hope this will be of use in looking for possible new directions for the institute.

One of the directions that could have been taken is to move to interdisciplinary research as I have raised many times (I'm also including areas within mathematics themselves). But what areas would be most natural for mathematicians to work with? The best way to answer is to see what available data there are on mathematical research publications. Some researchers have taken the time to look at this by constructing a map of mathematics. One map would be based on the paper submissions on the arXiv (see figure below):

The blue ones are those considered to be traditionally in mathematics. The purple ones are coming from physics. The red ones are computer science and also engineering. The green ones are statistics (but we will not consider this as distinct from mathematics for simplification). Physics has always been closed to mathematics and should have interacted more with mathematics. Reality on the ground is that most of our physics staff do very little theoretical work. Hence we should source more interactions from the other closely associated areas of computer science and engineering. In fact, I would have loved to see more computer scientists (particularly theoretical ones) to be more active in our institute. Engineering would be another but they are not quite well represented in the above map since arXiv is much biased on those areas that have established preprint culture and perhaps engineering has yet to be one. Another map that will show the engineering representation better is perhaps the map constructed from Zentralblatt data (see figure below).

The mathematical areas that are close to engineering are those in partial differential equations and numerical analysis. Note the heavy weighted lines from PDE and numerical analysis to fluid mechanics, thermodynamics and heat transfer, systems theory and control. Thus, engineering would be another source of interaction that the institute could place hope for. Having said that readiness and willingness of researchers (mathematicians, computer scientists and engineers) is a complex matter to deal with and we hope that we can overcome this challenge.

Back to carving niche areas, cryptography was one of the identified niche areas of the institute even before the current restructuring process takes place. This would however be too restrictive for the institute to base on and will exclude many other important established areas of the institute. To generalize this, one of us suggested disruptive technologies (see here and here) which will include further things like internet of things, data science and others often quoted for IR 4.0 (see for instance these examples). This would make the institute heavily based on (available) technologies (note most technologies are disruptive in nature). To be more far-sighted, one should also include the sciences that drive them. One particular area whose technology is expected to change the IT landscape is quantum sciences, an area which I dream of establishing firmly in UPM (time is running out since I will be retiring in a few years time). In fact, the idea was there when I joined ITMA last time and we suggested the field of study on Quantum Science and Technology (way back in 2002 - see figure below).

This initiative (experimental side) stopped just after I left ITMA. Today, the term Quantum Science and Technology is already a common term. IOP for instance has created a journal with such name. Here is a relatively recent interview on Japan initiatives in quantum science and technology.

Now, quantum information science fits well in the niche area of science that drives near-future disruptive technology. It actually took about 70-80 years from the beginning of quantum science to reach the point of precise manipulation of single quantum particles. Today there are many available technologies of doing so. However technologically we are far away from doing this in our labs (though there is a possibility of accessing such technology with external help, which we are considering). The best option for us is to take the theoretical (or mathematical) options for quantum information science research which I think we are able to do (provided support and enough manpower).

Having said this, it is not about serving my own interests in doing the institute's restructuring exercise. We have to take into account for what is natural and what serves the institute better. For instance, mathematical physics which is an important area that connects physics and mathematics (see map of mathematics in the figures above) will now take a back seat in favour of mathematical engineering. This is to attract more engineers to the institute (which I do hope will happen). An area that seems to fall out of favour in this action will be our work on hyperbolic spaces and cusp forms. Unless of course if we can still relocate such research by making it into something applicable like quantum money.

Another area that I hope the institute can pursue is the area of data science, which the institute had already eyed on before this. Now while most people are imagining the data explosion, my concern is actually more on some inherent (human or otherwise) limitations of obtaining useful finite information and that contextualization is probably needed (see here). On the other hand, we need new inventive ways to produce useful information. One such tool is the idea of complex networks (an area that I hope the institute pursue), which has already show many applications and some may even be counter-intuitive (see e.g. here).

All these ideas are ambitious of course. Ideas will remain ideas unless they are worked on by our researchers. It is here that we wish that we get all the support that we can to make our institute stronger and sustainable.