*A submission from the NSERC Mathematics and Statistics Liaison group, to the Review of Federal Support to Research and Development, on research funding of mathematics and statistics in Canada.*

**1. Mathematics and Statistics in our research system; revolution, or a golden age**

To the non-initiate, mathematics might seem frozen in time, rehashing the glories of earlier centuries. Nothing could be further from the truth. The last twenty years have seen a veritable explosion of results, as new horizons have opened. Centuries-old conjectures such as Fermat’s or Poincaré’s have been proven, and not just by seeing something long overlooked; on the contrary, the proofs have come through veritable revolutions, fundamental rebuilding of the mathematical arsenal, which are now having vast impacts way beyond their original application. These are exciting times; the reader might with profit, albeit at the expense of some time, consult the excellent “The Mathematical Sciences in 2025”, prepared for the US National Research Council.

*A vast increase in scope.* This change is happening just in time, too, as the challenges of our evolving world are with ever greater frequency arriving on the mathematical and statistical doorstep. The time from fundamental discovery to implementation is shorter and shorter: one needs only think of the applications of compressed sensing, or the complete statistical refit required by multiple gene or indeed whole genome statistical analysis. Simultaneously, the size of our remit is ever increasing, as these new tools give us the means of attacking problems thought inaccessible just a few years ago. Science is by definition quantitative; whether in biology, finance, imaging or a myriad of other subjects, the increased use of quite sophisticated technique is involving mathematicians and statisticians in areas that they have never touched before. A turning point in biology, for example, was the human genome project; a great success for biology of course, but a triumph for mathematics, statistics and computer science. Similarly, the mathematics of atmospheric science is rapidly evolving, as interest shifts from meteorology to long term climate behaviour. In statistics, the analysis of risk, whether climatic, or financial, or other, is developing at pace, with a renewed interest in heavy tail distributions and extreme events. Indeed, statistics is seeing a major transformation- while it has always been involved in many practical applications, the nature of these has changed, as people are struggling to deal with a flood of data. Big data’s other name, data science, in some sense says it eloquently, for what is data science at the core if not statistics? The computational and mathematical ramifications of big data are also enormous, of course: even as far afield as in topology, where we have a 10M$ startup at Stanford examining the topology of large dimensional data sets, with a view, of course, to commercial applications.

*Canada is there.* Canada’s math and stats community is particularly well placed to join in the effort. A 2011 study showed that half of the faculty at that time had been hired in the last ten years. More is to come; for example, a review of the CFREF allocations gives an impressive number of universities with big data or data science either at the heart, or crucial to, their development plans. Graduate students, those bellwethers of the future, are sensing this; the last ten years have seen a doubling of their population in Canadian Mathematics and Statistics departments. And they are getting jobs.

**Institutes**

One peculiarity of mathematics and statistics as a discipline across the world is a prevalence of mathematics and statistics institutes. A few explanatory words are perhaps in order.

In a theoretical science, with no laboratory to lug around, why not go directly to the source and work for a while with a kindred soul elsewhere in the world? The race, in our world, is often to the most international, and institutes have been the foci of these interactions for some time now. The format of these institutes varies; some focus on workshops, others on long term visitors, some have permanent staff, some not. Their common feature is to serve as an international meeting point for exchange and development of the discipline. The importance of the model is recognised internationally; the American NSF, for example, funds eight of them.

Canada has it own network of institutes. In mathematics, it was built over the years, starting in the late sixties; its current configuration was set in place in the nineties following a report of a committee chaired by Jean-Pierre Bourguignon, the current head of Europe’s ERC. There are three main mathematics institutes, CRM (Quebec), Fields (Ontario) and PIMS (West), reflecting Canada’s rather large geographic spread, and recognising the necessity of a good local university base for functioning. The geographic coverage of the mathematics network was then completed with the addition of the Atlantic Association for Research in Mathematical Sciences (AARMS).

The Banff International Research Station, a joint Canada-US (and now also Mexico) initiative, was added to this network in 2003. This unique joint Canada-US-Mexico initiative is a research infrastructure that addresses the imperatives of collaborative research and cross-disciplinary synergy, by facilitating intense and prolonged interactions among mathematical scientists from around the world. To this day, it is the only major joint scientific initiative of the three NAFTA countries, funded by NSERC, NSF, and Mexico’s CONACYT; it is subscribed several times over, and must make quite draconian choices in its selection of the 50 workshops it runs every year, which touch not only on mathematics and statistics, but also their vast array of applications in science and engineering.

While the Mathematics Institutes always had a certain amount of statistics in their scientific programme, the Statistics community felt that the only way to get the level of activity they need would be to have its own. Thus we now have CANSSI, which is in its inaugural cycle of workshops, collaborative research groups, and post-doctoral fellowships.

The institute model has many advantages, which could indeed apply to other theoretical disciplines:

*An international presence.* All of these initiatives have put Canada on the map, and indeed the institutes all have strong international ties, for example hosting international labs of France’s CNRS, or running joint programmes with their American, European, or Asian counterparts. They are natural hosts for academic visitors from abroad, whose presence greatly enhances our scientific enterprise.

*Drivers of the discipline.* Locally, they have been important drivers of the discipline; with their international scientific panels, they privilege in their scientific programs the most cutting edge choices, orienting allocations and keeping us on our toes. They also have an accelerator effect on the discipline, by bringing together all those researchers for concentrated work.

*Concentration of resources. *As a discipline, we are not rich in resources, and the institutes have enabled many crucial elements in our scientific system, beyond the budgets of individual researchers. For example, they have allowed very roughly a tripling of the number of post-doctoral fellowships available in the country. These fellowships are an ever more important element of mathematical training. Another example would be summer schools for graduate students, bringing the world’s best to Canada for our students.

*Leverage: * The institutes, collectively, have been able to leverage their federal funds several times over, from provincial, university and private sources. This extends to the international arena, with resources from several foreign countries often associated to their activities.

*Outreach: *The institutes very carefully position themselves as working in mathematical or statistical sciences, as opposed to mathematics and statistics, and have consistently explored disciplinary boundaries, into, for example, areas such as finance, quantum computing, theoretical chemistry, mathematical physics, and computational biology. Another form of outreach they have pursued is in the role of minorities and under-represented groups. In Canada, yet another role has been in tying together and developing geographically isolated locales.

*A focus for initiatives: *Either together, or individually, the institutes have served as organisational foci for important scientific initiatives. Of these the most notable was the creation of MITACS, Canada’s most successful NCE. One more recent example is the 2013 Mathematics of Planet Earth initiative, which started out at Montreal’s CRM, but expanded into a world-wide initiative, with recognition from UNESCO.

*The innovation agenda:* One of the forms of outreach that has had great success is the development of links to industry. The investment in time in developing these ties is not really feasible for the average researcher. The main vehicle for this outreach for several years was MITACS; it has been followed up with the Institutes Innovation Platform.

**2. Mathematics, statistics and research funding.**

We summarise several main features of the funding of mathematical and statistical research in Canada.

*NSERC is the main game in town.* Here we are omitting the Canada research chairs, of which the discipline has a share, and considering only the direct funding of research. A 2011 Canadian Mathematical Society survey had NSERC funding as 69% of the total portfolio in mathematics, with four fifths of that being from the discovery grant program; most of the rest was university or provincial money. The statistics figures are similar, except that they include some CIHR funding, though rarely as principal investigator. This is in contrast with the American model, in which a large number of private sources of funding (Sloane, Simons, Clay), as well as government agencies like the DOE complement NSF funding.

*We are, relatively speaking, resource poor. *The level of research funding has long been a sore point in our community. It is one of those things frozen in place many years ago, which nobody seems able or willing to change. Our main source of funding is in Discovery grants; while the lab disciplines might need extra resources, there is no reason why mathematics and statistics should be treated in any way differently from other theoretical disciplines; on the ground however, the average grant for mathematician/statistician is 19K, while for a computer scientist it is 28K (Numbers from the NSERC data base).

The main expenses in the theoretical disciplines are basically the same: people, in particular students. It is no wonder that researchers on the boundaries of our discipline try to leave the fold, if they can pass as something else. It would seem that equal grant for equal science should be a sensible principle, but no such principle is applied.

*Student pressure. *The pressure has increased considerably in recent years, as our graduate student cohort has doubled and NSERC has made graduate student training an important feature of evaluation (a good idea, by the way, but then one needs the money to train). The problem is exacerbated by the relative poverty, in comparison to our American cousins, of our Universities, which have come to rely on research funds for many things, in particular funding of graduate students. Our typical department requires a contribution of around 10K or more per year from the researchers grant to fund a student, making up the rest with teaching assistantships and internal scholarships. Our US colleagues face a much weaker requirement to fund from their grants. One could debate whether graduate students should all be funded; however that is the ‘market’ as we live it. In any case, with a research grant at the average of 19K a year, this makes for a maximum of 2 students at any time, and indeed the main constraint on accepting students is funding, not capacity.

*Resources, bis. * There is also a rather severe cash crunch at our Institutes. The total federal budget for the mathematics institutes is currently 4.9M. One should compare this with 4.5M (US) each from the NSF for Berkeley’s MSRI (Math) and North Carolina’s SAMSI (Stats). It is remarkable, and indeed a tribute to our leadership, that in our collaborative efforts these institutions treat us as equals. The amounts have also not moved substantially in over ten years; the situation was particularly difficult in the last round when room had to be found in the budget for CANSSI, resulting in severe cuts on the mathematics side for institutions that deserved much better.

**Our funding system: directions for improvement**

*1) The Discovery Grants. *Our community shares the general approval of the rest of the scientific community of NSERC’s Discovery Grants, a truly visionary system of which the country should be proud. The basic model of a university scientist as an ‘ideas entrepreneur’ is one that fits well with our mission, and the flexibility to assume one’ s responsibilities afforded by a general grant is one that is much appreciated. As a system, in general, it is highly cost effective.

The general scheme of funding careers rather than individual projects is a wise one, for example leaving room in the portfolio for more speculative work, and the approach reduces overhead. In this context, a high success rate is warranted, in particular in regards to ongoing responsibility for graduate students; and indeed is not vastly different from that of a individual project that is submitted to a program with a low success rate, but which goes back several times.

It is not a perfect system:

*It needs to evolve. *Disciplinary allocations are jealously guarded, and so have not evolved in time, as our community has found to its cost; the inevitable question from NSERC officials when one complains of grant level is to say fine, but who do we take it from? This is not unreasonable, but one could wish for some flexibility.

*It needs to be coherent.* The boundary effects are enormous: if you can ‘pass’ as a computer scientist, or as a theoretical physicist, your grant will increase dramatically. These labels should not be driving scientific choices; one would like equal grants for equal science, taking into account costs of research. Also, the protean nature of the discovery grant means that it is different things to different people, and evaluations of success and need vary accordingly. For us, as we said, it is the main game in town, while for a successful research engineer, it is the basic enabler, and a small element in a large research industry based portfolio.

*Drop the targeting. *One excellent innovation to the Discovery Grant program was the introduction of Accelerator supplements, giving a career boost to particularly deserving cases. Most of these supplements were unavailable to us, as they were “strategically targeted” to particular disciplines, an expression which in our neck of the woods means not for you. This a priori targeting should be dropped, and indeed this recommendation goes for the whole granting system (the example of the Canada Excellence Research Chairs comes to mind). The targeting choices, which seem to emanate from some think-tank, are yesterday’s hot idea emerging as today’s cliché and tomorrow’s leftovers. It is amazing how people who often are strong believers in a free market do not believe in the free market of ideas. Let the ideas compete.

*Dependence. *Finally, the system’s flexibility and the universities’ relative poverty have, as we said, lead universities to rely on NSERC funds to function, so for example, the loss of the Discovery grant leads to a severely handicapped faculty member. It is a rather sad waste of human resources. More generally, cut-backs in the federal system lead to fairly severe repercussions throughout the university system, going beyond what one would call the typical research sphere.

In any case, from our side, the recommendation is clear; good system, needs a bit of work, above all, given its protean role, needs money.

*2) Platforms. * The question of federal funding of research platforms seems to us particularly apposite. Our Network of Institutes has never really fit in, though we should say that NSERC has done its best to be accommodating. Early on, they were funded through some major facilities programs; they then, for a while, basically each had a status of individual Grant Selection Committees, and even participated as such in the various reallocations that took place. (Interestingly, they always did very well, coming in in the top four of the two dozen or so committees in the competition.) The latest format has them lumped in with Mathematics and Statistics Discovery grants, where they represent 20% of the budget; they are subject to a Long Range Plan, in which the community must allocate resources. Again, a source of stasis, as a community must decide whether to cannibalise an already stretched system of individual grants in order to fund institutes as they should be. As noted, individual American institutes basically each have the resources allocated to all of Canada’s network; while we run a lean operation, there are limits.

Our institutes have also played a role in fostering a wider, more interdisciplinary approach, going beyond the mathematics and statistics core. The institutes run workshops on the more mathematical and statistical aspects of biology, physics, chemistry and computer science. It is not really an ideal fit for them to be confined to the mathematics and statistics envelope.

It is thus that a separate platform system, that recognizes the contribution to our common intellectual project that these institutes make, seems to us most desirable.

*3) International grants. * There is a whole world out there waiting to partner with us. Our system is singularly badly adapted to providing the matching funds to participate in say, EU projects. Science is international, and we can only benefit from a wider arena. Some flexible (untargeted!) funds for matching to other countries’ programs would be desirable.

**Who we are**

These recommendations come from the NSERC Mathematics and Statistics Liaison Committee, a representative group of eleven senior mathematicians and statisticians from across the country constituted by NSERC as a consultative group on issues of funding in mathematics and statistics. Its current membership is

- Nassif Ghoussoub, OC, FRSC, Professor of Mathematics, UBC,
- Paul Gustafson, Professor of Statistics, UBC,
- Jacques Hurtubise FRSC, Professor of Mathematics, McGill University (Chair),
- Jeanette Janssen, Professor of Mathematics, Dalhousie University,
- Niky Kamran FRSC, James McGill Professor of Mathematics, McGill University,
- Barbara Keyfitz, Dr. Charles Saltzer Professor of Mathematics, Ohio State University,
- Mark Lewis FRSC, Canada Research Chair in Mathematical Biology, University of Alberta,
- Kumar Murty FRSC, Professor of Mathematics and Chair, University of Toronto,
- Christiane Rousseau, Professor, Univ. de Montréal and International Mathematical Union,
- Edward Susko, Killam Professor of Mathematics and Statistics, Dalhousie University,
- Changbao Wu, Professor of Statistics, University of Waterloo.