Jerome Droniou lecturers at Monash University, he spoke with us at length about his research, interests and questions that puzzle him on an annual basis. He will be speaking at Summer School 2016
What are the most interesting “big questions” in your field?
I can identify two of them:
i) The design of numerical methods that respect physical properties of the model. A number of physical processes are modelled through partial differential equations (PDEs). This is for example the case of fluid flows. These equations have a structure that naturally enforce the expected physical properties of the solutions.
For example, the equation describing the evolution of a the concentration of a component in a mixture ensures that this concentration remains non-negative (a negative concentration does not make any sense!). However, when “discretising” these PDEs, i.e. trying to find numerical algorithms to approximate them, this magical structure which enforces physical bound is often lost. It is not clear how to preserve this structure, while taking into account engineering constraints (such as very complex grids encountered in some applications).
ii) The analysis of numerical methods under real-world assumptions. After designing a numerical method, the second job is to assess their accuracy, that is demonstrate that the approximate solution is close to the solution of the model. This can be done by testing the method in a number of situations, but this method of assessing the accuracy is highly dependent on the situations we consider. A more satisfactory way, for the mathematician, is to establish general results that are independent of any particular case/situation. This is often done by assuming some very strong assumptions on the solution or the model; in situations encountered in applications, these assumptions are not satisfied, and the analysis done in the ideal mathematical world cannot be applied. There are however ways to conduct a rigorous mathematical analysis under the exact conditions encountered in applications, at least for models that are not too complex (but include nonetheless some very meaningful physics). This is a recent topic in my field, and I believe one of the most important ones, as this participates in bringing rigorous mathematical analysis closer to real-world applications.
What kind of problems are you interested in broadly in the field?
I study partial differential equations, both at the theoretical and numerical levels. These equations appear as models in physics, mechanics, biology, etc. Their theoretical study consists primarily in establishing the existence and uniqueness of a solution to these equations. Since the models are usually quite complex, it is extremely rare to find formulas for the solutions.
Numerical analysis, which pertains to the design and analysis of algorithms to approximate these solutions, is often the only way to obtain qualitative information on their behaviour.
What are your favourite applications of your work?
When some of the mathematical tools I helped develop are found useful for the analysis (theoretical or numerical) of real-world models, such as equations modelling oil recovery.
What are some other areas of maths that are particularly interesting to you?
I like differential and Riemannian geometries, although I don’t practice often enough to be familiar with all the concepts in this field.
Why did you become a mathematician?
Because maths is like a game to me, and thus it’s fun. Moreover, I particularly enjoy when a theory comes together nicely and every piece fit into the other one and, in the end, this is what maths is about.
Do you have any advice for future mathematicians?
I think the most important quality of a mathematician is rigour. Depending on the kind of mathematics you do this can take different forms, but in any case each argument in a mathematical reasoning should be clearly justified and understood. If you’re not convinced yourself of your reasoning, then it need to be re-worked. Apart from that, choose your mathematical field based in your own interest and motivation, not on some trend or perceived career opportunities. Maths is a difficult field, but also a very rewarding one if you do what you like.
Biggest mathematical/statistical regret?
Not having the time to write all the paper I have ideas for.
Biggest mathematical/statistical success?
I can’t really identify one. I’d say that every year or so (sometimes even more frequently) I obtain a mathematical result of which I’m very proud. I consider it my biggest success, until another one supplants it the year after…
