Interview with Dr. Weiran Sun

Assistant Professor, Department of Mathematics

Applied Analysis, Partial Differential Equations, Fluid Mechanics, Kinetic Theory

Dr. Sun’s research interests lie in applied mathematics. Within that part of the field of mathematics, she is drawn to the analysis of partial differential equations, especially equations arising from problems in physics.  The topics of kinetic theory and fluid mechanics also peek her interest, and as a result her research group explores a variety of questions in these areas, including problems from physics and biology.  Dr. Sun’s work is a perfect example of integrating creative problem solving and mathematics into other scientific fields to yield exciting outcomes.

What kind of research are you working on right now?
My broad field of study is the analysis of partial differential equations, and the types of equations I look at come from physics and biology – I bring a mathematician’s approach to problems arising in these areas. For instance, one important question in statistical physics is to bridge the microscopic and macroscopic views of gas dynamics. Using the behaviour of airflow in a room as an example, one can take the microscopic view and consider the air as a collection of particles. Alternatively, one can adopt the macroscopic view and treat the airflow like water. Although the connection between these two views can be argued in a formal way, the mathematician’s role here is to provide a solid foundation by rigorously justifying the relation between them in various parameter regimes. And the tools that we use are differential equations.

In both physics and mathematics, there are various types of differential equations. In the airflow case, there are ordinary equations that describe the motion of particles; stepping back a bit, there are partial differential equations that describe density distribution of particles, considering the number of particles present within a certain velocity and position range; leaving out even more details, there are equations that describe the evolution of macroscopic densities of mass, momentum, and energy. My interest is in a medium-detailed or mesoscopic view of the problem, the kinetic description. My research group is trying to understand properties of kinetic equations and connect them to the macroscopic fluid level view. Analyzing this so-called multiscale phenomenon involves a lot of advanced mathematical tools.

To what sort of problems in different fields do you apply your expertise in applied analysis of partial differential equations?
Kinetic equations started in physics. These days, however, researchers develop kinetic models that go beyond physics and math problems.

For example, working with my Simon Fraser University (SFU) colleague Razvan Fetecau on a mathematical biology project, we are looking at collective behaviour of individuals such as birds and fish. Using the birds as an illustration, we can treat them as particles, similar to treating the air as a collection of molecules. These 'particles' interact with each other, and we can use certain mathematical equations to describe how they interact and how the interaction changes their behaviour.

Can kinetic models be applied to other disciplines, as well?
Yes. In social science, for example, one can use kinetic models to study the social dynamics of a population. In this case, each individual is assigned an index denoting his/her social status. Instead of tracing the change of the status of each individual, one can use a density distribution function for the whole society and construct kinetic models that describe the evolution of the society due to interaction among people. Similar ideas are also applied to other topics like modelling traffic flows and studying economic inequalities.

Has your work with kinetic models introduced you to new areas of study?
In a very interesting recent collaboration I learned about using kinetic models to describe bacterial behaviour. One such example is the movement of E. coli, which is influenced by the environment that they live in. These bacteria can move directionally in their environment depending on the concentration of a chemical in the region; this response is called chemotaxis. One can assign a density function to these bacteria and set up a kinetic equation to describe how the environment acts on them and how they respond to the changes in the environment.  This is an exciting project for me: on the one hand, the biologists have data that can be used to check the validity of our model, and on the other, our analysis provides insight for understanding the behaviour of the bacteria.

What do you look for in prospective trainees?
I look at the background of students to see how well prepared and interested they are in analysis. Research in analysis requires a lot of creativity and patience. One needs to have a good understanding of both the problem and the tools involved. To achieve this, one has to sit down and do a lot of computation.

Because my area is applied math, I also look for students who have an open mind about physics and other fields beyond math. The educational background of students in my group is typically an undergraduate degree in math, although a current student of mine also has a background in physics.

What issue in your field of research concerns you the most and needs more attention?
For applied mathematicians, it is important to talk to people in other fields, because these people bring a very different perspective, one that can be surprisingly insightful for the mathematicians. For example, typically, analysis of partial differential equations (PDE) problems come from the applied fields. Those doing the PDE analysis tend to look at just the math part of the work instead of also asking what the variables are about or what the physical meaning is of the results. I think applied mathematicians should be more open-minded and aim to understand more fully the math together with the application field.

On the other side, I also feel that it is easy for people doing analysis to separate analysis from numerical computation. You don't have to do numerical computation if you're a pure analyst, but in my experience, working with numerical computation researchers is very helpful. The interaction between analysis and numerical computation is just as important as the link between PDE analysis and the applied field.

What sort of advances are going to be the next big thing that changes your field?
The area of stochastic analysis will continue to develop and using it can change the point of view of a lot of things. Many classical differential equations describing physical phenomena are deterministic. However, data from physics observations and experiments are usually affected by random effects. Nowadays, people are starting to involve stochastic effects in classical models, and uncertainty quantification also is becoming a popular subject. I am very curious about whether adding stochastic effects could fundamentally change properties of some classical PDEs that are difficult to analyze in the deterministic setting.

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Read more: Dr. Sun’s profile on the Department of Mathematics website, her personal website and interviews with other faculty members on the Featured Researchers page

Interview by Jacqueline Watson with Theresa Kitos