DiM Research Panel PART II – Professors (Winter 2025)

1. Intro

• Jessica Lin
  • Interface of PDE and probability
  • Partial differential equations are the primary mathematical models for physical phenomena
  • Research focuses on proving theorems related to these equations (more theoretical work, in nature)
  • PDE’s tend to pop up in probabilistic models (e.g. given a Brownian motion medium)
  • Other examples of probabilistic models:
    • Random walks
    • More general diffusion processes
    • Branching processes
  • Specific research interests:
    • Finding ways that PDE s can help studying stochastic processes
    • Proving new results in theory of PDEs
  • Past undergraduate student projects
    • Extending convergence results for finite difference schemes for a weak class of PDEs
    • Improving the rate of convergence in vanishing viscosity method
    • Proving an invariance principle for rescaled random walks in periodic lattices
  • Overall, a strong background in real analysis and a course in probability and PDEs for an undergraduate is necessary for working on a project
    • Also very important to be willing to learn background knowledge
• Adrian Vetta
  • Algorithmic economics, discrete math (math and cs)
  • In two departments: Mathematics and Computer Science
  • Teaches discrete math, algorithmic game theory, more 
  • Examples of undergraduate research projects
    • Kidney donations
      • Most kidney transplants are living donations in paired kidney donation programs
      • Want to minimize waiting times, do it in a fair way (there might be people who are hard to match)
    • Carbon pricing to offset climate change
      • Evaluating different options based on effectiveness and fairness
    • Fair decision making in risk assessment scores, job recruitment, and medical treatments
      • How do we design algorithms that are effective and fair?
      • A lot of computer programs have embedded racial, gender, and other biases in them
    • Fair division
      • Discrete objects and voting mechanisms
  • Overall, studies a lot of large combinatorial optimization problems
• Johanna Neslehova
  • intersection of statistics and probability (new methods that need to be evaluated rigorously)
  • Research is in statistics; if interested the math courses as an undergrad are a good place to start (Majors: MATH 324 and Honours: MATH 357)
    • McGill also offers grad classes that are open to undergrad students if you are looking for more!
  • Interested in risk management involving extremes
    • Example: water protection dams need to be strong or high enough to withstand a flood that happens every N years, with little data available. (N could be 10, 50, 100…)
  • There are nice theories behind extrapolating on minimal data like this
    • Maxima theory is a lot more complicated, but along the lines of the central limit theorem students may have seen if they took a probability course. 
  • Outputs from climate models are also used as data, but these still need to be extrapolated on because running these models is quite costly in terms of time and energy. So sometimes statistical and machine learning techniques are also used. 
  • Undergrad project examples
    • Rainfall in Florida:. There was a grid of data provided and they needed to extrapolate in space
  •  There is a lot of unexplored territory in the realm of predicting extremes (they are often underestimated)
  • It’s important to use different models, e.g. extreme value copulas, to model these extremes (they don’t have a normal distribution, for example)
• Linan Chen
  • Probability (two parts: random geometry objects and stochastic analysis view of modaling dynamics) formulating solutions and regularity problems. 
  • Research has two pillars: random geometry and stochastic processes
  • How do we put a probability measure on a space with infinite dimensions?
    • A lebesgue measure does not exist in infinite dimensions
    • A gaussian measure is a well-developed example
      • Brownian motion is a classic example of a model of this
  • Research is rooted in fundamental but abstract models
  • Did research with an undergraduate on random geometry:
    • Independent and identically distributed points on the domain randomly partitioned by cells.
    • The problem is highly non local (when you perturb it changes the diagram)
    • There are two levels of randomness- what cell contains a certain point and what is the measure of that cell?
      • It turns out that the volume of a cell will follow the same distribution!
  • Research with undergrad in stochastic processes:
    • The PDE that describes the dynamics of brownian motion is the heat equation
    • Putting a coefficient (dependent on x) in front of the diffusion term and only considering the interval [0,1]
    • This is called the Wright-Fisher Model, and it is used to measure a gene allele in a certain population.
    • A lot of analytical techniques fail here, and the goal of the research is to develop tools to understand this kind of equation
  • Needed for undergrad research: a strong background in analysis (ideally with some background in measure theory)

2. What math background are you looking for in potential students?

• Neslehova: Good basis in probability and statistics (at least MATH 323 and 324)
  • Some probability, some analysis.
  • Two components:
    • one more applied (RStudio, Python, course in stats and probability)
    • one more theory (good if you have more advanced probability)
• Lin: At least one year of analysis, two full years is great
• Chen: As said before, at least one year of analysis and a good understanding of measure theory 
• Vetta: Discrete math, probability, programming is not necessary. It’s also important to have an idea of how to prove things about algorithms.
  • Algebra 1, 2, 3 are helpful

3. Would there be opportunities for U1 students?

• Vetta: You really need a solid basis and to get the course background. Also it’s more competitive to get the scholarships/grants because they will go to the senior students; generally, you need tools to use in research. 
• Neslehova: If you don’t have the tools, the project is not as advanced or exciting. [I am] on the committee for awarding the undergraduate research awards, and very very rarely U1 get the awards. 
• Lin: Contact professors still, if you are willing to explore research without getting a grant, it’s not a bad idea. Another good idea: the Directed Reading Program! A good way to get a taste of research, when maybe it’s still early to apply for a summer grant. 
• Chen: You really need to know what you are interested in, your interest is what drives the research process. U1 is probably too early to tell what exactly you want to do. If you don’t have an idea, that is probably a sign that you need more background. 

4. When is a good time to be contacting professors?

• Lin: Around now (Mid to late January). Professors are finalizing their summer research plans right before the winter reading break. If you reach out earlier, they might not have everything figured out for the summer yet. 

5. What should be included in an email?

• Vetta: You don’t need to include much, it’s better to talk in person. There’s no key words that need to be said. 
• Lin: It can be helpful to attach your transcript or CV, and it can’t hurt. 
• Neslehova: The personal meeting is always important! You can always drop by and professors know that undergrads are not full-fledged researchers. As a professor I want the project to be fun for the undergraduates.

6. Does self-taught material (e.g. measure theory) supplement a course?

• Chen: Yes!

7. What is more important: passion or knowledge?

• Neslehova: Passion. It can motivate the learning experience, engagement, and involvement. 
• Vetta: It is correlated, if you have the passion you are more likely to get the knowledge. 

8. Have you supervised projects in other departments or majors?

• Vetta: Yes, [I have] supervised projects across several departments.
• Neslehova: [I had] a student in neuroscience who wanted to observe behaviour in epileptic seizures. We converged on a project and it was very enriching for me as a supervisor. 

9. Is there a minimum GPA needed for research?

• Vetta and Neslehova: No, absolutely not. But, for NSERC and SURA there are about 10 given out of each per year (very competitive) so there is definitely an advantage to having a high GPA (unfortunately the applications are limited, and there is not much for the committee to go off of). MATH 470 and 410 are good opportunities to do research without the summer grant. 

10. Is it possible to do research in the summer after you graduate?

• Vetta: Good jumping board onto a masters or PhD.
• Neslehova: There are restrictions on NSERC or SURA, so you would have to look into those.

11. DO you have recommendations on following a train of interest?

• Neslehova: Having taken a class that you really like, that is a good time to approach a professor about a research opportunity. 
• Vetta: If you have done a course that you like, and there is no higher level courses, that is a good time to take an independent study course (MATH 420 or 480)
• Lin: And you can try to do it in the Directed Reading Program!
• Chen: It is a valuable moment when you find something that you like. Talk to the professor and get more materials on it!

12. What is the difference between summer research verses project course?

• Chen: [I have] had summer students that continue their work into the fall semester as a project course. Don’t drop a topic, and continue pursuing it if you can.
• Lin: When I supervise 470 there is usually an expectation of a report due at the end. Same with the SURA and NSERC awards. If a professor is funding you individually, it can be different [expectations]. 
• Neslehova: SURA and NSERC are jobs. So you have to be there and they are paid. 410 and 470 are courses that you get a credit at the end. It all depends on the student [in the actual research that you do].
• Vetta: They are different in scale as well; MATH 410 and 470 are really small. You can’t get much done with three or four other courses. They are better when you combine them with a summer research project. 

13. What is the day to day during the summer? When are goals usually set?

• Lin: Probably varies professor to professor. I will meet with my group of undergrads about twice a week. There is an expectation of quite a bit of contact (if they have a question they can knock on my door). I encourage my students to LaTeX their progress throughout the summer. So, yes, I use a pretty regular schedule. 
• Chen: We work up a big-picture plan at the beginning of the summer. Have different meetings depending on what part of the project they are on. 

14. Is research more problem solving or theory? How often do students get results?

• Vetta: It really depends. Sometimes there is a paper that comes out of it. Hopefully you learn something at the end. 
• Neslehova: Typically there is always some sort of outcome. You always learn something. If there was something that you were trying to solve and you can’t solve it, at least there was some progress made.

15. How much can a student be expected to catch up on when starting a research project as opposed to contributing to research being done?

• Vetta: There is a learning phase, but you hope that they are contributing to research being done. 
• Neslehova: For most people these problems should be doable in the time that you have, sometimes the problem will turn out to be harder than expected, but in the end you will have learned quite a bit.