DiM Research Panel PART II – Professors (Winter 2023)

Intro + 5 min Research Presentations

1. Elliot Paquette (Probability, Random Matrix Theory)

• Very awesome drawings
• Combination of linear algebra and probability theory. Putting probability distributions onto matrices and studying the effect of the eigenvalues!

2. Jessica Lin (Stochastic Homogenization, Elliptic and Parabolic Nondivergence Form Equations, Reaction-Diffusion Equations, Partial Differential Equations, Probability)

• She dedicated her slides to us! So cool! Thank you!
• PDEs are differential equations with partial derivatives, models many physical phenomena and probability. She likes to prove theorems about the behaviour of solutions, wanting to develop more robust theories/theorems. Those will help us with better understanding and more accurate models.
• Looking for in summer undergrads:
  • Strong background in Real Analysis. Probability/PDEs are great but not essential
  • Willingness to learn background, patience. Collaborative attitude
  • If not this summer, then maybe in the future! So, reach out!

3. Brent Pym (Algebra, Geometry, Mathematical Physics)

• He said “Most abstract minded person on the panel today” but his background is in physics
• Works on quantization:
  • Link between classical and quantum mechanics, which is essentially all pure math.
  • Poisson manifolds are really cool and relate to abstracting classical physics via geometry.
  • Noncommutative algebra helps in quantum physics (order of measurement changes the value). These two are linked via semi-classical approximation.
  • Understand algebra given structure of poisson brackets, used to help with quantization problems.
• Research group: using geometry as a tool to understand lots of cool things.
• Get in touch by emailing CV and transcript, but he sticks to departmental deadlines! Which apparently is tomorrow, February 3rd. But apply in the future!

4. Sergey Norin (Graph theory and combinatorics, Discrete Mathematics)

• Graphs model networks, basically everything is a network, and they’re fairly simple structures. Rich theory!
• Example problem: burning a graph. At every step, we start a fire at some vertex, which spreads to the neighbours. Want to know which connected graph on n vertices needs the most turns to burn up? This is an open conjecture!
• What can we say about the structure of large graphs by sampling sub configurations and analysing the statistics?

5. Louigi Addario-Berry (Applied Mathematics, Discrete Mathematics, Probability, Statistics)

• Very cool photo on a random discrete structure!
• Research on random graphs, interested in patterns such as the largest connected cluster and how changing probabilities induces sharp phase transitions (change in behaviour). This has relations to random fractal structures, which have applications/inspiration in rust formation, random minimum spanning tree, etc
• Vague overview of how research works with him!
  • Winter: brainstorm bunch of research problems and readings w students,
  • Spring/early summer: code of conduct with students to be generous/kind/respect, students read and present problems, working groups form.
  • Summer is teamwork, meetings, mini courses and more!

6. Rustum Choksi (Applied Mathematics)

• Euler follower, everything is optimization → applied math
• Research group: energy driven pattern formation. A lot about patterns and functionals (functions of functions). Many applications in physics and biology, as well as image processing/data science.
• Common math themes/structures: Nonconvexity & Nonlocality
• More recently his research focuses on addressing specific questions from a scientific problem, allowing him to interact with many different mathematical communities and those outside math!
• Had some awesome examples of summer students doing great research; he agrees on the collaboration being a really good quality.

Research Q&A

7. What background do you look for in undergraduates?

• Elliot:
  • probability. Maybe also some CS (lots of computer exploration opportunities) or also more analytical methods so probability/analysis ≥ 2
• Brent:
  • foundations on geometry and topology, which he acknowledges aren’t undergrad courses, but lots of undergrads have coding experience and can help develop and explore research problems. Knowing some abstract algebra (rings) and also vector calculus!!
• Rustum:
  • students need appreciation for rigour in math (correctness). Depending on field, analysis/PDEs, and a willingness to code (not necessarily experience) in Python. Openness to read lots of things and will to explore and understand them
• Louigi:
  • depends on the year for what background he looks for. Commonly joint math & cs students because he does a lot of algorithmic approach (not coding though). Collaborative attitude is important
• Sergey:
  • basic knowledge of graph theory. He has lots of problems and can fit most backgrounds. Programming is a plus but many problems don’t require it. Desire to learn is the most important thing.

8. What can undergrads expect in terms of level of difficulty of research?

• Brent: doing research is very different from homework problems. There are open problems and nobody knows the answer. Part of the art is realising if you’ve asked the question in the wrong way. Need mindset to explore creatively and not just problem solve. Be okay with failure.
• “I fail to prove theorems every day without fail”
• Rustum: success of the summer / how much you get out of it is up to you (the student). You need to be quite motivated and want to think about the problems and discuss them. That is hard, can be frustrating (esp at beginning), but that is ok, you’re learning! Have a growth mindset. It’s in your hands!

9. Difference between U2/U3: is the extra year worth it? Can we apply as U3s?

• Students can work summer after graduation (U3+). U2s can do research! It depends on the prof, difficulty of the problem, etc. SURA / NSERC prefers U2 but U3 still could get it.
• Only way to find out is to get in touch! Brent advises to not be afraid to put yourself out there! Other Profs nod.

10. Expectations of how much should be student learning versus doing new “useful” things for a summer research project?

• It’s very okay to just be learning! Summer projects should have some student development (i.e. learning). It’s rare for undergraduates to get a publication, so it’s not an expectation, possible if they are very motivated. Sometimes these projects turn you into their masters’ students!
• Louigi: If students learn some math and leave feeling not driven away from math, he considers it a win. He wants them to learn and enjoy mostly, not expecting publication. This is why he brainstorms many problems and has students work together, then the experience is much more fruitful and equitable.
• Rustum: most profs don’t expect undergrad publication! Turning a project into a publication is mostly the student’s agency. Excellent way to continue is Honours research project after summer
• Jessica: projects often aren’t confined to one summer, where they continue the project / topic (as a publication or thesis)!