Arick Shao 邵崇哲

Teaching / Miscellaneous

I am also occasionally involved in some teaching and outreach related projects outside of standard university courses.

Undergraduate Research Opportunities Programme (UROP), 2015

  • Location: Imperial College London
  • Term: Summer 2015

UROP is a program delivering summer research opportunities to undergraduate students. In summer 2015, I supervised a second-year undergraduate student, Lo Chun Hong, in a summer project. The main topic of the project is the application of harmonic analysis in partial differential equations, in particular the role of Fourier transforms and Littlewood-Paley theory in solving nonlinear wave equations.

The results of this project were later presented at the Warwick Imperial Autumn Meeting.

Math Mentor Program, 2012

  • Location: University of Toronto
  • Semester: Spring 2012

I served as a volunteer mentor in the Math Mentor Program at the University of Toronto. For this program, I met and worked regularly with two high school students, discussing various mathematical topics that are not part of standard high school curricula.

Below are some brief notes summarizing some of the topics discussed thus far, although a quick Google search would probably reveal about a hundred better write-ups on any of these topics.

  1. Cardinality: Counting the Size of Sets (.pdf)
  2. Set Theoretic Paradoxes (.pdf)
  3. The Need for Real Numbers (.pdf)
  4. Extending Calculus: Limits (.pdf)
  5. Solving Differential Equations (.pdf)
  6. Extending Calculus: Derivatives (.pdf)

Finally, we discussed the brachistochrone problem, which dates back several centuries and is often credited as the historical starting point of the calculus of variations. The brachistochrone problem provides an application for our discussions involving extending the notion of derivatives to "infinite-dimensional" settings.

The problem can be roughly posed as follows:

  • Suppose we are holding a ball at a "starting point" \(A\). Once we release the ball, it rolls down some curve to a "destination point" \(B\). What shape must this curve have, so that the ball rolls from \(A\) to \(B\) along this curve in the shortest possible time?

Most interestingly, the solution curve that minimizes the time traveled is not a straight line (which minimizes the length of the curve), but instead is an upside-down cycloid. This can be shown using calculus of variations methods, in particular by extending the notion of directional derivatives in multivariable calculus to infinite-dimensional spaces of curves. Roughly speaking, if a differentiable functional has an extremum at a "point"/function, then all of its directional derivatives, i.e., its "first variation", must also vanish at that "point"/function.

The final project was a poster, created by student Michael Chen, on the brachistochrone problem and its solution.

This poster was displayed at the annual Poster Session for the mentorship program. Congrats for a job well done!

Thanks to Daniel Egli for suggesting the brachistochrone problem as a possible topic of discussion.

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