Stepping up your Math Game – 35 Summer Projects for Post-Secondary Math Students

Build your math muscles!

This summer invest in your math education.

Undergraduate students often ask me for advice for how to improve themselves as mathematicians. There are many answers to this question: you can focus on learning new mathematics, you can work on your programming skills, you can improve your career prospects by working on professional development, …

This post is the result of many conversations and coffees with undergrads; It is the advice that I would have given myself as a second-year undergrad at the University of Winnipeg. These 35 projects will also provide necessary skills for completing a Ph.D. in Mathematics and will increase your employability.

Some of these projects will take an afternoon (“Learn how to make a bibliography in LaTeX”) and some might take a month (“Make a Predator-Prey Visualizer”). There is no time limit and no test, so take your time and make it your own.

Table of Contents

  1. Math Skills
  2. Reading
  3. Math Projects
  4. Teaching
  5. Professional Development
  6. Other Skills

[B] Beginners

Some projects are marked with a [B]. These are especially well-suited for beginners.

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Future of Set Theory – Panel Discussion

The following is a (somewhat complete) transcription of a panel discussion held at the Fields Institute on April 1, 2015 as part of the “Forcing and its Applications Retrospective” workshop.

The speakers were

  • Stevo Todorcevic (University of Toronto)
  • Jindrich Zapletal (University of Florida)
  • Christina Brech (University of S√£o Paulo)
  • Assaf Rinot (Bar-Ilan University)
  • Matteo Viale (University of Torino)
  • Justin Moore (Cornell University, Moderator)

I took notes, so please bear in mind that some of this is paraphrased. I apologize if I misrepresented anyone, and I am happy to make corrections if I am emailed.

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Succeeding in First-Year Calculus: My guides to MAT135 and MAT137

Last year I wrote a guide for students taking the University of Toronto’s big (2000 student) first year calculus class MAT135. It was so successful that I wrote another guide to MAT137, the more specialized first year calculus class. Let me share them with you:

How to Succeed in MAT135.”

How to Succeed in MAT137.”

They are both links to Reddit, but you don’t need an account there to read them.

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Delta-Epsilon Magic

In my second year of undergrad I had a formative experience with Delta-Epsilon proofs that stuck with me for a long time. Last week I was able to provide a similar experience for some first year calculus students.

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Measuring the (Seemingly) Impossible

(This talk was given as part of the Adventures in the Classroom lecture series at the University of Toronto on August 7, 2014. It aims to connect math educators with researchers in the academic math community. A video of this talk will be available later.)

We will introduce two fun projects involving Geodesy, which is devoted to measuring things to do with the Earth.

From Harold Llyod’s “Safety Last!”

“Without doing anything dangerous, measure the height of your school”

I love this question. It is a bit affronting (how could I possibly do that?!), but there’s something magical about it that draws you in and gets you using your imagination. If you have never thought about this question before, please take the time now. (Seriously, go for a walk and think about it.)

I have posed this question to many people and have received many different solutions. When I talk to people about this problem I am struck by what solution they think is the most “obvious” or “natural” solution. Think about that as we examine this problem.

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How to Read, Understand and Study Proofs

(This talk was given on March 31, 2014, at the University of Toronto to a class of mostly MAT 137 students. It was standing room only!)

In my first year of undergrad I was bad at proofs. In my second year of undergrad I was terrible at proofs. In my third year I was okay at proofs, but I was terrible at studying proofs.

The way I used to learn proofs was by memorizing the words in the textbook’s proof, word by word, with almost no understanding. I knew math, and I was fairly good at problems, but I just couldn’t get any purchase when it came to learning proofs.

Eventually I started to pick up various “tricks” and strategies for learning proofs. This talk is aimed at me in first year, and what I needed to hear so that I could have studied proofs better. (“I no proof good.”)

We’ll look at the basics of proof reading, the idea of definition unwinding and clever ideas, and finally we’ll present a general method for reading proofs.

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High-Dimensional Ramsey Theory – Course Notes – Spring 2014

I am currently taking Stevo Todorcevic’s course (MAT1435HS Topics in Geometric Topology: High-Dimensional Ramsey Theory) at the University of Toronto in Spring 2014. I will be typing up notes and posting them here. Please contact me (by commenting below, or by email) to give me any feedback (typos, questions, clarifications, etc.)

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Absolute Chaos is Impossible!

(This is a talk I presented for the Classroom Adventures lecture series on January 17, 2014, for high school teachers.)

(Apparently the video of this talk will be made available online. I will link it when it is available.)

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Contest Pigeons!

(This is a talk I gave for the Canadian IMO team at their 2014 winter camp at York University on Jan 3, 2014.)

The pigeonhole principle is a remarkable combinatorial theorem that looks silly and obvious, but turns out to be quite powerful and useful, especially in the context of contest problem solving. I’m going to present a couple of statements of the pigeonhole principle, then I’ll give some broad applications of it. I’ll end off with a list of problems.

Found them!

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Van der Waerden’s Theorem is false for $\omega_1$

One of my main research problems involves something I think is related to arithmetic progressions in $\Z$. After learning a nice proof of Van der Waerden’s Theorem, and sharing it with colleagues, Daniel Soukup asked a nice question about Van der Waerden’s Theorem on $\omega_1$. We answered it, and the example was sufficiently nice that I would like to share it.

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