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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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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Using Dushnik-Miller to prove that every sigma-compact group is ccc

(This is the write-up for a talk I gave in the Toronto Student Set Theory and Topology seminar on May 2, 2013.)

A couple of weeks ago I gave a talk for the set-theoretic topology course I was in, on the topic of cardinal invariants of topological groups. While I was preparing that presentation I discovered the following fact:

Theorem [Tkachenko, 1983] Every \sigma -compact group is ccc.

I will present a proof that I have adapted from Tkachenko’s original paper (“Souslin property in free topological groups on bicompacta”) and the proof that appears in Arhangel’skii & Tkachenko’s big purple book (Section 5.3 of Topological Groups and Related Structures). Both proofs involve first proving a Ramsey result about covers of a space, then using this to prove that a particular space has “weak-precalibre \aleph_1 ” (i.e. Property K) which is a property that implies ccc. Learning this proof has been part of my ongoing attempt to learn how Ramsey results show up in topology.

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A Survey of Cardinal Invariants of Topological Groups

(This is a presentation I gave for Bill Weiss’ course Set-Theoretic Topology on April 19, 2013. In class we discussed some cardinal invariants and how they are related; here I will survey what happens when we look at the cardinal invariants of topological groups.)

This review follows very closely the discussion in section 5.2 of Arhangel’skii and Tkachenko’s book “Topological Groups and Related Structures“. Another good resource is section 3 of Comfort’s article “Topological Groups” in the Handbook of Set-Theoretic Topology. The only thing I claim to be my own are the (unsourced) pictures I have provided.

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Kangaroo Contest 2013 Talk about Cryptography

On Sunday March 24, 2013, I gave a talk on the History of Cryptography [PDF], at the University of Toronto (Scarborough) for the parents of students writing the Kangaroo Contest. I had many questions after my talk, so here are some answers to the questions I received.

 

Where did you get this information?

Most of this talk came from Elementary Number Theory by David M. Burton, the Wikipedia article for RSA, and the Wikipedia article for Diffie-Hellman. As a general rule of thumb, Wikipedia is a reliable source for things of a mathematical nature (as only experts tend to edit the articles).

My child is interested in codes, what are some resources for them to learn more?

Here is a great introduction to modular arithmetic which serves as the foundation for learning about the math of cryptography. Modular arithmetic is like “clock math”, where 4 hours after 10 o’clock is 2 o’clock.

Codecademy is a very good way to start learning computer programming. It is a very fun website and is very motivating, and fun!

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