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

One of my main research problems involves something I think is related to arithmetic progressions in \mathbb{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.

Continue reading Van der Waerden’s Theorem is false for \omega_1 

Ropes and Dwarfs in a Munich Subway

People in Germany like Math! After my recent trip to Bonn, Germany, I visited my friend Maria in Munich, Germany for a weekend. She was wonderful and showed me all around town. One thing about her that has always impressed me was how into mathematics she is. When I met her in Toronto last year, I distinctly remember thinking that her interest in mathematics (she is a PhD student in Classics) came from her German education. My trip to Munich only reinforced this.

Continue reading Ropes and Dwarfs in a Munich Subway

Secret Santa 4: The Surprise

Surprise!

After a long hiatus (7 months!) I am finally back to writing. This week I revisited an old problem that Sam Coskey told me a couple of years ago. Some of you will remember this “Secret Santa problem”, which I wrote about before. The problem is:

Sam’s Problem. Is it possible for two people to each choose a natural number so that both numbers are exactly 1 apart and neither person knows who has the larger of the two numbers?

Continue reading Secret Santa 4: The Surprise

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.

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

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.

Continue reading A Survey of Cardinal Invariants of Topological Groups

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!

Continue reading Kangaroo Contest 2013 Talk about Cryptography

Let’s Make Some T-sequences

(This was the basis for a talk I gave at the Toronto Student Set Theory and Topology seminar on January 15, 2013. The assumed knowledge is an undergraduate course in general topology. This is only a draft, and will be updated soon.)

Introduction

There are many questions in mathematics and sciences in general whose main object of study is the topological group. These objects are very versatile and can represent many of the structures we encounter. One question that I’ve been working on examines the (extreme) dynamical properties of topologies on the integers. On the recommendation of Vladimir Pestov (one of my advisors) I have been learning about T-sequences, which provide a rich method for producing topological groups with extreme behaviour. Here I will present two techniques involving T-sequences that help to answer two different questions about topological groups; one is about dynamics and the other is about combinatorial properties of \mathbb{N}^\N . These results all come from “Topologies on the Integers” by Protasov and Zelenyuk.

Continue reading Let’s Make Some T-sequences

Math Anxiety

The brain… in my head!

A study was done at the University of Chicago looking at whether math can cause physical pain. (The rather laughable title of the article is “Doing Math Really Does Make Your Head Hurt, Says Science”. Here‘s the actual published study.)

The idea is that for people with high levels of math anxiety, just knowing that they are about to do a math problem is enough to make the pain part of their brain go off. The actual solving of the problem wasn’t such a big deal. This suggests that presenting math is an important part of teaching it and discussing it.

Some other anecdotal examples are that people are notoriously good at calculating retail discounts – “It will be 12.50 for that 25 dollar shirt that is half off”. Compare this to the more difficult “What is 25 divided by 2?” or even worse “What is 25 x 0.5?”.

Similarly compare these two questions:

  1. After the half-hour 6 o’clock news you watch an episode of Jeopardy, then Survivor, then two episodes of Wheel of Fortune. You read until it’s a quarter to 10 and then go to bed. How long did you read before you went to bed?
  2. Here’s an easy question about fractions: Solve 6 + 1/2 + 1/2 + 1 + 2 (1/2) + R = 10 – 1/4.

I got a little bit anxious looking at that second question.

What experiences do you have about teaching people with math-anxiety?

Perceptions about Mathematics

“When someone asks you what you do as a mathematician, they take the most advanced training they have in mathematics and assume that you do a much harder version of that.”

This is a paraphrase of something I will attribute to Leo Goldmakher.

The idea is that if someone has only ever taken high-school math, they might think that mathematicians spend their time multiplying and dividing very large numbers; (“Are you even allowed to use calculators in your work?“). If someone has been lucky enough to take a first-year calculus course they might imagine that mathematicians solve harder and harder related rates questions. Going a step further, telling a mathematician that I study set theory often invokes brow-furrowing and questions like: “Don’t we already know everything about ordinals and cardinal arithmetic?“.