Yesterday Steven Strogatz posted links to the Breakthrough Prize lectures given by Jacob Lurie, Terry Tao, Maxim Kontsevich, Simon Donaldson, and Richard Taylor. I had a chance to watch the Lurie and Tao lectures and they were excellent. This morning I talked through the first 10 minutes of Lurie’s lecture with my kids. All of the links to the lectures are in this blog post from yesterday:
The inspiration for today’s Family Math talk, Jacob Lurie’s lecture, is here:
I began today’s talk by following Lurie directly – looking at number systems by starting with the counting numbers and expanding out to the complex numbers. Both kids have some familiarity with these different number systems, but it was still nice to talk through examples of each type of number. Then (again following Lurie) I gave some examples of equations that cannot be solved in each number system:
Next we moved on to talking (very simply) about Emmy Noether and rings. The only detail I went into about a ring is that there was some way to add and some way to multiply. We showed how you could both add and multiply in the number systems that we used in the previous video. We then talked for a little bit about clock arithmetic. Explaining multiplication in this setting confused the kids a little, but they eventually got it. Viewing this again just now, I wish that I would have taken a slightly different approach in explaining multiplication, but maybe this extra bit of discussion was still good anyway.
The next discussion used clock arithmetic to show that the equation has no solutions where both and are integers. This is a great example for kids of the power of modular arithmetic from Lurie. We started by just playing around trying to find a few solutions, but we were not able to find any. However, by just looking at the cases where and are odd and even, we can see that there cannot be integer solutions. Amazing! Lurie mentions this point quickly in his talk, but it was fun to dive into it in a little more detail with the boys:
I thought it would be fun to try a second modular arithmetic example just to show something that wasn’t in Lurie’s talk. I chose to show them that in a right triangle with integer sides, one of the side lengths is always a multiple of three. One of the reasons that I wanted to give this example is to make a connection with geometry. Another reason was that a few of the Pythagorean triples were already familiar (to my older son, mostly) and I wanted to show him that these simple ideas in Lurie’s talk allow us to see some structure that we may not have noticed before. We approached this problem by looking at a clock with three “hours.”
One really fun thing that came up in this part of our talk was that my younger son noticed another possible pattern in the Pythagorean triples that we’d written down – each one contained a prime number. He asked if one of the lengths in a Pythagorean right triangle would have to be a prime number. Good question!!
Finally we looked at the theorem of Fermat that Lurie mentions: every prime number of the form can be written as the sum of two squares. Reviewing the proof of this theorem last night, I decided it would be too difficult to prove to the kids. However, proving a slightly different theorem was quite a bit easier: No prime number of the form can be written as the sum of two squares. This idea is another nice example of the power of modular arithmetic. We first looked a few examples of both types of prime numbers and then moved on to the proof of the easier theorem. At the end of the video we looked at a few larger primes of the form and found how they could be written as the sum of two squares.
I’m really excited to see that the Breakthrough Prize lectures have been made public, and I love that parts of these lectures can be made accessible to kids. These lectures are a great way for kids (and anyone, really) to learn about some of the most interesting ideas in mathematics from some of the world’s top mathematicians. What an amazing opportunity!