[note: home sick with some stomach bug for the last two days – sorry for what is surely a bit of a rambling post]
Ed Frenkel published a nice piece in the New York Times today on the life and work of Alexander Grothendieck.
In addition to Frenkel’s perspective on Grothendieck, what caught my attention was an almost off-hand observation about complex numbers that is really fascinating. I know it would have been quite a head scratcher for me in high school so I thought it would be fun to write about. Here’s the comment about the equation :
“One can show that the solutions of  in complex numbers are points of an entirely different space; namely, a plane with one point removed.”
Students familiar with the equation probably have only thought about this equation when both of the variables and are real numbers (when the solution is the familiar unit circle). The extension to complex numbers is a nice mathematical surprise.
So how can you think about Frenkel’s example? An excellent starting point is Richard Rusczyk’s sample solution for problem #25 of the 2013 AMC 12. The video below is a great way for students to see the power of geometric reasoning with complex numbers:
An approach similar to what Rusczyk outlines above is also a good way to start thinking about Frenkel’s equation. Try a few examples first – if for example, what values of will satisfy the equation (remember that both and are complex numbers)?
Now, if you have a generic value of , what values of will solve the equation? You’ll find that there are 2 values of for most values of though importantly, not all.
Next is a real geometric leap – if every point in the complex plane paired with exactly two points in Frenkel’s equation, seems as though the solution to the equation would be equivalent to two copies of the complex plane (possibly glued together in some strange way). Though it is challenging for sure, it is fun to think about what’s different from the situation I just described – in what way is the situation Frenkel describes similar to a plane with a point missing?
Away from this fun example of geometry with complex numbers, it was nice to see Grothendieck’s work described to the public. Another recent article about mathematicians written for the public was Michael Harris’ piece in Slate about the Breakthrough Prizes in math:
One of Harris’ points caught me off guard:
“Tao—the only math laureate with any social media presence (29,000-plus followers on Google Plus)—was a guest on The Colbert Report a few days after the ceremony. He is articulate, attractive, and the only one of the five who has done work that can be made accessible to Colbert’s audience in a six-minute segment.”
I was surprised to hear that Harris thought that the work of Jacob Lurie, Richard Taylor, Maxim Kontsevich, and Simon Donaldson really could not be made accessible to the public. Surprised enough, actually, to ask Jordan Ellenberg on twitter if he agreed with the statement:
Though his answer was not really a shock, it still disappoints me a little that work of these researchers is so inaccessible to the general public. Hopefully Frenkel, or other mathematics writers, can find a way to bring the beauty of their work to the public. I’d love to know more about Lurie’s work, or any of their work, frankly.
More public lectures like the one Terry Tao gave at the Museum of Math would be great, too. I’ve already done three projects with my kids already based on that lecture. It is amazing for them to be able to learn from Terry Tao!
It would be wonderful if there were more opportunities like Tao’s public lecture to introduce kids to research mathematicians and more article’s like Frenkel’s, too. Despite being home sick, Frenkel’s article made my day today.