Alan Turing on Stage and Screen

A statue of Alan Turing at Bletchley Park, by artist Stephan Kettle. Photograph: Sjoerd Ferwerda, via Wikimedia Commons.

It was a big week for Alan Turing dramatizations. On Monday, the UK and US trailers for the Turing biopic The Imitation Game, starring Benedict Cumberbatch, were released, and on Wednesday, the 102nd anniversary of Turing’s birth, the Pet Shop Boys premiered A Man from the Future, an opera they wrote about Alan Turing, at the BBC Proms. Both works are based on Andrew Hodges’ biography Alan Turing: The Enigma.

I heard about the movie first from Christian Perfect at the Aperiodical. It seems to focus on Turing’s contributions to cracking the enigma code. Here’s the US trailer.

And here’s the UK trailer.

This morning, James Grime, a Turing and Enigma machine expert, wrote about the trailers, also for the Aperiodical. Grime has made several videos about Turing and the Enigma machine himself (see here, here, and here). His post is quite thorough, and it will help me go into the movie with a little better idea of what is factual and what is embellished. Grime’s review is largely positive. He writes, “Sure there are inaccuracies, but I think that is forgivable in a dramatisation of events. I think the film will actually inspire people to find out more about Turing, Enigma, and the work at Bletchley Park.”

Jeffrey Bloomer wrote on Slate’s Outward blog that the trailer was “disgraceful” in the way it failed to portray Turing’s sexuality, saying it, “frames the movie as a wartime epic and romance between Turing and his contemporary, Joan Clarke (Keira Knightley). For viewers in-the-know about Turing’s sexuality, there are some coy allusions to what’s really going on (‘What if I don’t fancy her in that way?’). But it’s not long before we’re back to tender scenes of the photogenic couple in duress.”

I felt like this criticism was overblown. First, I think Turing’s homosexuality is better-known than Bloomer realizes, making the allusion not so coy to a large number of viewers, particularly in the UK. (I could be wrong about that, of course.) Second, while I hope the movie doesn’t “straightwash” Turing, I also hope his accomplishments, not his sexuality and persecution, are the focus of the film. In a two-minute trailer, I thought the amount of time spent alluding to his relationships sexuality was about right. And while his relationship with Clarke is probably overstated, the two were engaged for a short time, and we can hardly expect a major movie not to exploit that fact a bit. I’m curious how much the script has changed since Hodges criticized it on this matter last year.

I’m more concerned with another inaccuracy the film, pointed out by Grime: “The film seems to be setting up [Commander] Denniston as an antagonist to Turing, which is probably a great disservice to Denniston, who by all accounts understood the difficultly of the work, deliberately recruiting the professor type, and was proud of their achievements.” Although Grime says that others in the military may have been less supportive of the Bletchley Park work than Denniston, this seems like a fairly big problem. It changes the way I will watch the film more than the embellished romance with Clarke.

A Man from the Future, on the other hand, focuses as much on Turing’s sexuality as it does on his science. The opera is not nearly as plot-oriented as a movie. It features snippets of narration drawn largely from Hodges’ biography interwoven with choral songs. Bits of Morse code bubble up here and there in the music. When work on the opera began, the Queen had not yet pardoned Turing for his so-called “gross indecency.” At the time, Neil Tennant, half of the Pet Shop Boys, was quoted as saying, “Of course the reason they won’t pardon Alan Turing is because they’d have to pardon all those homosexual men.” Now Turing has been pardoned, but the others have not been. The end of the opera notes, “an exception was made. The convictions for gross indecency of tens of thousands of other men, dead, and alive, remain unpardoned.”

You can listen to the BBC Proms concert online for the next 29 days. A Man from the Future starts around the 37 minute mark. The Imitation Game will be showing up at some film festivals early this fall, but most of us will have to wait until November to see it. I’m looking forward to it!

Mathematician Presents Flawed Proof – in a work of fiction

Following Evelyn’s last post about the new Breakthrough Prize in Mathematics, I will now discuss the opposite of wild mathematical success.

Depending on how excited you are about public speaking, the moments before giving a talk at a math conference may be full of anticipation or anxiety.  So what happens when the speaker says something incorrect? At best, it’s embarrassing – like messing up in the middle of a recital in front of other musicians who know that music. What if the speaker presents an argument that is somehow fundamentally flawed? We talk a lot about how to handle students’ mistakes and how valuable they are – see Evelyn’s February blog post. But what about colleagues’ mistakes? How can we take it ins stride when we make mistakes in front of peers?

That’s one topic of the entertaining short story “The Penultimate Conjecture” by the late celebrated writer Leonard Michaels. One of a series of seven stories featuring fictional mathematician named Nachman, this story is read by Rebecca Curtis at the New Yorker’s monthly podcast posted on the first of this month. In just 45 minutes listeners can soak up some literature with a mathematical flavor and a dark sense of humor. According to Alex Kasman, who maintains the site Mathematical Fiction, this is the most mathematical of all of Michaels’ stories about Nachman, who, after attending fellow mathematician Lindquist’s much-anticipated presentation of the proof of the long-outstanding Penultimate Conjecture, realizes that the presenter’s proof is flawed. Should he say something to Lindquist? It’s not clear what he will do, and we share his uneasiness at being the messenger of bad news. Kasman files all the Nachman stories under the “anti-social mathematician” banner, but unlike the clichés in other stories about mathematicians, this one seems more true-to-life to me. Like the New Yorker’s fiction editor who chats with Ms. Curtis at the end of the reading, I am interested to know what the Ultimate Conjecture might be, and I tend to agree with this blogger that it is probably meant to be that of Nachman. However, one thing that isn’t discussed during the New Yorker podcast is the remaining possibility that Lindquist’s work will be fruitful in other ways besides what he aimed. My thoughts turn to the Math Overflow post concerning the Most Interesting Mistakes in Mathematics , in which many fascinating examples of mistakes made by preeminent mathematicians led to innovations. The most recent example mentioned concerns the Perko Pair, a pair of knots once thought to be non-isomorphic (due to a theorem that was later disproven), and later shown to be isomorphic (by a lawyer named Perko).

Most of us lack preeminence, and for the pessimists among us, the quality of famous mathematicians’ mistakes might just be a reason to save the contents of every Field’s medalist’s wastebasket. But for the optimists among us, it’s also encouraging to think that exercising ones intellect is bound to be fruitful even if it’s not in the manner intended. An interesting side note – in an effort to find a discussion about how mistakes during talks should be handled, I found pretty much nothing, which reinforces my belief that mathematicians are a very polite (or perhaps just confrontation-averse) bunch. I did, however, find a post on Math Overflow about how to correct mistakes in published work. Of course, step one is always to email the author. I also ran into this recent nice post by Orr Shalit at his blog Noncommutative Analysis, in which Shalit discusses how he handled the discovery of a mistake in a 16-year-old paper.  What are your thoughts on mistakes in our field? On mathematical literature?

The Inaugural Breakthrough Prizes in Mathematics

Image: Julie Rybarczyk, via flickr.

Last month, the inaugural Breakthrough Prizes in mathematics, founded and partially funded by internet billionaires Yuri Milner and Mark Zuckerberg, were awarded to five people: Simon Donaldson, Maxim Kontsevich, Jacob Lurie, Terence Tao, and Richard Taylor. The prize is $3 million per person, and the first five winners will be on the committee for the selection of future winners. (In the future, there will be only one prize awarded per year.) I was a bit surprised that there hasn’t been much talk on blogs about the prizes, but there has been a bit. Peter Woit wrote about the prize on Not Even Wrong, and the comments to his post are interesting. “Shecky Riemann” also has a post on Math-Frolic. I must admit that I am somewhat cynical about the prize. (Now might be a good time to reiterate the disclaimer that appears on the sidebar of this blog: my opinions do not necessarily reflect the opinions of the AMS.) The five winners are all productive, brilliant mathematicians who have enhanced their fields immensely, and they deserve to be recognized. But$3 million is just so much money! It’s hard for me to see how concentrating that much money in the hands of so few people is an efficient way to support mathematics.

Woit’s post voices some similar concerns. He writes,

“…it’s still debatable whether this is a good way to encourage mathematics research. The people chosen are already among the most highly rewarded in the subject, with all of them having very well-paid positions with few responsibilities beyond their research, as well as access to funding of research expenses. The argument for the prize is mainly that these sums of money will help make great mathematicians celebrities, and encourage the young to want to be like them. I can see this argument and why some people find it compelling. Personally though, I think our society in general and academia in particular is already suffering a great deal as it becomes more and more of a winner-take-all, celebrity-obsessed culture, with ever greater disparities in wealth, and this sort of prize just makes that worse. It’s encouraging to see that most of the prize winners have already announced intentions to redirect some of the prize moneys for a wider benefit to others and the rest of the field.”

In fact, the New York Times reports that Tao, one of the winners, has similar feelings:

“Dr. Tao tried to talk Mr. Milner out of it, and suggested that more prizes of smaller amounts might be more effective in supporting mathematics. ‘The size of the award, I think it’s ridiculous,’ he said. ‘I didn’t feel I was the most qualified for this prize.’

“But Dr. Tao added: ‘It’s his money. He can do whatever he wants with it.’

“Dr. Tao said he might use some of the prize money to help set up open-access mathematics journals, which would be available free to anyone, or for large-scale collaborative online efforts to solve important problems.

As a young academic who has seen postdoc positions seem to dry up since the beginning of the financial crisis, I can’t help but do a little arithmetic. $50,000 is a nice round salary for a postdoc. Before benefits are factored in, that makes each$3 million prize the equivalent of 60 postdoc years. Even if we add another $50,000 a year for health insurance, travel, and other research expenses, that money could fund 30 postdocs a year, or create 10 three-year postdoc positions each year. But 30 postdocs a year wouldn’t make a good press release. The New York Times wouldn’t write an article about their multimillion dollar minds. And the funders of the Breakthrough Prize want to encourage mathematical celebrity, which supposedly will lead to public awareness, not to fund worthwhile math research in the most efficient way possible. In a Scientific American article about the prize, Ben Fogelson writes, “Milner’s goal, however, is to increase the popularity of science by celebrating the scientists. ‘Dividing [money] in small pieces and distributing it widely has been tried before and it works,’ Milner says. ‘I think the idea behind this initiative is to really focus on raising public awareness.’” A commenter on Woit’s post suggested that each year, the prize money could be used to endow a research position at a university, noting that at MIT, you can endow a professorship for$3 million. Would that be high-profile enough? I think you could still write a press release about it!

I had some interesting discussions about the prize on Twitter after the prizewinners were announced, mainly focused on the utility of mathematical celebrity. Those discussions helped me frame a few questions about celebrity and public awareness. I’ve tried to figure out some analogous questions about movies and the Oscars because the Breakthrough Prizes have been described as the Oscars of science.

• Will people think mathematics is more valuable because a few people can earn giant prizes from it? (Do people think filmmaking is more valuable because the Oscars exist?)
• Will people want to become mathematicians because they think they could earn a big prize from it? (Do people become actors or filmmakers because they think they could win an Oscar?)
• If the ultimate goal of the prize is to raise public awareness of math, what is a more effective way to do that: tell them about a successful mathematician, or tell them about an idea in math? (If someone doesn’t know much about cinema, would it be more effective to tell them about an Oscar-winning actor or show them a movie?)
• Are these even the right questions and analogies?

This post might sound like I’m saying, “I don’t like this new prize because I’m never going to get it, but I would like it if it funded people more like me.” But I don’t think I’m quite there either. I have reservations about the suggested alternate uses of the prize money as well, thanks largely to two posts by Cathy O’Neil about billionaire money in mathematics and in academia in general.

On a lighter note, if you are a mathematician who is a bit embarrassed about a recent windfall, Persiflage suggests that “a bottle of Chateau d’Yquem 1967 does wonders to wash away any last remaining vestiges of embarrassment…”

As a college student in the ‘90’s with a penchant for “visual learning” I was never drawn to computer science. My one computer science class focused mostly on syntax and basic logic. Had shuffling and sorting been presented as eye-catching animations and colorful braids, I might have taken more CS classes. Mike Bostock presents such eye-catching images, specifically concerning four different goals: sampling of images, shuffling, sorting, and maze generation. For each goal, Bostock chooses several algorithms and explains how they each work visually. Even a brief look at the visualizations, some of which are animated and some static, gives a sense of the differences between the algorithms. As Bostock points out, visualizing algorithms is entertaining, makes understanding the algorithm more intuitive to others, helps in debugging your algorithm, and deepens your own personal understanding. In short, Bostock wants you to visualize not just data, but the algorithms used to analyze the data.  From a mathematical perspective, this is similar to insisting that we find ways to visualize the proof mechanism itself and not just the concept that it justifies.  In other words, we can draw a right triangle with appropriate labeling and call that a visualization of the pythagorean theorem, but it’s not nearly as instructive as this short animation or this geogebra applet.

Sampling is as ubiquitous in digital computing as the Pythagorean theorem is in geometry.  And while it is easy to understand as a concept, sampling is difficult to execute well. Bostock takes as an example an impressionistic painting. How can we choose samples in a way that will best digitally reproduce the original image? It turns out that we can learn form our own biology; the eye’s photoreceptors “sample” the world around us and are distributed in an irregular but uniformly dense manner. Such a sampling helps combat aliasing, a phenomenon whereby regularly spaced samples distort data with high levels of periodicity (Imagine a picture of a striped sweater that is sampled at a resolution lower than the spacing between stripes). This is where Poisson Disk sampling comes in. It is computationally more complex than other options, but produces a uniformly dense yet random sampling. As Bridson’s algorithm for Poisson Disk sampling plays, you immediately notice how the samples emanate from an initial seed rather than populating the entire sample space over time. This is one example of how visualization can helps us distinguish between algorithms.

Bostock works for the New York Times, and has many wonderful examples and posts on his site. Being a delinquent topologist, I was drawn to “How To Infer Topology”. Bostock is very interested in Cartography, and this leads to his writing about how to think about the data in a map more efficiently using a topological perspective. Several animations break down how one might encode a map by recording a sequence of arcs and special intersection points.

After a bit of perusal I realized that I had seen some of the work of one of Bostock’s collaborators, Jason Davies on Math Munch’s page in the past. Looking over Davies’ site, I can’t help but recommend that you take a look at his simple but beautiful graphic of a tiling of the Poincare disc. Zoom in and you can see the upper half plane model of the hyperbolic plane!

The Human Side of Computer Science

Charles Dickens, the inspiration for Dick Lipton and Ken Regan’s joint alter ego, Pip. Image: public domain, via Wikimedia commons.

Dick Lipton is a computer science professor at Georgia Tech who thinks P=NP, and Ken Regan is a computer science professor at the University of Buffalo who thinks P≠NP. Together, they are “Pip,” a Dick-Kens character.

Today I want to tell you about their blog, Gödel’s Lost Letter and P=NP.

Lipton and Regan write mainly about computer science, mathematics, and the history of those subjects, but they always put people first. Most posts start with a brief bio sketch of the person or people whose research they discuss, followed by a little background on the question they’re writing about and the technical content. In the “About” page, Lipton writes, “my real reason for making “who” central is to explain what researchers do when they work on hard open problems. They make mistakes, they go down dead-ends, they make progress. Also students may not know, but even experts sometimes forget, that ideas that we now see as easy were once unknown.”

The blog’s archives are extensive, and I’ve fallen down quite a few rabbit holes browsing through them. Rather than try to give any kind of a comprehensive overview, here’s an assorted (I dare not say “random”) collection of posts you might enjoy if you like creative, personal writing about some pretty technical topics.

I’m writing about this blog now because of how much I’ve enjoyed several of the most recent posts:
Avoiding Monsters and Non-Monsters, on Weierstrass’s “monsters” (continuous, non-differentiable functions) and some of their monstrous and non-monstrous friends
How to Avoid Leaving Clues, on how to have your memory and use it too
Is This a Proof? On the Weak Goldbach Conjecture and computer-aided proofs.
Proof Envy, on the theorems they wish they could use. “Call the phenomenon theorem envy: There are theorems that I have long known, but have never been able to use in any actual paper, in any proof of my own. I find this curiously unsatisfying, to know a cool result and yet be unable to use it in one of my papers.”
The Problem of Catching Chess Cheaters, on Regan’s recent work on the topic. Another good post on chess cheating is the Crown Game Affair.
In Praise of P=NP Proofs, on why reading P=NP proofs is a good idea, even if (so far) they’re all wrong.
I also stumbled on an entertaining post from 2010 about how well mathematicians have done at guessing whether conjectures are true.

Lipton and Regan suggested a few other posts to me, including some that are more computer-focused:
Perpetual Motion of the 21st Century, a double guest post from Gil Kalai, who does not think large-scale quantum computers are possible, and Aram Harrow, who does.
Quantum Chocolate Boxes, on using linear extensions to figure out whether your chocolate box is full of pralines or not.
Hedy Lamarr the Inventor, on amateurs in science and technology. In a similar vein, it takes guts to do research.
Finally, check out their Gödelinterviewseries, which gets more and more imaginative as it goes on.

Posted in Mathematics and Computing | | 2 Comments

When Life Hands You Lemons, Make Fibonacci Lemonade

I’m so glad I found Andrea Hawksley’s blog earlier this year! Hawksley is a software developer, mathematical artist, co-founder of the Octahedral Group, an organization of Bay Area mathematical artists. She works on the eleVR project, where she helps make 3-D VR videos for viewing on a spherical viewer. (They have a blog, too. I haven’t browsed it much yet, but it deals with the geometry and neuroscience of creating VR videos. Cool!)

Hawksley posts about recreational math and the mathematical art she and others make. I am indebted to her for posting a template and instructions for making a 6-card ball with pyritohedral symmetry group out of playing cards. I made one while my students were taking their final exam last semester. Here’s my finished product:

A 6-card ball I made using Andrea Hawksley’s template and instructions. Image: Evelyn Lamb.

The ball was her gift to attendees of the Gathering 4 Gardner conference, which she posted about here. If you’re like me, you’ll be swooning at the hair tie creations and the G4G gifts from other participants. You’ll also swoon over her other posts about mathematical art, particularly her “topological” origami.

Unukalhai, an origami sculpture in Andrea Hawksley’s Star Polyhedra series. Image: Andrea Hawksley.

Two of my favorites are her posts on non-Euclidean chess. What happens if we design “chutes” between random squares on the board? Could we handicap better players to make more interesting games? How does a bishop move if we tile our hyperbolic chessboard with squares that meet six to a vertex? How does a rook move on a hyperbolic chessboard tiled with pentagons that meet four to a vertex? It’s interesting to think about the ways that the game would be different with these different choices. I’d love for someone to write a program that plays chess on these boards to figure out how the strategy changes as we change the board.

Finally, one of Hawksley’s most recent posts is on making Fibonacci lemonade, a layered drink that gets sweeter as you go down the glass, with lemon to sugar ratio gradually approximating the Golden ratio. As she writes, “This drink may be the world’s first tastable example of the relationship between the Fibonacci sequence and the golden ratio!” It’s not bad to look at, either.

Having an industry job, I will not have any real change in my routine as summer hits. But I still think of summer as the season of reading for pleasure. So what are some new books out there that I’m thinking of reading, and where on the web can you find some excellent reviews of them?

1)   Love and Math by Edward Frenkel

• NY Times review by Leonard Mlodinow

2)   How Not to Be Wrong by Jordan Ellenberg—wasn’t on my list until today after having read the following two reviews:

3)   The Best Writing in Mathematics edited by Mircea Pitici includes an article by my academic sister, Kelly Delp, on Topology and Fashion — that’s reason enough for me to buy the book J.  This title has been a yearly publication of Princeton University Press since 2010.

4)   50 Visions of Mathematics

I’m drawn to any book that focuses on the visual beauty of mathematics.

5)   Beautiful Geometry by Eli Maor, Eugen Jost

Again, pretty pictures!!! This is coauthored by an artist and a mathematician.

6)   Doing Data Science by Cathy O’Neil and Rachel Schutt

• Revolution DataConsidering my current profession and the fact that blogger and Data Scientist Cathy O’Neil is one of the authors, it’s no wonder that this book is currently laying in my house…

7)   Our Mathematical Universe Max Tegmark

There are certainly no guarantees that I will get around to reading all of these—but in my experience, the first step to reaching a goal is setting it J. What’s on your summer reading list?

Fermi Estimation with Liquid Mercury Splash Fights

The semester is over (sorry, quarter system folks, but you can get your revenge in August and September), and you just want to put your feet up and surf the Internet. Of course, there are lots of ways you might accidentally learn something while you do that. One of them is reading the xkcd “what if?” blog by Randall Munroe. Of course, xkcd is a favorite comic for a lot of math nerd types. “What if?” takes the more data-driven side of the xkcd comic and runs with it, figuring out answers, or at least reasonable guesses, to bizarre questions Munroe’s readers ask.

My favorite “what if?” so far is about extreme boating. “What would it be like to navigate a rowboat through a lake of mercury? What about bromine? Liquid gallium? Liquid tungsten? Liquid nitrogen? Liquid helium?” I learned that liquid mercury may be the least dangerous of all the options (but still, you should not get into a splash fight on a liquid mercury lake), that aluminum absorbs gallium (to the detriment of the structural integrity of any aluminum boats on gallium lakes), and that tungsten has such a high melting point that it’s hard to study in liquid form because it would melt any container we put it in. For some reason, I find that hilarious.

Everybody jumping on Rhode Island. Image: Randall Munroe.

Or another gem: what happens if everyone on earth stands in Rhode Island and jumps at exactly the same time? Munroe does not stop with the physics, he goes on to the aftermath, the nightmarish logistics of having 7 billion people together in Rhode Island.

“Any two people who meet are unlikely to have a language in common, and almost nobody knows the area. The state becomes a patchwork chaos of coalescing and collapsing social hierarchies. Violence is common. Everybody is hungry and thirsty. Grocery stores are emptied. Fresh water is hard to come by and there’s no efficient system for distributing it.
Within weeks, Rhode Island is a graveyard of billions.”

There’s how fast we could drain the ocean, whether soda cans would be effective for carbon sequestration, and exactly how many world economies Au Bon Pain would need to pay a \$2,000,000,000,000,000,000,000,000,000,000,000,000 lawsuit it’s facing. (Answer: a lot.)

“What if?” is not exactly mathematical, although it does have a lot of numbers in it. But I see it as a great way to play with Fermi estimation problems, which often show up in physics and engineering. Munroe’s answers are more thoroughly researched than a “real” Fermi problem, but the big idea of making carefully chosen, well-thought-out simplifying assumptions is there. Along with a lot of interesting chemistry and physics trivia. I could see these problems as part of a general education math or physics class. Students could look at the questions, figure out their own strategies for solving them, and compare their solutions to Munroe’s.

Whether or not you want to use the posts deliberately to induce learning, they’re always entertaining. (They might even help you get a friend to like math!) If browsing through the blog archive is not enough for you, there’s a book coming in September.

Crowd-Funded Mathematics

1. Gary Antonick at the New York Times Numberplay blog recently featured Primo, a mathematical game designed by Dan Finkel, who blogs at Math For Love. The game is based off of thinking of prime factors as corresponding to different colors, allowing even younger children to play the game and learn basic operations as well as logical strategies for controlling their two pawns.
2. Similarly, the Moebius Noodles blog is hosting a crowdfunding campaign for Camp Logic, a book that introduces older children to logic via games and puzzles. You can preview the book for free, which is written by Mark Saul and Sian Zelbo from the Courant Institute’s Center for Mathematical Talent.

Seeing the success enjoyed by these campaigns so far made me think about how this could be a partial solution to the problems discussed by Cathy O’Neil at Mathbabe concerning the declining number of research projects funded by government funds. One example of a research project involving mathematics that seems to have engaged many individuals enough to garner their dollars is OpenWorm. This is a project that aims to create a digital worm from scratch by using scientists’ knowledge of the molecular structures within the worm. The idea would be that in the future, some research on an animal could be conducted by simply “downloading” the animal. By programming low-level interactions within the worm, the project organizers have seen the movements that one might expect arise “organically”. Of course all of this modeling requires a ton of mathematics. The model is open source so that anyone can view the code using GitHub.

Discovering Proofs

Patrick Stevens is an undergraduate mathematics student at the University of Cambridge, and I’ve really been enjoying his blog recently. He’s been doing a series of posts about discovering proofs of standard real analysis theorems. He writes that the series is “mostly intended so that I start finding the results intuitive – having once found a proof myself, I hope to be able to reproduce it without too much effort in the exam.” When I teach analysis, one of my main goals is for students to start developing their mathematical intuition, to learn how to “follow their noses.”

Hector the dog is following his nose, perhaps sniffing out a proof of the Heine-Borel theorem. Image: SaudS, via Wikimedia Commons.

It’s fun for me to watch Stevens follow his nose and figure out these proofs, especially because I’ve done most of them with my students recently. In addition to figuring out the proofs, Stevens also writes about figuring out statements of theorems themselves.

“A little while ago I set myself the exercise of stating and proving the Contraction Mapping Theorem. It turned out that I mis-stated it in three different aspects (“contraction”, “non-empty” and “complete”), but I was able to correct the statement because there were several points in the proof where it was very natural to do a certain thing (and where that thing turned out to rely on a correct statement of the theorem).”

I’m trying to figure out how to incorporate these posts into my analysis class the next time I teach it. My instinct is to make them recommended reading for my students, but I’m not sure the best way to make that an active learning moment for them, rather than just another time for them to watch someone else do math. Perhaps a writing assignment where they walk through the details like Stevens does would be better. Or I could suggest that they work along with Stevens and try to figure out what the next step will be. If they come up with different steps, it would be good for them to figure out how they are different and whether they are both valid ways to continue the proof.

In a post about making topology simpler, Stevens tackles the eternal confusion that the words “open” and “closed” create. I talked about that confusion on my other blog last September, and there is a pretty great video about it called Hitler Learns Topology. (I don’t usually like Hitler parody videos, but this one cracks me up.)

I’ve been reading Stevens’ blog for a while, and I would be remiss if I did not highlight my favorite of his posts so far, Slightly silly Sylow pseudo-sonnets. Yes, they are poems about the Sylow theorems. Here’s the first one:

“Suppose we have a finite group called G.
This group has size m times a power of p.
We choose m to have coprimality:
the power of p‘s the biggest it can be.
Then One: a subgroup of that size do we
assert exists. Two: conjugate are Sy-
low p-subgroups. And m‘s nought mod np
And np=1(modp); that’s Three.”

Posted in Math Education, Theoretical Mathematics | | 3 Comments