This article was published in Scientific American’s former blog network and reflects the views of the author, not necessarily those of Scientific American
On this episode of My Favorite Theorem, I was happy to talk with Ami Radunskaya, a math professor at Pomona College and co-director of EDGE, a program that helps prepare women for graduate school in mathematics. We recorded the episode at MathFest, the summer meeting of the Mathematical Association of America. You can listen to the episode here or at kpknudson.com.
Dr. Radunskaya’s favorite theorem is the Birkhoff ergodic theorem. This theorem comes from dynamical systems, the study of how spaces change over time when certain transformations are applied. You can think of particles bouncing around inside a box or billiards careening around a pool table.
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In any of these transformations, there will be sets of points that are invariant; that is, any point in the set will go to another point in the set. An easy, though unsatisfying, example of an invariant set would be the entire space. If there were some kind of barrier built into the box of particles or pool table, the points on either side of that barrier would have to stay on their side, so each side would be an invariant set. These invariant sets sort of measure the mixiness of the function. A function in which the right side always stays on the right is less mixy than a function in which a point on one side can end up anywhere. A function is ergodic if it is very mixy. Of course, the mathematical definition does not contain the phrase “very mixy.” It is a bit more technical, saying that the only invariant sets are either very small or very large. (To
further confuse the issuemake everything more fun, mixing is also a technical term in dynamical systems, and it is not quite the same as ergodicity.)
The Birkhoff ergodic theorem says that ergodic functions have another useful property: the time average of the path of one point in the space is equal to the average of all the values in the space. A real-world example of this property might be something like: if any one person has a 1 in 500 chance of contracting a particular disease in a particular year, it’s also the case that about 1 in 500 people in the population will contract this disease in any given year. The “time average” of getting the disease is one person’s risk in a given year. The “space average” is the entire population’s rate of disease in the year.
I know just enough about ergodic theory to be dangerous, and as a result I spoiled the theorem in this episode! My working definition of the word ergodic is actually the characterization of the property from the Birkhoff ergodic theorem, and I opened my big mouth and jumped in with that description before Dr. Radunskaya actually got to it. How embarrassing! Dr. Radunskaya was very gracious despite my clumsiness, and we had a nice chat about ergodic theory and the Birkhoff ergodic theorem.
Dr. Radunskaya, who had a career as a cellist before going back to school to get her Ph.D. in math, chose to pair her theorem with Violin Phase, a composition by Steve Reich, and the Italian pasta dish paglia e fieno, or straw and hay. You’ll have to listen to the episode to hear why she thought they were the perfect accompaniments to the Birkhoff ergodic theorem.
I thought this was a particularly good episode of the podcast for the end of one year and beginning of the next. As we reflect on this year and think about our goals for next year, we can look to the ergodic theorem for guidance. As Dr. Radunskaya describes it, “You are the point, and you’re going around in your life. If your life is ergodic, and a lot of the time it is, it says that you’ll keep bumping into certain things more often than others. What are those things you’ll bump into more often? Well, the things that have higher measure for you, have higher meaning.” I want to make sure the things I keep bumping into are indeed the things that have the highest measure for me.
Happy New Year! May your life be ergodic in 2018!
You can find more information about the mathematicians and theorems featured in this podcast, along with other delightful mathematical treats, at kpknudson.com and here at Roots of Unity. A transcript is available here. You can subscribe and review the podcast on iTunes and other podcast delivery systems. We love to hear from our listeners, so please drop us a line at myfavoritetheorem@gmail.com. Kevin Knudson’s handle on Twitter is @niveknosdunk, and mine is @evelynjlamb. The show itself also has a Twitter feed: @myfavethm and a Facebook page. Join us next time to learn another fascinating piece of mathematics.
Previously on My Favorite Theorem:
Episode 0: Your hosts' favorite theorems Episode 1: Amie Wilkinson’s favorite theorem Episode 2: Dave Richeson's favorite theorem Episode 3: Emille Davie Lawrence's favorite theorem Episode 4: Jordan Ellenberg's favorite theorem Episode 5: Dusa McDuff's favorite theorem Episode 6: Eriko Hironaka's favorite theorem Episode 7: Henry Fowler's favorite theorem Episode 8: Justin Curry's favorite theorem