On this very special Valentine’s Day episode of our podcast My Favorite Theorem, my cohost Kevin Knudson and I talked with Nikita Nikolaev and Beatriz Navarro Lameda, two mathematicians who recently married each other. You can listen to the episode here or at kpknudson.com, where there is also a transcript.
We heard about Nikolaev and Navarro Lameda last month when part of their wedding went a little viral. “Math-shaming” reads the beginning of a Reddit post. “Bride's wedding day math quiz for guests stirs debate online” says another headline. The couple designed a delightfully nerdy and thoughtful mathematical game for their wedding guests. Rest assured, despite what some Redditors seemed to assume, no one left the wedding hungry because they couldn’t solve a math problem. (And as a fact check, despite Yahoo’s headline and the fact that the post was in the “Bridezilla” subreddit, it was the groom, not the bride, whose Facebook post started the whole thing.)
The podcast is about theorems, not about weddings, and our guest chose the Intermediate Value Theorem as their joint favorite theorem. After listening to their case, I’m tempted to change my loyalty as well. The Intermediate Value Theorem states that if a continuous function has the value, say, 0 at the point a and 1 at the point b, then somewhere between a and b, it must assume the value 1/2, and somewhere between a and b, it must assume the value 1/3, and 1/π, and every other number between 0 and 1. (Of course, 0 and 1 aren’t special here. Whatever values the function takes on two endpoints, every intermediate value must be attained somewhere between them.)
As Nikolaev and Navarro Lameda describe it, the theorem seems completely obvious when you think about it. How could a continuous function avoid hitting everything between two of its values? But it ends up being extremely powerful. For an introduction to the Intermediate Value Theorem, click here.
As we discussed, it is something of a surprise that the Intermediate Value Theorem does not actually yield a definition of continuity. But there are non-continuous functions that satisfy the property that all intermediate values are attained. One example of such a function is the topologist’s sine curve (more precisely, the function defining the y-value of the topologist’s sine curve. Another even more exotic function, discontinuous at every point, is the Conway base 13 function.
We also talked about the relationship between the Intermediate Value Theorem and the least upper bound property, which you can read about here. Another theorem they mentioned that relies on the Intermediate Value Theorem is Sharkovskii's theorem. One version of that theorem is described as “period three implies chaos.” Sharkovskii’s theorem starts with a re-ordering of the positive whole numbers. Instead of 1,2,3,4,…they go 3,5,7…6,10,14,...,12,20,28,…. In this ordering, if a particular type of mathematical transformation has a period of length n, it has a period of length m for any number m that appears after n on the list. In particular, if it has a period of length 3, it has a period of all other lengths because 3 is the first number on the list.
In each episode of the podcast, we invite our guest (in this case, guests) to pair their theorem with something: food, beverage, art, music, or any other delight in life. You’ll have to listen to the episode to hear their Valentine’s Day-appropriate pairing.
Whether you are newlyweds, oldlyweds, happily single, or anywhere in between, we hope you find some theorem love in your life this Valentine’s Day.
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 to 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 firstname.lastname@example.org. 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
Episode 9: Ami Radunskaya's favorite theorem
Episode 10: Mohamed Omar's favorite theorem
Episode 11: Jeanne Clelland's favorite theorem
Episode 12: Candice Price's favorite theorem
Episode 13: Patrick Honner's favorite theorem
Episode 14: Laura Taalman's favorite theorem
Episode 15: Federico Ardila's favorite theorem
Episode 16: Jayadev Athreya's favorite theorem
Episode 17: Nalini Joshi's favorite theorem
Episode 18: John Urschel's favorite theorem
Episode 19: Emily Riehl's favorite theorem
Episode 20: Francis Su's favorite theorem
Episode 21: Jana Rordiguez Hertz's favorite theorem
Episode 22: Ken Ribet's favorite theorem
Episode 23: Ingrid Daubechies's favorite theorem
Episode 24: Vidit Nanda's favorite theorem
Episode 25: Holly Krieger's favorite theorem
Episode 26: Erika Camacho's favorite theorem
Episode 27: James Tanton's favorite theorem
Episode 28: Chawne Kimber's favorite theorem
Episode 29: Mike Lawler's favorite theorem
Episode 30: Katie Steckles' favorite theorem
Episode 31: Yen Duong's favorite theorem
Episode 32: Anil Venkatesh's favorite theorem
Episode 33: Michèle Audin's favorite theorem
Episode 34: Skip Garibaldi's Favorite Theorem
Episode 35: Nira Chamberlain's Favorite Theorem