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The Most Addictive Theorem in Applied Mathematics

On this episode of our podcast My Favorite Theorem, my cohost Kevin Knudson and I had the privilege of talking with Erika Camacho, an applied mathematician at Arizona State University. You can listen to the episode here or at kpknudson.com, where there is also a transcript.

Arizona State University applied mathematician Erika Camacho. Credit: Erika Camacho

Dr. Camacho was inspired to go into applied mathematics when she did a summer research project modeling HIV as an undergraduate. (You can hear more about that in the interview I did with her for the Lathisms podcast series, which features conversations with Hispanic and Latinx mathematicians.) Since deciding to go into applied math, she has studied a few different topics, including the dynamics of how fanaticism spreads in communities and mathematical modeling of the eye. Currently one of her main research projects is on retinitis pigmentosa, which can cause blindness.

There’s one theorem Dr. Camacho says has been a steady companion in her work: the Hartman-Grobman theorem. This theorem is useful for understanding systems that are nonlinear. In mathematics, nonlinear basically means “hard to analyze;” linear systems respond proportionally to changes in variables, whereas nonlinear systems have more complicated relationships. A key way to understand nonlinear systems is to figure out where and when they can be well-approximated by linear systems. The Hartman-Grobman theorem does just that. As Dr. Camacho describes, this theorem applies to systems that have a certain type of equilibrium, or steady state. The theorem states that under certain conditions, those systems can be well-described by linear systems near the equilibrium point. And there is much rejoicing.

In each episode of the podcast, we ask our guest to pair their theorem with something: food, beverage, art, music, or some other delight in life. For this addictive, layered theorem, Dr. Camacho chose a special dessert. You’ll have to listen to the episode to hear why Tennessee whiskey cake is the perfect accompaniment to the Hartman-Grobman theorem.

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 [email protected] 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

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