On this episode of our podcast My Favorite Theorem, Kevin Knudson and I were pleased to have the opportunity to talk with Moon Duchin, a mathematician at Tufts University. You can listen to the episode here or at kpknudson.com, where there is also a transcript.

Dr. Duchin has appointments not only in the math department but also in the college of civic life. For the past few years, she has been working on applying mathematics to the thorny issue of what voting districts should look like. She is a co-leader of the Metric Geometry and Gerrymandering Group, which studies computational and geometric aspects of redistricting and creates software others can use to draw voting districts and helps mathematicians learn how to serve as expert witnesses for court cases involving gerrymandering.

But Dr. Duchin’s favorite theorem is not about voting districts, at least not directly. She told us about Gromov’s gap, which is neither a mountain pass in the Urals nor a chasm in the Mines of Moria in Middle Earth. Instead, it is a surprising fact from the mathematical area of geometric group theory.

Geometric group theory is a branch of mathematics that unites geometry and algebra. A group in mathematics has more structure than the English word implies. It is a collection of numbers or other mathematical objects along with a rule for how to combine them. For example, the integers along with the operation of addition form a group. A priori, groups are abstract, but geometric group theorists explore ways they can be understood as more tangible geometric and topological objects or through the way they can act on geometric spaces.

Geometric group theorists have standard ways to visualize groups as graphs. In this case, we mean the kind of graph that is a collection of vertices connected by edges, not an illustration of a function. In the graph corresponding to the group, one can measure the relationship of area and perimeter. That relationship can be quantified (think of the way the area of a disc grows: the area changes proportionally to the square of the diameter), and it is always the number 1 or a rational number greater than 2. Numbers between 1 and 2 are not achievable. That chasm is Gromov’s gap.

If you're interested in digging deeper, here are a few places get started: Metric Spaces of Non-Positive Curvature by Martin Bridson and André Haefliger; Office Hours with a Geometric Group Theorist, edited by Matt Clay and Dan Margalit and featuring a chapter on hyperbolic groups by Dr. Duchin; and Statistical Hyperbolicity in Groups by Dr. Duchin, Samuel Lelièvre, and Christopher Mooney.

In each episode of the podcast, we ask our guest to pair their theorem with something. Dr. Duchin went with politics. You’ll have to listen to the episode to learn why they make such a good pairing.

You can find Dr. Duchin at her website and the Metric Geometry and Gerrymandering Group website. She recently appeared on the 538 politics podcast and gave a public lecture on political mathematics at the 2018 Joint Mathematics Meetings. 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 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: