On a recent episode of our podcast My Favorite Theorem, my cohost Kevin Knudson and I were happy to have Ruthi Hortsch join us. You can listen to the episode here or at kpknudson.com, where there is also a transcript.
Hortsch is a mathematician and senior program manager at Bridge to Enter Advanced Mathematics (BEAM), a program that helps prepare students from underserved communities for future study and jobs in mathematics. On a personal note, BEAM is one of the math-related charities I support. They currently have programs in both New York City and Los Angeles, and they are working on expanding to more cities across the US. A substantial proportion of their students are Black, and during this time when people all over the country are thinking about how they can combat racism and inequality, supporting Black students through programs like BEAM is just one of many ways to step up.
Hortsch told us about Faltings’s theorem (also called the Mordell conjecture), which goes nicely with a past episode of My Favorite Theorem, when we talked with Matilde Lalín about the congruent number problem and Mordell’s theorem. Louis Mordell made the conjecture in the same 1922 paper in which he proved Mordell’s theorem, and Gerd Faltings proved the conjecture in 1983.
Both Mordell’s theorem and Mordell’s conjecture have to do with the rational points that appear on certain polynomial-defined shapes called algebraic curves. An algebraic curve is the zero set of a polynomial in two variables, something like the set of points where x2+y2=1, which you might recognize as the equation of a circle in the plane. To make things a little more complicated, in this branch of mathematics, the variables can take complex values, not just real values, so instead of a circle in the plane, we would actually need to go to four-dimensional space to fully visualize the shape.
But even though it’s simplified, the circle example can help us understand the basic question, which is how many rational points are on the curve. A rational point is a point with all rational coordinates, like the point (1, 1/2) in the plane or (−3, 2, 1/5) in three-dimensional space. For the equation x2+y2=1, there are many points with rational coordinates that satisfy the equation: (1, 0), (3/5, −4/5), and (5/13, 12/13) are three examples. But some curves only have a few rational points.
Algebraic curves can be classified by something called genus, which you can think of as how many holes are in the object (a donut has one and most pretzels have three). Mordell’s theorem describes an algebraic structure for rational points on curves that come from genus one curves, while Faltings’ theorem states that for higher-genus curves, there are only finitely many rational points.
On each episode of the podcast, we ask our guest to pair their theorem with food, beverage, or other delight in life. You’ll have to listen to the episode to hear why Hortsch thinks a New York City bagel is the perfect accompaniment to Faltings’ theorem. For more about Hortsch and BEAM, follow Hortsch on Twitter and check out the BEAM website.