It was a productive holiday break for me. I launched my Patreon account for my podcast The Cartesian Cafe to enable fans to get additional benefits should they want to support the endeavor. One notable feature I’d like to give a shout out to is being able to provide content requests before episode recordings. Given that some of my guests will be quite prominent figures that will provide a rare in-depth talk on infinitely rich subjects, it’d be good to “open the floor” to what people want to hear.
Two days ago I also released my 8th episode of The Cartesian Cafe with the amazing Greg Yang. His biographical story is the stuff of legends – you’ll have to listen to find out why. The final production spans three hours of us talking about Greg’s work on Tensor Programs, which in addition to being a powerful mathematical theory, has valuable applications to the rigorous understanding of neural networks. It was quite a grueling session to record: food break and nap break over the course of 5+ hours. Lots of fun nevertheless!
I recently recorded a nearly 4 hour long conversation with theoretical computer scientist Scott Aaronson, a leading figure in quantum computing and also a famed blogger. We spent most (a lot!) of the time talking about quantum computing, complexity theory, and quantum supremacy, but during the biographical introduction we had a long digression about what Scott and I have in common: refuting “theories of everything”. In Scott’s case, it was Wolfram’s proposal and in my case, Eric Weinstein’s. So I ended up chopping our conversation into two parts: the main body which became my most recent podcast episode for the Cartesian Cafe
and a 24 minute digression of our catching up over the details and controversies involved with our refutations.
You’ve seen the quadratic formula but did you know there is no quintic formula for solving degree 5 polynomials? Join me in learning why this classical result is true at the Cartesian Cafe with the one and only Grant Sanderson:
Grant Sanderson is a mathematician who is the author of the YouTube channel “3Blue1Brown”, viewed by millions for its beautiful blend of visual animation and mathematical pedagogy. His channel covers a wide range of mathematical topics, which to name a few include calculus, quaternions, epidemic modeling, and artificial neural networks. Grant received his bachelor’s degree in mathematics from Stanford University and has worked with a variety of mathematics educators and outlets, including Khan Academy, The Art of Problem Solving, MIT OpenCourseWare, Numberphile, and Quanta Magazine.
In this episode, we discuss the famous unsolvability of quintic polynomials: there exists no formula, consisting only of finitely many arithmetic operations and radicals, for expressing the roots of a general fifth degree polynomial in terms of the polynomial’s coefficients. The standard proof that is taught in abstract algebra courses uses the machinery of Galois theory. Instead of following that route, Grant and I proceed in barebones style along (somewhat) historical lines by first solving quadratics, cubics, and quartics. Along the way, we present the insights obtained by Lagrange that motivate a very natural combinatorial question, which contains the germs of modern group theory and Galois theory and whose answer suggests that the quintic is unsolvable (later confirmed through the work of Abel and Galois). We end with some informal discussions about Abel’s proof and the topological proof due to Vladimir Arnold.