In a 1967 letter to the number theorist André Weil, a 30-year-old mathematician named Robert Langlands outlined striking conjectures that predicted a correspondence between two objects from completely different fields of math. The Langlands program was born. Today, it's one of the most ambitious mathematical feats ever attempted. Its symmetries imply deep, powerful and beautiful connections between the most important branches of mathematics. Many mathematicians agree that it has the potential to solve some of math's most intractable problems, in time, becoming a kind of “grand unified theory of mathematics," as the mathematician Edward Frenkel has described it. In a new video explainer, Rutgers University mathematician Alex Kontorovich takes us on a journey through the continents of mathematics to learn about the awe-inspiring symmetries at the heart of the Langlands program, including how Andrew Wiles solved Fermat's Last Theorem. Read more at Quanta Magazine: https://www.quantamagazine.org/what-is-the-langlands-program-20220531/ 00:00 A map of the mathematical world 00:25 The land of Number Theory" 00:39 The continent of Harmonic Analysis 01:20 A bridge: the Langlands Program 01:46 Robert Langlands' conjectures link the two worlds 02:40 Ramanujan Discriminant Function 03:00 Modular Forms 04:36 Pierre Deligne's proof of Ramanujan's conjecture 04:47 Functoriality 05:03 Pierre De Fermat's Last Theorem 06:13 Andrew Wiles builds a bridge 06:30 Elliptic curves 07:07 Modular arithmetic 08:56 Infinite power series 09:20 Taniyama - Shimura - Weil conjecture 10:40 Frey's counterexample to Frey's last theorem 11:30 Wiles' proof of Fermat's Last Theorem - VISIT our Website: https://www.quantamagazine.org/ - LIKE us on Facebook: https://www.facebook.com/QuantaNews - FOLLOW us Twitter: https://twitter.com/QuantaMagazine Quanta Magazine is an editorially independent publication supported by the Simons Foundation https://www.simonsfoundation.org/