Summary of The Biggest Project in Modern Mathematics

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The Langlands Program is a grand unified theory of mathematics that connects two disparate continents of mathematics - number theory and harmonic analysis. It was developed by Robert Langlands in the 1960s and has since been used to solve difficult problems in both fields. Two mathematicians, Srinivasa Ramanujan and Pierre Delign, are credited with making significant contributions to the theory.

• 00:00:00 The Langlands Program is a grand unified theory of mathematics that connects two disparate continents of mathematics - number theory and harmonic analysis. It was developed by Robert Langlands in the 1960s and has since been used to solve difficult problems in both fields. Two mathematicians, Srinivasa Ramanujan and Pierre Delign, are credited with making significant contributions to the theory.
• 00:05:00 In 1637, the French lawyer turned hobbyist mathematician Pierre de Fermat scribbled an equation in his copy of Diophantus's Arithmetica. Fermat never offered a proof, claiming that it was too marvelous for the narrow margins to contain. The theorem featured a polynomial equation, a fundamental object native to the continent of number theory. These are some of the most basic equations you can write down. Just variables, with exponents, that are positive whole numbers. You may remember the Pythagorean theorem from school. The theorem was proved in the 1990s by Princeton mathematician Andrew Wiles. He used a tool from modular arithmetic to find solutions to the equation. The elliptic curve graphed on the cartesian plane shows all the pairs of real numbers that satisfy the equation, but we're also interested in the rational and integer solutions. To study these solutions, it's useful to consider the elliptic curve from another perspective. Using modular arithmetic, we can count the solutions to the equation. The graph of all of the modular solutions is displayed. The elliptic curve has 24 solutions when n equals 31.
• 00:10:00 The Langlands Program is a vast and ambitious project aimed at solving some of the most difficult problems in mathematics. German mathematician Gerhard Frey made a key connection between Fermat's last theorem and elliptic curves, which proved that every elliptic curve produces an infinite power series which is modular. This proved that Fermat's last theorem cannot have a solution, and thus Fermat's equation is solved.