Thousands of years ago, humans crossed the temporary Bering Land Bridge to migrate from one side of the world to another. In the world of mathematics, Langlands Program ambitiously aims to build a permanent bridge between two distinct branches of mathematics: number theory and harmonic analysis. Mathematician Alex Kantorovich illustrates a wonderful map of the mathematical world. On one side of the mathematical world, there is number theory where theorists speak of integers, prime numbers, and rational numbers [Kontorovich]. This land is well known for its most renowned Pythagorean Theorem and attracts new middle schoolers each year. Although not as appealing to middle schoolers, elliptical curves are also prominent in number theory. Theorists like Marcus Banks closely study elliptic curves to find rational solutions to these “boomerang”-like curves which leaves mathematicians “frustrated” and “fascinated” [Banks]. On the other side of the mathematical world is harmonic analysis where the landscape depicts periodic phenomena like waves and repeating patterns [Kontorovich]. This land likes to deal with shapes and symmetry: take the trigonometric function  f(x) = cos(x) which is symmetrical when flipped along the y-axis and capable of infinite transformation variations [Cepelewicz]. Take one step further and we have modular forms similar to trigonometric functions “but on steroids” in the words of mathematician Ken Ono [Cepelewicz]. These modular forms use real numbers and complex numbers (“imaginary” ones). When analysts deal with complex numbers, they can represent modular forms as reference graphs full of colors in beautiful patterns and captivating symmetries [Cepelewicz].  

Although it seems like one side is engrossed in numbers while the other side is enthralled by periodic waves, mathematician Andrew Wiles has constructed the beginning framework of a bridge between these two lands: Wiles had proved a connection between elliptic curves and modular forms when dealing with rational numbers. He proved the Taniyama-Shimura-Weil conjecture which predicted that each modular form had a corresponding curve [Cepelewicz]. The consequence of this groundbreaking discovery allowed Wiles to indirectly prove the 358-year unproven theorem. Pierre de Fermat had left a marginal note in his copy of Diophantus’s Arithmetica commenting that “it is impossible […] for any number which is a power greater than the second to be written as the sum of two powers [x^n + y^n = z^n for n>2]. I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain” [Fermat]. In 1984, mathematician Gerhard Frey shared his rearrangement of Fermat’s Last Theorem: y^2 = x^3 + (A^N – B^N)x^2 – A^N B^N. This follows the form of elliptic equations, y^2 = x^2 +ax^2 + b^x+ c where a=A^N-B^N, b=0, and c = A^N B^N [Singh]. Frey’s elliptical equation “was so weird that it was not modular” [Singh]. If the Taniyama-Shimura-Weil conjecture was true and every elliptical curve “must be modular” except for Frey’s “weird” elliptical equation, which would have no modular form, it must mean there are no solutions to x^n + y^n = z^n for n>2, thus proving Fermat’s statement to be true [Singh]. After seven laborious years of research, Wiles had killed three birds with one stone in a 129 page long proof: he proved the Taniyama-Shimura-Weil conjecture, indirectly proved Fermat’s Last Theorem, and contributed to the world’s largest and most ambitious mathematics project. His proof contributed to Langland’s vision of a “Grand Unified Theory of Mathematics” with ever-expanding bridges stretching between the two mathematical lands [Crowell].

Works Cited:

Banks, Marcus. “Elliptic Curves: Simple Equations Still Shrouded in Mystery.” Simons Foundation, 19 Nov. 2021, www.simonsfoundation.org/2021/09/27/elliptic-curves-simple-equations-still-shrouded-in-mystery/.

Cepelewicz, Jordana, et al. “Behold Modular Forms, the ‘fifth Fundamental Operation’ of Math.” Quanta Magazine, 16 Mar. 2024, www.quantamagazine.org/behold-modular-forms-the-fifth-fundamental-operation-of-math-20230921/.  

Crowell, Rachel. “The Evolving Quest for a Grand Unified Theory of Mathematics.” Scientific American, Scientific American, 20 Feb. 2024, www.scientificamerican.com/article/the-evolving-quest-for-a-grand-unified-theory-of-mathematics/.  

“Fermat’s Last Margin Note.” Lapham’s Quarterly, www.laphamsquarterly.org/miscellany/fermats-last-margin-note. Accessed 8 Sept. 2024.

Kontorovich, Alex. “The Biggest Project in Modern Mathematics.” YouTube, YouTube, 1 June 2022, www.youtube.com/watch?v=_bJeKUosqoY.  

Singh, Simon. Fermat’s Last Theorem. HarperCollins Publishers, 2012.