In this web content, the author discusses the concept of dualities in mathematics, specifically the pairing of automorphic forms and Galois groups. These dualities allow mathematicians to study different classes of objects by examining them in terms of each other. While mathematicians have been working to prove Langlands’ hypothesized duality, only limited cases have been established so far. However, even these limited cases have produced remarkable results, such as Andrew Wiles’ proof of Fermat’s Last Theorem. The author also mentions the study of dualities between arithmetic objects related to, but distinct from, those that Langlands focused on. One particular duality examined by Sakellaridis and Venkatesh involves periods and L-functions, two seemingly unrelated mathematical objects. The author explains that periods and L-functions have emerged as objects of interest in mathematics due to their connections to various mathematical questions. Sakellaridis and Venkatesh developed a machine that, given a period, could compute an L-function. Although there were limitations to their approach, their work advanced the understanding of the relationship between periods and L-functions. However, they were unable to explain why certain periods yielded certain L-functions or determine which L-functions had associated periods. The author provides a helpful
https://www.quantamagazine.org/echoes-of-electromagnetism-found-in-number-theory-20231012/