A Rosetta Stone for Mathematics | Quanta Magazine

🌈 Abstract

The article discusses the significance of André Weil's 1940 letter, in which he proposed a "Rosetta stone" for mathematics that linked number theory, geometry, and the study of finite fields. Weil's idea presaged the Langlands program, a major initiative in contemporary mathematical research, and has guided progress in connecting disparate fields of mathematics.

🙋 Q&A

[01] Introduction

1. What was the context in which André Weil wrote the letter described in the article?

• Weil was serving time in a jailhouse in Rouen, France in 1940 for refusing to join the French army.
• He filled his days by writing letters to his sister, Simone, an accomplished philosopher living in London.

2. What was the main idea Weil proposed in his letter?

• Weil sketched his idea for a "Rosetta stone" for mathematics, linking three fields: number theory, geometry, and the study of finite fields.
• He aimed to find connections and translations between these seemingly disparate mathematical domains.

3. How did Weil's idea relate to the famous Rosetta Stone engraving?

• Just as the Rosetta Stone made ancient Egyptian writing legible to Western readers through translation into Ancient Greek, Weil's Rosetta stone sought to link and translate between different areas of mathematics.

[02] Number Theory and Geometry

1. What is the central concern of number theory, and what tools do number theorists use?

• Number theory focuses on the integers and functions that rely on them.
• Number theorists try to prove results about the distribution of prime numbers and study mathematical worlds called number fields.
• They use tools from various esoteric branches of mathematics.

2. How are geometric shapes related to complex numbers and polynomial equations?

• Geometric shapes like spheres, doughnuts, and pretzels are the solution sets of certain polynomial equations with two variables.
• These shapes can be represented using "complex" numbers, which have both a real and an imaginary part.
• The structure of these shapes allows for the use of techniques from complex analysis, a form of calculus.

3. What was the motivation for 19th-century mathematicians to connect geometry and number theory?

• They wanted to prove theorems about "Riemann surfaces" (the geometric shapes Weil was interested in) and then translate those theorems into number theory.
• However, Weil acknowledged that the theory of Riemann surfaces was "too far from the theory of numbers" and a "bridge" was needed between the two.

[03] Finite Fields

1. What are finite fields, and how do they relate number theory and geometry?

• Finite fields are small number systems that resemble the real numbers, but with a prime number of elements.
• Over finite fields, polynomials and whole numbers can both be represented as strings of zeros and ones, allowing for connections between number theory and geometry.
• Finite fields are a place where number theory and geometry begin to blend.

2. How did Weil see the relationship between finite fields and number fields?

• Weil declared that "the analogy with number fields is so strict and obvious that there is neither an argument nor a result in arithmetic that cannot be translated almost word for word to the function [or finite] field."
• He believed that while the distance between Riemann surfaces and finite fields was greater, "a patient study would not teach us the art of passing from one to the other."

3. What was Weil's grand ambition in deciphering his "Rosetta stone"?

• Weil saw his work as "deciphering a trilingual text" where he had only "disparate fragments" of each of the three columns (number theory, geometry, and finite fields).
• He aimed to find the "great differences in meaning from one column to another" and develop precise methods to connect these seemingly disparate mathematical domains.

[04] The Impact of Weil's Rosetta Stone

1. How did Weil's ideas influence the development of the Langlands program?

• Weil's Rosetta stone guided progress in the Langlands program, a grand project to unify disparate fields of mathematics.
• The geometric version of the Langlands program, developed in the early 1980s, expanded on Langlands' original number-theory vision to encompass a connection between number theory and geometry.
• Recent advances in the Langlands program have involved translations between the number-theory and geometric versions, following the approaches set out in Weil's Rosetta stone.

2. What were some of the key developments that built on Weil's ideas?

• In the late 1950s and early 1960s, Alexander Grothendieck made foundational contributions to algebraic geometry in pursuit of Weil's conjectures.
• In 1973, Pierre Deligne used Grothendieck's techniques to prove Weil's finite-field version of the Riemann hypothesis in higher dimensions.
• In 2021, Laurent Fargues and Peter Scholze finalized work on the Fargues-Fontaine curve, providing one of the first direct translations between the geometric and number-theory versions of the Langlands program.

3. How does the article characterize the nature of Weil's original idea?

• The article suggests that Weil's idea was more than just a dream, as it was not only articulated in the letter but also converted into something concrete through his subsequent work.
• The article contrasts Weil's idea with other important mathematical ideas that "seem to come from thin air" and are "not easily traceable," highlighting the originality and significance of Weil's Rosetta stone concept.