Andrew Wiles

Definition

Andrew Wiles is a British mathematician best known for proving Fermat's Last Theorem, a longstanding problem in number theory. His groundbreaking work involved the deep connections between elliptic curves and modular forms, which ultimately led to a resolution of this famous theorem. This achievement not only solved a centuries-old mathematical question but also provided significant advancements in various areas of mathematics.

5 Must Know Facts For Your Next Test

1. Wiles first announced his proof of Fermat's Last Theorem in 1993, after working in secrecy for nearly seven years.
2. The proof relies heavily on the Taniyama-Shimura-Weil conjecture, which links elliptic curves to modular forms.
3. Wiles' work received widespread recognition and was awarded the prestigious Abel Prize in 2016.
4. The initial proof contained an error, but Wiles, along with his former student Richard Taylor, corrected it within a year.
5. Wiles' achievement has had a profound impact on modern number theory and has opened up new avenues of research.

Review Questions

• How did Andrew Wiles connect elliptic curves to modular forms in his proof of Fermat's Last Theorem?
  • Andrew Wiles established a link between elliptic curves and modular forms through the Taniyama-Shimura-Weil conjecture. This conjecture posited that every rational elliptic curve is also a modular form. By proving this connection, Wiles could apply techniques from the study of modular forms to demonstrate the validity of Fermat's Last Theorem, thus resolving a centuries-old problem.
• Discuss the significance of Wiles' proof on the broader field of mathematics beyond just solving Fermat's Last Theorem.
  • Wiles' proof of Fermat's Last Theorem has significantly impacted the broader field of mathematics by bridging various areas such as number theory, algebraic geometry, and arithmetic geometry. It provided insights into the relationships between different mathematical structures and encouraged further exploration into modular forms and elliptic curves. This cross-disciplinary approach has inspired new research directions and contributed to advancements in areas like cryptography and arithmetic algebraic geometry.
• Evaluate how the initial error found in Wiles' proof affected perceptions of mathematical rigor and collaboration in modern mathematics.
  • The discovery of an error in Wiles' initial proof brought attention to the importance of rigorous peer review and collaboration in modern mathematics. Although initially concerning, Wiles' swift correction with Richard Taylor showcased the collaborative spirit within the mathematical community. This incident reinforced the idea that mathematics is a collective endeavor, where sharing ideas and working together can lead to breakthroughs despite setbacks. It also emphasized that even highly regarded mathematicians can face challenges, fostering an environment that values persistence and teamwork.

Related terms

Fermat's Last Theorem: A statement made by Pierre de Fermat in 1637 that asserts there are no three positive integers a, b, and c that can satisfy the equation $$a^n + b^n = c^n$$ for any integer value of n greater than 2.

Elliptic Curves: Algebraic curves defined by equations of the form $$y^2 = x^3 + ax + b$$, which have important applications in number theory and cryptography.

Modular Forms: Complex functions that are highly symmetric and exhibit certain transformation properties, playing a crucial role in number theory and the proof of Fermat's Last Theorem.