Andrew Wiles

Sir Andrew John Wiles (born 11 April 1953) is a British mathematician and a Royal Society Research Professor at the University of Oxford, specializing in number theory. He is globally renowned for proving Fermat's Last Theorem, a monumental achievement that earned him numerous accolades, including the Abel Prize and a knighthood, and is widely considered one of the greatest mathematical breakthroughs of the 20th century.

Early Life and Education

Andrew Wiles was born in Cambridge, England, to Maurice Frank Wiles, a professor of divinity at the University of Oxford, and Patricia Wiles. He received his early education at King's College School in Cambridge and later attended The Leys School.

Wiles pursued his undergraduate studies at Merton College, Oxford, where he earned a Bachelor of Arts degree in mathematics in 1974. He subsequently attended Clare College, Cambridge, for his postgraduate studies. Under the supervision of John Coates, Wiles completed his PhD in 1980. His doctoral research focused on Iwasawa theory, specifically concerning the arithmetic of elliptic curves with complex multiplication, a foundational area that would later prove crucial to his most famous work.

Academic Career

Following his doctoral studies, Wiles held several prestigious academic positions. He was a Junior Research Fellow at Clare College, Cambridge, and later held a Benjamin Peirce Assistant Professorship at Harvard University. He also spent time as a visiting scholar at the Institute for Advanced Study in Princeton, New Jersey.

In 1982, Wiles joined the faculty at Princeton University, where he was promoted to full professor in 1988. He remained at Princeton for nearly three decades, shaping the institution's mathematics department. Concurrently, from 1988 to 1990, he served as a Royal Society Research Professor at the University of Oxford. In 2011, Wiles returned to the UK to take up a permanent position at Oxford. In 2018, he was appointed the Regius Professor of Mathematics at the University of Oxford, a rare and highly distinguished royal chair.

Proof of Fermat's Last Theorem

Wiles is best known for his proof of Fermat's Last Theorem, a problem that had remained unsolved for over 350 years. The theorem states that no three positive integers $a$, $b$, and $c$ can satisfy the equation $a^n + b^n = c^n$ for any integer value of $n$ greater than 2. Wiles first encountered the theorem in a local library at the age of ten and became determined to solve it.

The Strategy

In the 1980s, mathematicians Gerhard Frey, Jean-Pierre Serre, and Ken Ribet established a link between Fermat's Last Theorem and the Taniyama-Shimura-Weil conjecture (now known as the modularity theorem). Ribet proved that if the Taniyama-Shimura-Weil conjecture were true for semistable elliptic curves, then Fermat's Last Theorem would also be true. Recognizing this, Wiles dedicated himself to proving this specific case of the modularity conjecture.

Secrecy and Announcement

Understanding the magnitude of the challenge and wishing to avoid distractions, Wiles worked in near-total secrecy for seven years, sharing his progress only with his wife and a few trusted colleagues. In June 1993, he presented his proof in a series of three lectures at the Isaac Newton Institute in Cambridge. The announcement was met with global acclaim and widespread media attention.

The Flaw and Resolution

During the peer-review process, mathematician Nick Katz discovered a subtle but critical error in a key part of the proof involving the Kolyvagin-Flach method. Wiles spent nearly a year attempting to fix the error, initially on his own and later in collaboration with his former student, Richard Taylor.

In September 1994, Wiles experienced a profound breakthrough. He realized that the very reason the Kolyvagin-Flach method failed could be used to complete an alternative approach using Iwasawa theory, a subject he had studied during his PhD. This insight allowed him and Taylor to patch the gap. The corrected proof was published in the Annals of Mathematics in May 1995, accompanied by a second paper co-authored by Taylor and Wiles that established the necessary properties of certain Hecke algebras.

Awards and Honors

Wiles's proof of Fermat's Last Theorem brought him unprecedented recognition in the scientific community. Because the Fields Medal is restricted to mathematicians under the age of 40, and Wiles was 41 at the time of his proof's completion, he was ineligible for the award. However, in 1998, the International Mathematical Union (IMU) awarded him the first-ever IMU Silver Plaque in recognition of his outstanding contribution.

His other major honors include:

• The Wolf Prize in Mathematics (1995)
• The Royal Medal (1996)
• Knight Commander of the Order of the British Empire (KBE) in the 2000 New Year Honours, granting him the title "Sir"
• The Shaw Prize in Mathematical Sciences (2005)
• The Abel Prize (2016), awarded by the Norwegian Academy of Science and Letters "for his stunning proof of Fermat's Last Theorem by way of the modularity conjecture for semistable elliptic curves, opening a new era in number theory"
• The Copley Medal (2017), the Royal Society's oldest and most prestigious award

Legacy

Andrew Wiles's work fundamentally transformed modern number theory. By proving the modularity theorem for semistable elliptic curves, he not only solved a centuries-old puzzle but also forged deep, unexpected connections between disparate areas of mathematics, specifically elliptic curves and modular forms. This breakthrough provided a massive impetus to the Langlands program, a vast web of conjectures that seeks to unify number theory and representation theory.

Beyond his technical contributions, Wiles's dedication, perseverance, and the dramatic narrative of his proof have made him a cultural icon in the realm of science. His story has been the subject of numerous books, documentaries, and articles, serving as an enduring inspiration to aspiring mathematicians worldwide.