





By BBC News Online Science Editor Dr David Whitehouse



One of the most difficult problems in mathematics has finally been solved.



It is called the Shimura-Taniyama-Weil (STW) conjecture, and it has baffled and defeated some of the greatest minds in maths over the last 40 years.



Now an international team is claiming victory.



"This is one of the crowning achievements of mathematics in the 20th Century," said number theorist Professor Henri Darmon of McGill University in Canada.



The STW conjecture links two seemingly unrelated areas of mathematics: the theory of numbers and the theory of shapes or, as mathematicians prefer to call them, elliptic curves and modular forms.



For decades, mathematicians have studied these subjects realising that there are deep connections between them but without ever being able to pin down the exact relationship.



Andrew Wiles used the STW conjecture to provide the proof for the famous mathematical puzzle Fermat's Last Theorem. But before Wiles cracked the theorem in 1993, nobody even knew where to begin to tackle the STW conjecture.



In fact, Fermat's Last Theorem is only a particular part of the more profound STW conjecture. In technical terms, Andrew Wiles, a professor at Princeton, proved the STW conjecture for what are called semi-stable elliptic curves.



Final proof



It is the full proof of STW that will now be published in the December issue of the Notices of the American Mathematical Society.



And the work, perhaps not surprisingly, has come from three former Wiles students: Brian Conrad and Richard Taylor of Harvard University and Fred Diamond of Brandeis University. The team also included Christopher Breuil of the University of Paris.



Only front-line mathematicians will really understand the STW conjecture. But you could say "there exists a modular form of weight two and level N which is an Eigenform under the Hecke series and has a Fourier series".



The brilliant but ill-fated Japanese mathematician Yukata Taniyama was the first person to propose some of the ideas behind the STW conjecture in 1955. A few years later, at the age of 31, he committed suicide. Others extended his revolutionary work.



The conjecture goes deeper than just bridging the world of numbers with that of shapes. It is also part of what is known as the Langlands Program, a vast mathematical vision formulated by Robert Langlands. This is an attempt to unite whole areas of mathematics.



The US National Science Foundation sponsored the STW work. Its assistant director for mathematical sciences, Philippe Tondeur, said: "This is a breakthrough for mathematics. This proof will have far-reaching consequences because of the abundance of new mathematical tools developed in the process."



