Summary Article
Mathematics -- The Science of Patterns and Algorithms
Beyond Fermat
One of the most widely publicized mathematical results of the last few years is Wiles' proof of Fermat's last theorem: that the sum of two nth powers can not be another nth power if n > 2. This had been an unsolved problem for 350 years, and that reason alone would make it of great interest. But mathematicians see it as one result in a huge web of connected conjectures, some proven and some supported by extensive computations which together link arithmetic, geometry, analysis, group theory and even physics. These conjectures, and the powerful methods that are being developed to deal with them, have been some of the most exciting themes in fundamental mathematics in the last fifty years. Among these is the ABC conjecture which generalizes Fermat's theorem and expresses a fundamental tension between addition and multiplication: If A and B are any two numbers with many repeated large prime factors, then their sum cannot have many repeated large prime factors (here "many" can be made precise). Other conjectures relate to the distribution of prime numbers, 2, 3, 5, 7, 11, ... . This tantalizingly irregular sequence is encoded in a kind of generating function, the Riemann zeta function, whose analytic properties express the hidden patterns of the primes in a way which is worked out in this complex of ideas. The Riemann zeta function is the paradigm of a class of functions whose analytic properties are expected to shed light on such classical Diophantine questions as: how many rational solutions does a (given) cubic polynomial in two variables have? This vision connects to the theory of group representations, to the theory of transcendental numbers, and to the study and classification of polynomial equations. The study of polynomial equations and the locus of their zeroes is nowadays being energetically furthered by methods from physics, partial differential equations, and differential geometry. It is even expected that work in this area will shed light on the structure of finite simple groups (the "Monster group" in particular).
The breadth and unity of this development is hard to exaggerate; it makes for a particularly interesting time in number theory, with many challenges ahead.