Explicit Methods for Rational Points on Curves (07w5063)
Description
Many problems in mathematics are concerned with describing the solutions to an equation in which the variables are constrained to represent integers (like -34 or 7) or rational numbers (like -5/3 or 2/7). An example is Fermat's last theorem, proved in the 1990s: it states that the equation x^n+y^n=z^n has no solutions if n>2 and the variables are constrained to be positive integers.
For most equations, determining all the integer or rational solutions turns out to be very hard. No method we know of has been proved to find all solutions reliably. But we do have a large toolkit of methods that happen to work in many particular situations. Many of these are based on geometric ideas.
From February 4 to 9, the world's experts on these methods will descend on the Banff Centre to compare notes. They will work together to extend current methods as far as possible and hopefully solve the age-old problem of finding rational points on curves.
The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is by Canada's Natural Science and Engineering Research Council (NSERC), the US National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT).
For most equations, determining all the integer or rational solutions turns out to be very hard. No method we know of has been proved to find all solutions reliably. But we do have a large toolkit of methods that happen to work in many particular situations. Many of these are based on geometric ideas.
From February 4 to 9, the world's experts on these methods will descend on the Banff Centre to compare notes. They will work together to extend current methods as far as possible and hopefully solve the age-old problem of finding rational points on curves.
The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is by Canada's Natural Science and Engineering Research Council (NSERC), the US National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT).
