Date & Time: Tuesday, November 25, 2014, 14:30-15:30.
Venue: Ramanujan Hall
Title: On the abc conjecture and some of its consequences
Speaker: Michel Waldschmidt, Institut de mathématiques de Jussieu
Abstract: According to
Nature News, 10 September 2012, quoting Dorian Goldfeld, the abc Conjecture is "the most important unsolved problem in Diophantine analysis". It is a kind of grand unified theory of Diophantine curves: "The remarkable thing about the abc Conjecture is that it provides a way of reformulating an infinite number of Diophantine problems," says Goldfeld, "and, if it is true, of solving them." Proposed independently in the mid-80s by David Masser of the University of Basel and Joseph Oesterlé of Pierre et Marie Curie University (Paris 6), the abc Conjecture describes a kind of balance or tension between addition and multiplication, formalizing the observation that when two numbers a and b are divisible by large powers of small primes, a + b tends to be divisible by small powers of large primes. The abc Conjecture implies -- in a few lines -- the proofs of many difficult theorems and outstanding conjectures in Diophantine equations-- including Fermat's Last Theorem.
This talk will be at an elementary level, giving a collection of consequences of the abc Conjecture. It will not include an introduction to the Inter-universal Teichmüller Theory of Shinichi Mochizuki.