Diophantine Equations after Fermat's Last Theorem

A Diophantine problem is an equation where one is interested in finding solutions that are whole numbers. The study of Diophantine problem goes back at least to the time of Diophantus in the third century BC. The most famous Diophantine problem of all time is 'Fermat's Last Theorem'. This problem attracted the attention of huge numbers of both professional and amatuer Mathematicians for over 350 years, and was finally solved by Andrew Wiles in 1994. At first, it seemed that very few Diophantine problems can be solved using Wiles' technique. However, a few years ago, the Principal Investigator proposed that to successfully solve other interesting Diophantine problems, Wiles' ideas must be combined with other, unrelated, methods from what is called 'Diophantine analysis'. The strategy proposed by the Principal Investigator was carried out by a team consisting of himself, and Bugeaud and Mignotte; this has lead to spectacular successes including the resolution of several famous unsolved problems. The best known of these is to find all the perfect powers in the Fibonacci sequence. This was an unsolved problem for over 50 years and was finally solved in 2003 by the above-mentioned team.Very recently, the Principal Investigator suggested a refinement of the technique used in the proof of Fermat's Last Theorem which he called 'Multi-Frey'. This gives far more information about the solutions of Diophantine problems than the 'Single-Frey' that is used in the proof of Fermat's Last Theorem. Using this approach the above-mentioned team successfully solved several Diophantine equations involving 5 and 6 unknowns; a feet believed to be without parallel.The first part of the proposed research builds on the recent successes. It is expected to investigate the ideas in the most general context possible instead of looking just at particular cases. The 'multi-Frey' technique needs to studied throughly and the Diophantine equations that it applies to have to be classified.Another aspect of the Principal Investigator's work is 'Arithmetic of Curves'. Diophantine equations are classified according to something called 'dimension'. Those having dimension one are called curves. An obvious question about to ask about any Diophantine problem (even curves) is: does it have solutions? If it seems that a Diophantine equation does not have solutions, another question is: how can we make sure of this? Many methods have been proposed for showing that certain curves do not have solutions. The Principal Investigator, in joint work with Martin Bright suggested a very simple method for showing that some curves do not have solutions. This method appears to be the simplest method yet and requires the least amount of information and computation. However, we have yet to understand if it gives the same information as the other methods in all cases, and we have yet to gain a conceptual understanding of the ideas involved. These are both directions of proposed research.The subject of Diophantine equations has long ago fragmented to several sub-disciplines with little or no interaction between them. This project aims to combine techniques from several of these sub-disciplines to study interesting Diophantine problems. It is expected to be a step forward toward re-unifying the subject of Diophantine equations.