Diophantine geometry via analytic number theory

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Polynomial equations are extremely commonplace in nature, and can be used to describe a myriad of physical and mathematical phenomena. For example, the theorem of Pythagoras states that a right-angled triangle with sides of lengths a<=b<=c has the property that these lengths always satisfy the quadratic equation a^2+b^2=c^2. It is a very natural step to try and determine under what circumstances a given polynomial equation admits integer solutions. For the Pythagorean equation this was answered completely by Diophantus in 250 AD, from whom we have inherited the term 'Diophantine equations'. Diophantus actually managed to write down the general solution in integers to Pythagoras' equation. For a general Diophantine equation, there are 3 basic possibilities: either we can show that there are infinitely many solutions (as above), or we can show that there are only finitely many solutions (as Wiles famously did for Fermat's equation a^k+b^k=c^k, when k>2), or we have trouble showing anything at all! The propensity for the third outcome lies at the heart of the enduring appeal that the subject of Diophantine equations enjoys.The equations mentioned so far have only involved 3 variables, and these are the equations that have been most closely studied. By contrast the solubility in integers of equations in 4 or more variables is still an untamed frontier, with only a scattering of results and conjectures on the map. One of the major outcomes of this project will be that many of the conjectural waypoints become established fact. The tools that I will use are rooted in analytic number theory, but will also take advantage of methods from algebraic geometry and the theory of descent. There is a useful interplay between Diophantine equations and the underlying geometry of the equation. This sort of connection provides a very useful source of extra leverage, and often reveals quite beautiful relations. My research makes essential use of this point of view.