Title |
## Quantitative Aspects of Number Theory |
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Group | Number Theory |

Supervisor | |

Description |
The aim of this project is to investigate problems in number theory that have a quantitative component. Typically, the problems would involve some algebraic objects, such as as algebraic number fields or rational points on algebraic varieties. The questions that we ask are, however, of analytic nature. For example, how many number fields are there of a given type and with given properties, such that their discriminant is bounded by a (large) number B? The answer that we are seeking would then be an asymptotic formula or, if this is too hard, bounds for this number as B tends to infinity. Quantitative results of this kind are of interest by themselves, but moreover they are useful in other proofs, often to establish the existence of objects with certain properties. This holds, for example, for many applications of the Hardy-Littlewood circle method from analytic number theory. A candidate for this project would have some background in analytic or algebraic number theory, as well as an interest in learning the other field. An interest in algebraic geometry could be useful, but is by no means required. The concrete problems to work on will be chosen to match the student's interests. |

Title |
## Rational points on algebraic varieties |
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Group | Number Theory |

Supervisor | |

Description |
Diophantine equations are a classical object of study in number theory. During the course of the 20th century, it was realised that one obtains a more powerful conceptual framework by considering them through a more geometric lens, namely viewing a solution to a Diophantine equation as a rational point on the associated algebraic variety. Given an algebraic variety over a number field, natural questions are: Is there is a rational point? If yes, are there infinitely many? If also yes, can one obtain a finer quantitative description of the distribution of the rational points? These problems are very difficult in general, but in this project the aim is to make some progress for some special classes of varieties (in particular solve some new cases of Manin’s conjecture). A popular current research theme is to consider these problems in families, such as studying the distribution of varieties in a family with a rational point, or controlling failures of the Hasse principle in families. To solve these problems one usually uses a combination of techniques from algebraic geometry and analytic number theory, but the project could be tailored towards the preferences of the student (e.g. for a student without much knowledge of algebraic geometry). |

Title |
## Additive combinatorics and Diophantine problems |
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Group | Number Theory |

Supervisor | |

Description |
The study of Diophantine equations encompasses a diverse portion of modern number theory. Recent years have seen spectacular progress on solving linear Diophantine equations in certain sets of interest, such as dense sets or the set of primes. Much of this progress has been achieved by breaking the problem down into a structure versus randomness dichotomy, using tools from additive combinatorics. One tackles the structured problem using techniques from classical analytic number theory and dynamical systems, whilst the ‘random' problem is handled using ideas informed by probabilistic combinatorics and Fourier analysis. The consequences of this rapidly developing theory for - Existence of solutions to systems of Diophantine equations in dense sets. To what extent can Szemerédi’s theorem be generalised to non-linear systems of equations? - Quantitative bounds for sets lacking Diophantine configurations. Can one obtain good quantitative bounds in the polynomial Szemerédi theorem? What about sets lacking progressions with common difference equal to a prime minus one? - Partition regularity of Diophantine equations. Can one generalise a Ramsey-theoretic criterion of Rado to systems of degree greater than one? - Higher order Fourier analysis of non-linear equations. Is it possible to count solutions to hitherto intractable Diophantine equations by developing the Hardy—Littlewood method along the lines of Green and Tao? What are the obstructions to uniformity for such equations? |

Title |
## Analytic Number Theory and mean values of L-functions |
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Group | Number Theory |

Supervisor | |

Description |
The Riemann zeta-function and other L-functions play a central role in analytic number theory and in mathematics in general. For example, the Riemann zeta-function satisfies an Euler product, which underlines a connection between the natural numbers and the prime numbers. The problem of determining the properties of prime numbers has a long history, from the ancient theorem of Euclid that there are infinitely many primes, to the celebrated eight page paper of Riemann on the zeta-function in the mid-nineteenth century. Since that time, several important problems in analytic number theory have been solved, and Riemann's ideas have been the inspiration behind much of this progress. Investigating the properties of the Riemann zeta-function and L-functions in various contexts leads to many other interesting problems, which now represent major challenges in modern mathematics. In fact both the Riemann Hypothesis, which asserts that all the non-trivial zeros of the Riemann zeta-function lie on a particular line, and the Birch and Swinnerton-Dyer Conjecture, which concerns some properties of the L-functions associated to elliptic curves, have been included in the seven Millennium Prize Problems. The aim of the project is to study various questions related to the moments of the Riemann zeta-function and L-functions, which are the mean values over certain families of these functions. These questions have applications to the distribution of zeros of the Riemann zeta-function (partial answers to the Riemann Hypothesis), the order of magnitude of L-functions (partial answers to the Lindelof Hypothesis), order of vanishing of L-functions at the central point (analytic progress towards the Birch and Swinnerton-Dyer Conjecture), and many others. There is a remarkable connection between the subject and Random Matrix Theory, an area of Mathematical Physics used to describe complex quantum systems. |