New techniques for old problems in number theory

As Gauss said, mathematics is the queen of sciences and number theory is the queen of mathematics. The theory of numbers is a broad subject, and an ancient one. Many of the oldest conundrums in mathematics come from number theory. Using recent developments in mathematics, we revisit some of these problems.

Take diophantine approximation for instance. This is about how well one can approximate real numbers by rational numbers, i.e. fractions. Through modern combinatorics, we understand the structure of the set of 'good denominators'. This provides a crucial link to the infamous Littlewood conjecture. Using similar tools, I will also revisit other famous problems in diophantine approximation. In this strand of the proposal I furthermore seek out new phenomena.

An old question asks the following. Consider the Pythagorean equation. Colour each positive integer red, blue or green. Is there a solution with all three variables having the same colour? This property is typical in the subject of arithmetic Ramsey theory. We are able to establish it for many other equations. In some cases, we can characterise which equations in a family have this property. The key ingredient is Fourier analysis.

Finally, I am very excited to study the frequency of Galois groups of polynomials. One can think of the Galois group as the set of "natural" ways in which to permute the roots of the polynomial. Using algebraic criteria, the problem can be recast as a diophantine equation problem. From there one can deploy a wide variety of tools. This investigation has already uncovered surprising and powerful hidden symmetries. One of the challenges will be to discover more of these gems.

The main beneficiaries of the research will be those in the academic community. The proposal is to bring a wide range of innovative new techniques to bear on fundamental problems in number theory. Aside from theoretical and technological advances, an immediate impact will be to bring together different groups of mathematicians. These include the analytic number theory, additive combinatorics, diophantine approximation, algebraic number theory, harmonic analysis, geometry of numbers, uniform distribution, and extremal combinatorics communities. The timeliness and novelty of these connections is intended to spur an avalanche of further research, and also to inspire the next generation of mathematicians.

Beyond academia, there are potential applications to compressed sensing, wireless communication, and cryptography. The bridge to compressed sensing is the study of partial circulant matrices, which we investigate in relation to the lonely runner conjecture from diophantine approximation; a better understanding of the structure of the kernels of partial circulant matrices may lead to improvements in their use in compressed sensing. Meanwhile, the metric theory of diophantine approximation has recently proved to be surprisingly useful in multiple-input and multiple-output, which is state-of-the-art in wireless communication. Finally, diophantine equations and Galois theory have long been broadly-applicable to cryptography.

In conclusion, this work will lead to major scientific advances, and the development of novel intradisciplinary methods. These will provide a strong burst of research, which will unite people from different mathematical communities and teach them valuable skills. Furthermore, applications to compressed sensing, wireless, and cryptography could boost the UK economy and eventually improve our quality of life.