New techniques for old problems in number theory

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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.