New techniques for old problems in number theory

正如高斯所说，数学是科学女王，数论是数学女王。数论是一门广泛的学科，也是一门古老的学科。数学中许多最古老的难题都来自数论。利用数学的最新发展，我们重温其中的一些问题。以丢番图近似为例。这是关于一个人能多好地用有理数，即分数来逼近实数。通过现代组合学，我们了解了“好分母”集合的结构。这为臭名昭著的利特尔伍德猜想提供了一个关键环节。使用类似的工具，我还将重温丢番图逼近中的其他著名问题。在这一串提案中，我进一步探索了新的现象。一个老问题问了下面这个问题。考虑毕达哥拉斯方程。将每个正整数涂成红色、蓝色或绿色。有没有三个变量的颜色都相同的解决方案？这一性质在算术Ramsey理论中是典型的。我们能够为许多其他方程建立它。在某些情况下，我们可以刻画一个族中的哪些方程具有这种性质。关键成分是傅里叶分析。最后，我非常兴奋地研究伽罗瓦多项式群的频率。我们可以把伽罗华群看作是对多项式的根进行置换的一组“自然”方法。利用代数准则，该问题可以转化为丢番图方程问题。从那里可以部署各种各样的工具。这项调查已经发现了令人惊讶和强大的隐藏对称性。挑战之一将是发现更多这样的宝石。

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.