Model theory around the j-invariant

每个人都熟悉自然数0，1，2，3。以及还包括-1、-2、-3、.我们将“有理数”形成为整数的分数，如2/3和-3/5。通常，查看新数字的下一步是查看小数，例如2.17182818284590。.整数和有理数是代数和精确数学的一部分，而小数真正进入了近似数学的领域，因为任何给定的数字列表都只是十进制数字的完整无限列表的近似。这种近似的研究通常被称为数学分析。然而，还有其他一些数字，如2的平方根，它们超出了有理数，但可以用精确的方法来理解。这些数称为代数数。每一个都是一个方程的解，比如x.x = 2，其解正好是2的平方根。将代数数与十进制数和分析方法联系起来是一项令人惊讶的微妙而困难的任务。希尔伯特在1900年著名的演讲中列出的23个数学问题中的第12个问题大致是如何通过分析方法捕获这些代数数的某些家族。更确切地说，它要求一种计算任意数域K上的阿贝尔数的解析方法。一个合适的方法已经知道希尔伯特在最简单的情况下，当K是领域的有理数。下一个最简单的情况是当K只是有理数加上一个二次方程的解时。在某些情况下（“虚的”二次方程）的方法也是已知的希尔伯特，但在其他情况下（“真实的”二次方程）仍然没有已知的方法，虽然有些已经提出。在这个项目中，我们将使用数理逻辑的方法，特别是模型论，图表之间的代数和分析方法，这是“精确的”，但将允许一些权力的分析方法。我们希望这种新的方法将提供新的见解希尔伯特的问题，特别是有关的真实的二次的情况。

Everyone is familiar with the natural numbers 0, 1, 2, 3... and the integers which include also -1, -2, -3,.... We form the "rational" numbers as fractions of integers such as 2/3 and -3/5. Often the next step in looking at new numbers is to look at decimal fractions such as 2.17182818284590... . Whereas integers and rational numbers are part of algebra and exact mathematics, decimal fractions really come into the realm of approximate mathematics, since any given list of digits is only an approximation of the full infinite list of decimal digits. The study of such approximations is often called mathematical analysis. However there are other numbers such as the square root of 2 which lie beyond the rational numbers but can be understood with exact methods. These numbers are called algebraic numbers. Each is the solution to an equation such as x.x = 2, whose solutions are exactly the square roots of 2. Connecting the algebraic numbers with the decimal numbers and analytic methods is a surprisingly subtle and difficult task.The twelfth of Hilbert's list of 23 mathematical problems from his famous lecture in 1900 asks roughly how certain families of these algebraic numbers can be captured by analytic methods. A little more precisely, it asks for an analytic method for computing the abelian numbers over any number field K. A suitable method was already known before Hilbert in the simplest case when K is the field of rational numbers. The next simplest cases are when K is just the rational numbers together with a solution to a quadratic equation. In some cases (the "imaginary" quadratics) a method was also known by Hilbert, but in the other cases (the "real" quadratics) there is still no known method, although some have been proposed.In this project we will use the methods of mathematic logic, specifically model theory, to chart a new way between the algebraic and analytic methods which is "exact" but will allow some of the power of the analytic methods. We hope this new method will give new insights into Hilbert's problem, particularly relating to the real quadratic case.