On counting integral orbits and related applications

数论中最基本的问题之一是：给定一个具有整数系数的方程(可能包含多个变量)，它有多少个整数解？例如，给出一个变量的多项式，人们可能会问它多久取一次平方值。这个问题的一般答案将对数学、密码学和计算机科学领域产生巨大影响。然而，这个问题被证明是极其困难的。例如，即使当多项式的次数为3次时，我们也不知道如何回答它。*如果我们不只考虑一个方程，而是考虑整个方程式，并询问随机和平均的解的数量的行为，问题就会变得更容易处理。对一系列方程式的解的研究称为算术统计。除了要求取平方值的多项式外，还可以要求数值尽可能不是平方，即不是任何平方的倍数。计算这些数字相当于筛选出平方的倍数，就像经典的埃拉斯托芬筛子用来列举素数，这是解析数论的核心。*我的建议是解决算术统计学和解析数论两个领域的基本问题，结合这些领域和许多其他数学分支的思想：代数几何，研究复数中多项式方程的解的集合；算术不变理论，从复数到有理数和整数的通道；以及分析，或者更准确地说，是计算由不等定义的区域中的整点的数字几何方法。这将对整个数论产生广泛的影响。

One of the most fundamental questions in Number Theory is: given an equation (in possibly many variables) with integer coefficients, how many integer solutions does it have? For example, given a polynomial in one variable, one may ask how often does it take a square value. A general answer to this question will have a tremendous impact across the fields of Mathematics, Cryptography, and Computer Science. However, this question turns out to be extremely difficult. For example, we do not know how to answer it even when the polynomial has degree 3.******The question becomes somewhat more manageable if instead of considering only one equation, we consider an entire family of equations, and ask about the behavior of the number of solutions at random and on average. The study of solutions across a family of equations is called Arithmetic Statistics.******Instead of asking for polynomials taking square values, one may also ask for values that are as far from being squares as possible, numbers that are not a multiple of any square. Counting these numbers amounts to sieving out multiples of squares much like the classic sieve of Erastothenes for enumerating prime numbers, which lies at the heart of Analytic Number Theory.******My proposal is to tackle fundamental problems in both the fields of Arithmetic Statistics and Analytic Number Theory combining ideas from these fields and many other branches of mathematics including: Algebraic Geometry, a study of the set of solutions to polynomial equations in complex numbers; Arithmetic Invariant Theory, a passage from complex numbers to rational numbers and integers; and Analysis, or more precisely the method of geometry-of-numbers to count integer points in a region defined by inequalities. This will have a broad impact across Number Theory.