On counting integral orbits and related applications

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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.