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The representation of integers by quadratic forms: The famous “Four Squares Theorem” of Lagrange asserts that any positive integer can be expressed as the sum of four square numbers. That is, the quadratic form $a^2 + b^2 + c^2 + d^2$ represents all (positive) integers. When does a general quadratic form represent all integers? When does it represent all odd integers? When does it represent all primes? We show how all these questions turn out to have very simple and surprising answers. In particular, we describe joint work with Jonathan Hanke that led to a proof of Conway’s “290-Conjecture”. Recording available below.

How likely is it for an integer polynomial to take a square value? Understanding whether (and how often) a mathematical expression takes a square value is a problem that has fascinated mathematicians since antiquity. After giving a survey of this problem, we will then concentrate on the case where the mathematical expression in question is simply a polynomial in one variable. The main result in this case – proved just recently – is that if the degree of the polynomial is at least 6, then it is not very likely to take even a single square value! Recording available below.

The density of squarefree values taken by a polynomial: It is well known that the density of integers that are squarefree is $6/\pi^2$, giving one of the more intriguing occurrences of $\pi$ where one might not a priori expect it! A natural next problem that has played an important role in number theory is that of understanding the density of squarefree values taken by an integer polynomial. We survey a number of recent results on this problem for various types of polynomials – some of which use the ABC Conjecture and some of which do not. Recording available below.