The representability of natural numbers by sums of squares is a beautiful and rich question in number theory with a long history.  It naturally lead to the question of representing natural numbers by other positive definite quadratic forms and the famous recent 15 and 290 Theorems.  In this talk we begin by discussing some of this story with an aim to understanding what kinds of subsets of the natural numbers can be the image of a positive definite quadratic form.  We will the discuss the main ideas going into our main theorem, which in this language, is a statement about how well, or efficiently, the Hassett subset can be described as the image of a positive-definite quadratic form, namely that the minimal rank of a quadratic form supporting it is four.  Here the Hassett subset is the set of natural numbers d larger than 8 and congruent to 0 or 2 modulo 6.  Our interest in this question for this particular subset is motivated by a question about moduli spaces of cubic fourfolds in algebraic geometry.  When rephrased in algebraic geometry terms, we will explain how our result about quadratic forms proves that the intersection all the Hassett divisors C_d for d in (you guessed it!) the Hassett subset has dimension 16.  Time permitting, we will explain why this interesting for other question in algebraic geometry. Advisor: Dr. Howard Nuer