The relationship between quadratic forms and modular forms has been a rich source of mathematical questions and answers for a long time, through the theory of theta functions. This connection has been used recently by the PI to understand the failure of a local-global principle for integral ternary quadratic forms, to give a complete answer of what numbers are represented by any given positive definite integral quadratic form in four variables, and (jointly with M. Bhargava) to prove Conway's 290-Conjecture which characterizes all positive definite integer-valued quadratic forms representing all positive integers as those representing 29 critical numbers (the largest of which is 290). The PI proposes to continue this work by understanding the local-global principle for a totally definite ternary quadratic form over a number field in terms of the structure of its theta function, and investigating (jointly with M. Bhargava) finiteness theorems (like the 290-Conjecture) for quadratic forms representing all odd numbers and also all prime numbers. He is also interested in enumerating all totally definite quadratic forms of class number one, since such forms are the basis of many arithmetic results about quadratic forms.
The question of how many ways a number can be expressed as a sum of two, three, or four square numbers is one of the oldest and most interesting problems in number theory -- and the discovery of some of the most fundamental symmetries of the arithmetic of whole numbers. Understanding these symmetries has allowed for some of the most celebrated results in number theory, for example Wiles's proof of Fermat's last theorem, and Lagrange's proof that every positive number is a sum of (at most) four square numbers. The PI hopes to use these symmetries to continue to answer simple questions about whole numbers -- in particular sums of square numbers -- and to provide computational tools to allow others to do the same. These results could be used to create better encryption systems, but more importantly, could allow for a deeper understanding of the nature of arithmetic and its underlying structure.