Let $q,Q$ be two integral quadratic forms in $m < n$ variables. One can ask when $q$ can be represented by $Q$ - that is, whether there exists an $n \times m$-integer matrix $T$ such that $Q \circ T = q$. Naturally, a necessary condition is that such a representation exists locally, meaning over the real numbers and modulo $N$ for every positive integer $N$. In the absence of local obstructions, does a (global) representation of $q$ by $Q$ exist? This question is particularly delicate when the codimension $n-m$ is small, with codimension $2$ being the most challenging.

In the first talk, we will give a broad introduction to Linnik-type equidistribution problems (e.g. for periodic geodesics on the modular surface) in the context of the above question for quadratic forms. We will also explain some ideas of Linnik’s ergodic approach to such problems.

Note a second talk in the Geometry Seminar on Thursday.