We describe our recent program (ongoing with Lior Fishman, David Simmons and Mariusz Urbanski) to resolve certain conjectures and questions regarding the study of various systems of linear forms in metric Diophantine approximation. The talk will be mostly accessible to students and faculty, in the hope of inspiring (at least one or more of) them to undertake a deeper study of this predominantly unexplored yet incredibly verdant mathematical landscape.

In more detail, we extend the parametric geometry of numbers (initiated by Schmidt-Summerer, and deepened by Roy) to Diophantine approximation for systems of m linear forms in n variables, and establish a new connection to the metric theory via a variational principle that computes the fractal dimensions of a large number of sets of number-theoretic interest. In particular, we compute the Hausdorff and packing dimensions of the set of singular systems of linear forms and show they are equal, thereby resolving a conjecture of Kadyrov-Kleinbock-Lindenstrauss-Margulis (2014) as well as answering a question of Bugeaud-Cheung-Chevallier (2016). As a corollary of the Dani correspondence principle, this implies that the set of divergent trajectories of a one-parameter diagonal action on the space of unimodular lattices with exactly two Lyapunov exponents with opposite signs has equal Hausdorff and packing dimensions. Other applications include dimension formulas with respect to the uniform exponent of irrationality for simultaneous and dual approximation in two dimensions, completing a slew of partial results due to Baker, Bugeaud, Cheung, Chevallier, Dodson, Laurent, Rynne, and Yavid (1977-2016).