Title: Successive minima of lattice trajectories and topological games to compute fractal dimensions
Speaker: Tushar Das – University of Wisconsin – La Crosse
Abstract: We present certain sketches of a program, developed in collaboration with Lior Fishman, David Simmons, and Mariusz Urbanski, which extends the parametric geometry of numbers (initiated by Schmidt and Summerer, and deepened by Roy) to Diophantine approximation for systems of m linear forms in n variables, and establishes a new connection to the metric theory via a variational principle that computes the fractal dimensions of various sets of number-theoretic interest. Applications of our results include computing the Hausdorff and packing dimensions of the set of singular systems of linear forms and showing they are equal, resolving a conjecture of Kadyrov, Kleinbock, Lindenstrauss and Margulis, as well as a question of Bugeaud, Cheung and Chevallier. As a corollary of the correspondence principle (initiated by Dani, and deepened by Kleinbock and Margulis), the 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. Highlights of our program include the introduction of certain combinatorial objects that we call templates, which arise from a dynamical study of Minkowski’s successive minima in the geometry of numbers; as well as a new variant of Schmidt’s game designed to compute the Hausdorff and packing dimensions of any set in a doubling metric space. The talk will be accessible to students and faculty interested in some convex combination of homogeneous dynamics, Diophantine approximation and fractal geometry. I hope to present a sampling of open questions and directions that have yet to be explored, some of which may be pursued by either following or adapting the technology described in my talk.