Pure Maths Colloquium - A variational principle in the parametric geometry of numbers, with applications to metric Diophantine approximation - Dr David Simmons (University of York) Seminar

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Abstract: The parametric geometry of numbers, introduced by Schmidt and Summerer, is a framework for analyzing the Diophantine properties of a vector in terms of the successive minima of a certain one-parameter family of convex regions (or equivalently of a certain family of lattices) defined in terms of that vector. We generalize this framework to the setting of matrix approximation, and we calculate the Hausdorff and packing dimensions of certain sets defined in terms of the parametric geometry of numbers. One of the many applications of our theorem is a proof of the conjecture of Kadyrov, Kleinbock, Lindenstrauss, and Margulis stating that the Hausdorff dimension of the set of singular $m\times n$ matrices is equal to $mn(1-\frac1{m+n-1})$. This work is joint with Tushar Das, Lior Fishman, and Mariusz Urbański.