Shrinking targets on homogeneous spaces and improving Dirichlet’s Theorem

I will talk about an interesting direction inDiophantine approximation -- metric theory for improvements to Dirichlet's Theorem -- which was started by Davenport and Schmidt in the 1960s. Wewere able to obtain optimal results in the one-dimensional case via continued fractions\; while for simultaneous approximation the problem is still open. I will describe a reduction of the problem to dynamics (diagonalflows on the space of lattices), and then talk about an inhomogeneous version, which happens to be easier than the homogeneous one. Joint work with Nick Wadleigh.