The hyperbolic geometry of numbers

In his thesis, Andrey Markov classified the unimodular indefinite binary quadratic forms whose values on the integer lattice stay farthest away from zero. This is closely linked to the classification of the worst approximable irrational numbers. The main tool in Markov's classical theory have always been continued fractions. Geometrically, they describe the symbolic dynamics of geodesics in the Farey triangulation of the hyperbolic plane. This talk will be about a new geometric approach that eliminates the complicated symbolic dynamics of continued fractions in favor of considering not only the Farey triangulation but all ideal triangulations of the modular torus. We will see how Markov's classification problem boils down to the question: How far can a geodesic that crosses a triangle stay away from the vertices? This geometric approach can also be used to classify the worst approximable rational (sic!) numbers.