The hyperbolic geometry of Markov's theorem on Diophantine approximation and quadratic forms

Abstract: Markov's theorem classifies the worst irrational numbers and the most non-zero quadratic forms. This talk is about a new proof using hyperbolic geometry. The main ingredients are a dictionary to translate between hyperbolic geometry and algebra/number theory, and some very basic tools borrowed from modern geometric Teichmüller theory. Simple closed geodesics and ideal triangulations of the modular torus play an important role, and so does the problem: How far can a straight line crossing a triangle stay away from the vertices?

dynamical systems

Audience: researchers in the topic

Bremen Online Dynamics Seminar

Series comments: Talks are approx. 55 min plus discussion. Talks are not recorded.

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