Tuesday, July 23, 2019 - 15:00
Incontri di Algebra e Geometria allo SBAI
Daniel Labardini-Fragoso (Università Autonoma del Messico)
Cluster algebras and hyperbolic geometry
Sergey Fomin and Andrei Zelevinsky invented cluster algebras almost 20 years ago. In less than two decades, cluster algebras have found connections with many areas of Mathematics, e.g. Hyperbolic geometry and Teichmüller theory.
I will start this talk by giving a brief overview of what a cluster algebra is (a ring whose generators are produced recursively by applying a very simple combinatorial operation, called mutation, on oriented graphs). Then I will present a beautiful identity discovered to hold in the hyperbolic plane by Robert Penner, and describe how this identity allows cluster algebras to appear as coordinate rings of Teichmüller spaces of punctured surfaces, as discovered by Sergey Fomin, Michael Shapiro and Dylan Thurston, and Vladimir Fock and Alexander Goncharov. Finally, I will present 'Hyperbolic GeometPy', a program which, motivated by the above, I am writing to be able to interact with the hyperbolic plane and visualize things like orbits and invariant curves of Möbius transformations, hyperbolic convex hulls, hyperbolic geodesics with constant rapidity, circular motion with constant angular rapidity, and tessellations of the hyperbolic plane by Fuchsian group actions.