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Équations différentielles motiviques et au–delà

Quantum character varieties

IHP
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In this talk I will discuss a quantisation procedure for a class of affine del Pezzo surfaces that appear in several contexts of mathematics. Most of the talk will be based on a specific example, namely the affine del Pezzo surfaces defined as
character variety of a torus with one disk removed. I will show the relation between this surface, singularity theory, Painlev\'e differential equations, and introduce a cluster algebra structure on it which is related to Markov numbers. I will discuss quantisation of the Markov cubic in terms of basic orthogonal polynomials and relate it to Sklyanin algebra.