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In this talk, we will analyze the topological conjugacy problem for minimal homeomorphisms of the Cantor space from the point of view of descriptive set theory. We shall not assume any prior knowledge on the topic and the first half of the talk will be devoted to the mathematical framework provided by descriptive set theory to analyze the relative complexity of classification problems. In the second half of the talk, we shall focus on the topological conjugacy relation on the space of minimal homeomorphisms of the Cantor space and show that the Friedman-Stanley jump of the identity relation on reals is a lower bound for the Borel complexity of this relation.