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Explore the mathematical classification problem through the lens of computability theory in this 56-minute conference talk that examines how isomorphism problems can be formalized as equivalence relations. Delve into the connections between isomorphism problems and structural definability, including the relationship between index sets and Scott analysis, as well as recent findings by Harrison-Trainor, Melnikov, Miller, and Montalban on computable functors and effective interpretability. Focus on a specific variant originating in algebra: the isomorphism problem of algebraic presentations within given varieties, particularly examining finitely generated computably enumerable presentations. Compare the complexities of these isomorphism problems using computable reducibility as an effectivization of Borel reducibility, with detailed analysis of varieties including commutative monoids, commutative semigroups, abelian groups, and unary structures. Discover general results that establish connections between the algebraic properties of varieties and the computational complexities of their corresponding isomorphism problems, providing insights into how algebraic structure influences computational difficulty in classification tasks.
