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Institute for Advanced Study in Mathematics (IASM) in Hangzhou, China

Recursion theory measures the complexity of mathematical objects in terms of what is computable, in other words, what can be determined by a computer with no space or time limitations. We can define, for example, computable sets of natural numbers, computable (continuous) functions on the real numbers, or computably enumerable open subsets of the reals. Adding an oracle---an outside source of information---allows us to extend the reach of recursion theory beyond the computable. For example, every continuous function on the real numbers is a computable function relative to some oracle. For another example, a set of real numbers has Hausdorff dimension zero if and only if there is an oracle relative to which every real in the set can be significantly compressed. In this way, recursion theory offers a fine-grained way to analyze some of the most central notions in mathematics. This has recently led to deep applications to analysis, number theory, and set theory. The focus of this workshop is on understanding and extending these applications of recursion theory.