![]() ![]() I will present the history of the Duffin-Schaeffer conjecture and the main ideas behind the recent work of Koukoulopoulos-Maynard that settled it. We prove the local-in-time existence the solution of the compressible Euler equations in $3$-D, for the Cauchy data of the velocity, density and vorticity $(v,\varrho, \omega) \in H^s\times H^s\times H^^\infty\varphi(q)\Delta_q<\infty$ and almost no irrationals are approximable. I will talk about my work on the compressible Euler equations. The unexpected feature here is that this can be proved without settling the Ultimate-L Conjectures. By recent theorem, if V = Ultimate-L then the Ultrapower Axiom holds. For example, it implies the Generalized Continuum Hypothesis must hold above the least strongly compact cardinal and it implies the least strongly compact cardinal is supercompact. The Ultrapower Axiom has deep structural consequences in the context of large cardinal axioms. Goldberg's Ultrapower Axiom holds in all the current generalizations of L which have been constructed in the Inner Model Program which is another major program of Set Theory to identify generalizations of L. A key part of this program is to develop the structure theory of Ultimate-L. This involves a series a rather specific conjectures, which is the family of Ultimate-L Conjectures. ![]() The Ultimate Program is the program to show that the axiom V = Ultimate-L is not refuted by large cardinal axioms. The axiom V = Ultimate-L is the leading candidate for the maximum possible generalization of Gödel's axiom, V = L.
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