Abstract: I will review the strategy of the proof of the conservativity conjecture for the classical realisations of Voevodsky motives over a characteristic zero fields. I will also mention some other consequences of this proof such as the nilpotence of endomorphisms acting by zero on cohomology.
Abstract: The "equivalence principle" says that meaningful statements in mathematics should be invariant under the appropriate notion of equivalence of the objects under consideration. In set-theoretic foundations, the EP is not enforced; e.g., the statement "1 ϵ Nat" is not invariant under isomorphism of sets. In univalent foundations, on the other hand, the equivalence principle has been proved for many mathematical structures. In this introductory talk, I give an overview of univalent foundations and the equivalence principle therein.