Vladimir Voevodsky

"Infinity-groupoids encode all the paths in a space, including paths of paths, and paths of paths of paths. They crop up in other frontiers of mathematical research as ways of encoding similar higher-order relationships, but they are unwieldy objects from the point of view of set theory. Because of this, they were thought to be useless for Voevodsky’s goal of formalizing mathematics. Yet Voevodsky was able to create an interpretation of type theory in the language of infinity-groupoids, an advance that allows mathematicians to reason efficiently about infinity-groupoids without ever having to think of them in terms of sets. This advance ultimately led to the development of univalent foundations."

"Voevodskys construction makes it possible to obtain an “incarnation” of the motivic cohomology but it does not, however, find a solution to the standard conjectures, which are still today – along with the Hodge conjecture – the fundamental open question in modern algebraic geometry."

"Within mathematics itself, Voevodskys proposal, if adopted, will create a new paradigm. In his “fairy tale” and some of his other papers, Langlands made deft use of categories and even 2-categories, but number theory is only superficially categorical, and so is the Langlands program. In the event that Univalent Foundations could shed light on a guiding problem in number theory — the Riemann hypothesis or the Birch Swinnerton-Dyer conjecture, which is not so far removed from Voevodskys motives — then we could easily see Grothendiecks program absorbing the Langlands program within Voevodskys new paradigm."

"Today we face a problem that involves two difficult to satisfy conditions. On the one hand we have to find a way for computer assisted verification of mathematical proofs. This is necessary, first of all, because we have to stop the dissolution of the concept of proof in mathematics. On the other hand, we have to preserve the intimate connection between mathematics and the world of human intuition. This connection is what moves mathematics forward and what we often experience as the beauty of mathematics."

"A technical argument by a trusted author, which is hard to check and looks similar to arguments known to be correct, is hardly ever checked in detail"

"Mathematical research currently relies on a complex system of mutual trust based on reputations. By the time Simpsons paper appeared, both Kapranov and I had strong reputations. Simpsons paper created doubts in our result, which led to it being unused by other researchers, but no one came forward and challenged us on it."