This article describes a formal proof of the Kepler conjecture on dense sphere packings in a combination of the HOL Light and Isabelle proof assistants. This paper constitutes the official published account of the now completed Flyspeck project.

We analyze the pointwise convergence of a sequence of computable elements of L 1 (2 ω) in terms of algorithmic randomness. We consider two ways of expressing the dominated convergence theorem and show that, over the base theory RCA 0 , each is equivalent to the assertion that every G δ subset of Cantor space with positive measure has an element. This last statement is, in turn, equivalent to weak weak König's lemma relativized to the Turing jump of any set. It is also equivalent to the conjunction of the statement asserting the existence of a 2-random relative to any given set and the principle of Σ 2 collection.

Abstract. Let B be a p-uniformly convex Banach space, with p ≥ 2. Let T be a linear operator on B, and let Anx denote the ergodic average 1 n i

We correct Miyabe's proof of van Lambalgen's theorem for truth-table Schnorr randomness (which we will call uniformly relative Schnorr randomness). An immediate corollary is one direction of van Lambalgen's theorem for Schnorr randomness. It has been claimed in the literature that this corollary (and the analogous result for computable randomness) is a "straightforward modification of the proof of van Lambalgen's theorem." This is not so, and we point out why. We also point out an error in Miyabe's proof of van Lambalgen's theorem for truth-table reducible randomness (which we will call uniformly relative computable randomness). While we do not fix the error, we do prove a weaker version of van Lambalgen's theorem where each half is computably random uniformly relative to the other. We also argue that uniform relativization is the correct relativization for all randomness notions. arXiv:1209.5478v2 [math.LO]

Abstract. Unlike Martin-Löf randomness and Schnorr randomness, computable randomness has not been defined, except for a few ad hoc cases, outside of Cantor space. This paper offers such a definition (actually, several equivalent definitions), and further, provides a general method for abstracting "bit-wise" definitions of randomness from Cantor space to arbitrary computable probability spaces. This same method is also applied to give machine characterizations of computable and Schnorr randomness for computable probability spaces, extending the previously known results. The paper contains a new type of randomness-endomorphism randomness-which the author hopes will shed light on the open question of whether Kolmogorov-Loveland randomness is equivalent to Martin-Löf randomness. The last section contains ideas for future research.