NOTE: Tuesday meeting

This is the last lecture in an introductory survey of the theory of algorithmic randomness. The primary question we wish to answer is: what does it mean for a set of natural numbers, or equivalently an infinite binary sequence, to be random? We will focus on three intuitive paradigms of randomness: (i) a random sequence should be hard to describe, (ii) a random sequence should have no rare properties, and (iii) a random sequence should be unpredictable, in the sense that we should not be able to make large amounts of money by betting on the next bit of the sequence. Using ideas from computability theory, we will make each of these three intuitive notions of randomness precise and show that the three define the same class of sets.

We give an introductory survey of the theory of algorithmic randomness. The primary question we wish to answer is: what does it mean for a set of natural numbers, or equivalently an infinite binary sequence, to be random? We will focus on three intuitive paradigms of randomness: (i) a random sequence should be hard to describe, (ii) a random sequence should have no rare properties, and (iii) a random sequence should be unpredictable, in the sense that we should not be able to make large amounts of money by betting on the next bit of the sequence. Using ideas from computability theory, we will make each of these three intuitive notions of randomness precise and show that the three define the same class of sets.

 This talk will be concerned with compactness phenomena in set theory. Compactness is the phenomenon by which the local properties of a mathematical structure determine its global behaviour. This phenomenon is intrinsic to the architecture of the mathematical universe and manifests in various forms. Over the past fifty years, the study of compactness phenomena has been one of the flagships of research in set theory. This talk will present recent discoveries spanning classical themes like the tree property and stationary reflection while also forging new connections with other topics, such as Woodin's HOD Conjecture.

We prove from ZF + AD + DC that there is no sequence of distinct $\Gamma_{1,m}$ sets of length $\aleph_{m+2}$. This is the optimal result for the pointclass $\Gamma_{1,m}$ by earlier work of Hjorth. We also get a bound on the length of sequences of $\Gamma_{2n+1,m}$ sets using the same techniques.