We study maximal independent families (m.i.f.) in the projective hierarchy. We show that (a) the existence of a Σ 1 2 m.i.f. is equivalent to the existence of a Π 1 1 m.i.f., (b) in the Cohen model, there are no projective maximal independent families, and (c) in the Sacks model, there is a Π 1 1 m.i.f. We also consider a new cardinal invariant related to the question of destroying or preserving maximal independent families. ∀X ∈ [ω] ω ∃a 1 , . . . , a n , b 1 , . . . , b ℓ ∈ I s.t. σ(ā;b) ⊆ * X or σ(ā;b) ∩ X = * ∅.

We provide a list of open problems in the research area of generalised Baire spaces, compiled with the help of the participants of two workshops held in Amsterdam (2014) and Hamburg (2015).

We prove the consistency of together with the existence of a -definable mad family, answering a question posed by Friedman and Zdomskyy in [7, Question 16]. For the proof we construct a mad family in L which is an ℵ1-union of perfect a.d. sets, such that this union remains mad in the iterated Hechler extension. The construction also leads us to isolate a new cardinal invariant, the Borel almost-disjointness number, defined as the least number of Borel a.d. sets whose union is a mad family. Our proof yields the consistency of (and hence, ).

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We investigate the sample path properties of Martin-Löf random Brownian motion. We show (1) that many classical results which are known to hold almost surely hold for every Martin-Löf random Brownian path, (2) that the effective dimension of zeroes of a Martin-Löf random Brownian path must be at least 1/2, and conversely that every real with effective dimension greater than 1/2 must be a zero of some Martin-Löf random Brownian path, and (3) we will demonstrate a new proof that the solution to the Dirichlet problem in the plane is computable.