The talk is devoted to two related problems. The isomorphism problem for analytic discs: Suppose V is the unit disc D embedded in the d-dimensional unit ball Bd and at- tached to the unit sphere. Consider the subspace HV of the Drury-Arveson space, which consists of functions on V and its multiplier algebra MV = Mult(HV ). The isomorphism problem is the question whether V1 ∼= V2 is equivalent to MV1 ∼= MV2 . We discuss what is known for V without self-crossings on the boundary and consider what happens when there are self-crossings. The embedding dimension for complete Pick spaces: A Theorem of J. Agler and J. E. McCarthy states that any complete Pick space can be realized as HV , for some V in Bd, where d can be infinite. The smallest such d is called the embedding dimension. Given a complete Pick space can we find its em- bedding dimension? Can we at least determine if it is finite or infinite? We look into this problem for radial spaces on the unit disc D, prove a general result and provide answers for classes of such spaces, e.g., the weighted Hardy spaces. Advisor: Prof. Orr Shalit