Abstract: After reviewing very quickly some algebraic geometry, I will define the Hilbert scheme which parameterizes closed subschemes of projective space $P_{k}^{n}$ and state its basic properties, for k an algebraically closed field of characteristic 0. I will then define various notions of deformations (deformations sheaves, deformations over the dual numbers etc). Finally, I will use obstruction theory for a local ring to prove a lower bound on the dimension of irreducible components of the Hilbert scheme of Cohen-Macauley curves of genus g and degree d in $P_{k}^{3}$

Abstract: Every category $C$ looks locally like a category of sets, and further structure on $C$ determines what logic one can use to reason about these "sets". For example, if $C$ is a topos, one can use full (higher order) intuitionistic logic. Similarly, one expects that every higher category looks locally like a higher category of spaces. A natural question then is: what sort of logic can we use to reason about these "spaces"? It has been conjectured that such logics are provided by variants of Homotopy Type Theory, a formal logical system, recently proposed as a foundation of mathematics by Vladimir Voevodsky. After explaining the necessary background, I will report on the progress towards proving this conjecture.