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 $P_{k}^{3}$

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.

I shall discuss two elementary inequalities relating capacity to coverings. They provide an approach to determining whether a set has positive capacity and, if so, to estimating the value of the capacity. (Joint work with Quentin Rajon, Jeremie Rostand and Alexis Selezneff).

(Note the unusual day, time, and room.) I will talk about homotopy type theoretic proofs of a well known topological fact that the fundamental group of the circle is the set of integers. There are two closely related proofs. The homotopy-theoretic proof follows a similar reasoning used in the classical proof in topology, whereas the encode-decode proof is more type theoretic.

I will begin by discussing how and when the action of a Hopf algebra $H$ on an algebra $A$ can be viewed as a grading of $A$. For example, if $G$ is a finite group and $H$ is the dual of the group algebra $K[G]$, then $A$ is an $H$-algebra precisely when $A$ is group-graded by $G$. I will then discuss the identical relations of an algebra with a Hopf algebra action. In particular, I will address the following question: when does the existence of an $H$-identity on $A$ imply the existence of an ordinary polynomial identity on $A$?

We will survey some characterization theorems for locally presentable categories and variants thereof. Then we will discuss some applications of locally presentable categories to homological and homotopical algebra.