Mixed-norm Lebesgue spaces generalize standard Lebesgue ($L^p$) spaces. A mixed norm of a function of multiple variables is computed similarly to an $L^p$ norm, but with a different exponent associated with each variable.

When two Lebesgue spaces are defined (with different exponents and measures) for functions on a common domain, there are well-known conditions to determine whether any embedding exists between them, ultimately depending on properties of the Radon-Nikodym derivative. For most cases, we've found necessary and sufficient conditions for embeddings of mixed-norm spaces, relying on the inequalities of Holder and Minkowski.

This talk should be accessible to anyone familiar with Lebesgue spaces. However, the necessary ideas will be summarized, hopefully to make it somewhat understandable given only knowledge of normed vector spaces.

We will discuss in more detail the rules that define types, along with examples. The focus will be on inductive types, which are characterized by their constructors. An important example of inductive types is given by identity types, which play the role of path spaces in the homotopical interpretation. We will discuss path induction and the higher groupoid structure obtained from iterated identity types.

A natural way to study nilpotency in a group $G$ is to consider the colimit of the nilpotent subgroups of certain class. For a fixed nilpotency class $q$ there groups appear as the fundamental group of a certain subspace $B(q,G)$ of the usual classifying space $BG$, which are introduced in a paper by Adem, Cohen, and Torres Giese. I will discuss some properties of these colimits, and describe them for certain groups in the case of $q=2$ which corresponds to the colimit of the abelian subgroups.