We study the test model structure (as well as a “projective" variant) on several categories of cubical sets (cubical sets with or without connections / symmetries / reversals), study their monoidal properties, and establish Quillen equivalences among them and to simplicial sets and topological spaces.

With this groundwork in place, we then consider (on most of these cubical sites) cubical subdivision — an analog of simplicial subdivision (which is even better because it is strong monoidal!). This leads to a cubical analog of Kan’s $\mathrm{Ex}^\infty$ functor which we show to be a functorial fibrant replacement with good properties. As a corollary, we obtain cubical approximation theorems for various flavors of cubes.

The set $M(A)$ of multiplicative seminorms on a commutative ring $A$ carries a natural topology. By considering completions of $A$ with respect to these seminorms, Berkovich obtained a structure sheaf of $\mathcal O_A$ of "analytic" functions on $M(A)$.

We consider (the rudiments of) a categorification of Berkovich’s theory, associating to every symmetric monoidal stable infinity category $\mathcal A$ a topological space $M(\mathcal A)$ and sheaf of categories $\mathcal O_{\mathcal A}$. We emphasize the concrete nature of these objects. For example, for $k$ a field and any $r > 0$ the function $$N_{r,k}: \mathsf{Ob}(\mathsf{Sp}^\mathrm{fin}) \to \mathbb{R}_{\geq 0}$$ $$ X \mapsto \sum_i \mathrm{dim} H_i(X; k) r^i$$ is a point in $\mathcal M(\mathsf{Sp}^\mathrm{fin})$. Moreover, the skeletal filtration of a spectrum is often a "Cauchy sequence."

We carry through the theory far enough to compute $M(\mathsf{Sp}^\mathrm{fin})$ (which the reader familiar with chromatic homotopy theory may now guess).

Given a smooth projective variety X over F_q, a natural question is what is the size of X(F_q^n) for all n. This question leads to the famous Weil conjectures. In this talk, we will talk about the first tiny element of it called the Grothendieck-Lefschez trace formula, which finds X(F_q^n) in terms of alternating sums of traces of Frobenius maps acting on l-adic étale cohomology of X.