We will explain our recent joint work with G. Cortinas, M. Walker and C. Weibel concerning properties of the algebraic $K$-theory of toric varieties in positive characteristic. The results are proved using trace methods and a variant of the cyclic nerve construction that provides a homotopy theoretical model of the so-called Danilov sheaves of differentials. Most of the technical work happens completely within the world of monoid schemes (which are a particular manifestation of what goes by the name of "schemes over the field with one element").

In two subsequent less formal talks, I will endeavour to explain the theory of presheaves of spectra on monoid schemes, and the way topological cyclic homology is used in the constructions and proofs.

In this talk I'll explain my recent results on the spinor genus of the integral trace form of a number field. I'll show how from them one can decide in terms of finitely many ramification invariants, and under some restrictions, whether or not a pair of number fields have isometric integral trace forms. Inspired by the work of R. Perlis on number fields with the same zeta function, I'll define the notion of weak arithmetic equivalence, and I'll show that under certain hypothesis this equivalence determines the local root numbers of the number field, and the isometry class of the integral trace form.

I will start by defining the Hochschild cohomology of an algebra $A$ with coefficients in an $A$-bimodule $M$. Some examples will be given and special attention will be paid to the case $M=A^*$, the linear dual of $A$. I will then show that Hochschild cohomology is a derived functor, and will use this technique to compute the Hochschild cohomology of noncommutative tori. Following this, I will introduce the cyclic cohomology of an algebra $A$, and will derive Connes' long exact sequence, which provides a powerful tool for computing cyclic cohomology.

A basic problem in algebraic topology is to write down all homotopy classes of maps between spheres. This problem, simple to state, is impossible to solve -- we don't even have a working guess. However, we've gotten very good at using some very specialized bits of algebraic geometry (the theory of abelian varieties and $p$-divisible groups, to be be precise) to get some hold on large scale patterns in these groups. After talking about how this connection works, I'll review some of the basic examples, going back even into the 1960s, then talk about the work of Hopkins-Miller-Behrens in the 2000s that uncovering some remarkable patterns using modular forms. This is only a start, and I'll end with some current vistas.