We given an elementary proof of Waldhausen Additivity using key ideas from earlier proofs. Then we discuss how to prove the quasicategorical version. Model category arguments do not play a role, nor do any technical results about quasicategories. This is joint work with David Gepner and Wolfgang Lueck.

This talk will discuss further properties of categories, such as opposite functors and changing variance, products and bifunctors, adjoint functors, and exactness.

Last time, we gave a precise definition for a tropical variety as the closure of the image of a classical variety under an evaluation map. We'll continue the analysis by giving an equivalent description in terms of initial ideals and show that a tropical variety is a subcomplex of the Groebner complex. Next interesting topics in the line are the zero tension condition and Bezout's theorem as an intro to tropical intersection theory.

Every profinite group is a Galois group, but which one is also an ${\textit{absolute}}$ Galois group? The cohomological implications of the Bloch-Kato conjecture -- positively solved by M.~Rost and V.~Voevodsky -- allows us to define ${\bf{Bloch-Kato}}$ ${\bf{pro-}}$$p$ ${\bf{groups}}$, which play a crucial role, since they arise naturally as maximal pro-$p$ quotients and Sylow pro-$p$ subgroups of absolute Galois groups. In this seminar I will present the state of the art of the research on Bloch-Kato groups, with a particular mention of the 'Elementary Type Conjecture' of maximal pro-$p$ Galois groups. Yet, there's still a lot of work to do: indeed every maximal pro-$p$ Galois group is equipped with an ${\textit{orientation}}$ $G_F(p)\rightarrow\mathbb{Z}_p^\times$, arising from the action on the group of the roots of unity of $p$-power order. The study of such orientation for Bloch-Kato groups will provide hopefully new results.