Much like operads and monads, Lawvere theories are a way of encoding algebraic structures, such as those of modules over a ring or sets with a group action. In this talk, we discuss the algebraic K-theory of Lawvere theories, which contains information about automorphism groups of these structures. We'll discuss both particular examples and general constructions in the K-theory of Lawvere theories, including examples showing the limits of Morita invariance and the construction of assembly-style maps. This is joint work with Markus Szymik.

The class number formula is an inspiring pillar of number theory. By the work of many mathematicians, notably Deligne, Beilinson, Bloch, Kato, Fontaine, Perrin-Riou, Jannsen, and many others, we now have a quite general (conjectural) class number formulas for motives, i.e., the Tamagawa number conjecture of Bloch-Kato. Recently, Kato has proposed a new approach to this problem using heights of motives. In this talk, we will give an overview of this approach. In particular, we will show a precise relation between heights to Tamagawa numbers of motives. We also partially answer some of Kato's questions about the number of mixed motives of bounded heights in the case of mixed Tate motives.