The stable homotopy groups of spheres are highly interesting, also from a geometric point of view (they participate in the description of the group of exotic spheres) but also very complicated to describe. From Waldhausen’s ``brave new algebra’’ point of view, they are given by the homotopy groups of the sphere spectrum, the initial commutative ring in this brave new algebra. One approach to studying these stable homotopy groups is through a filtration on the sphere spectrum called the chromatic filtration, related to the stratification of the moduli of formal groups by height. Ravenel conjectured a number of structural results about stable homotopy theory from this ``chromatic’’ point of view, all of which except for one, the telescope conjecture, have been affirmatively solved around 35 years ago.\\ A priori unrelated is algebraic K-theory: It is also highly interesting from many points of views, and also very complicated to compute in general. Waldhausen realised that the K-theory of rings in higher algebra (like the sphere spectrum) contain yet more geometric information and suggested to use the chromatic filtration to study the K-theory of the sphere spectrum. Ever since, the relation between algebraic K-theory and chromatic homotopy theory has been the context of number of deep results and conjectures.\\ Recent advances in precisely this interaction between chromatic homotopy theory and algebraic K-theory solve a number of these conjectures in the positive, and quite surprisingly, have even lead to a disproof of the telescope conjecture.\\ After a gentle introduction to the above, I will survey these recent advances, and how they can be used to even say something about the structure of the stable homotopy groups of spheres themselves.

We discuss unsolved problems of linear algebra and how they can be encoded in moduli spaces. We then review some typical results and conjectures on invariants of such moduli spaces, resulting in potential applications to Tamari lattices.

Given a ring R, a Sylvester rank function on the category of finitely presented right R-modules is an isomorphism-invariant function which is additive on coproducts, subadditive on right-exact sequences, monotone on quotients, and taking the value 1 on R. In this talk I will start by observing that any Sylvester rank function can be uniquely extended to a so-called length function on the category of functors from finitely presented left R-modules to Abelian groups.\\ This enlargement of the setting comes with many advantages, in fact, the richer structure of the functor category makes it possible to use tools like torsion-theoretic localizations, rings of definable scalars and the Ziegler spectrum in the study of rank functions. To illustrate this, I will provide examples of results about rank functions whose initial proofs are technically demanding, yet can be derived almost effortlessly within the expanded framework.

Given a ring R, a Sylvester rank function on the category of finitely presented right R-modules is an isomorphism-invariant function which is additive on coproducts, subadditive on right-exact sequences, monotone on quotients, and taking the value 1 on R. In this talk I will start by observing that any Sylvester rank function can be uniquely extended to a so-called length function on the category of functors from finitely presented left R-modules to Abelian groups. This enlargement of the setting comes with many advantages, in fact, the richer structure of the functor category makes it possible to use tools like torsion-theoretic localizations, rings of definable scalars and the Ziegler spectrum in the study of rank functions. To illustrate this, I will provide examples of results about rank functions whose initial proofs are technically demanding, yet can be derived almost effortlessly within the expanded framework.