The field of algebraic topology has exposed deep connections between topology and algebra. One example of such a connection comes from algebraic K-theory. Algebraic K-theory is an invariant of rings, defined using tools from topology, that has important applications to algebraic geometry, number theory, and geometric topology. Algebraic K-groups are very difficult to compute, but advances in algebraic topology have led to many recent computations which were previously intractable. Equivariant homotopy theory, a branch of algebraic topology which studies topological objects with a group action, has been particularly important in the study of algebraic K-theory. In this talk I will introduce algebraic K-theory and its applications, explaining the interesting role of equivariant homotopy theory in this story. I will also discuss recent advances in the study of algebraic K-theory.

In this talk, I investigate the link between two little-known properties of rings and certain measure-theoretic properties of groups, particularly amenability and supramenability. The first ring-theoretic property, introduced by Paul Cohn in the 1960s, is the property of having unbounded generating number (UGN). A ring \(R\) is said to have \(UGN\) if, for every positive integer \(n\), there is no \(R\)-module epimorphism \(R^n\to R^{n+1}\). My focus is on determining conditions under which a ring graded by a group possesses UGN if its base ring has UGN---a problem originally posed by Peter Kropholler in May of 2020.

One condition that turns out to be especially relevant to Kropholler's problem is a specific measure-theoretic property of the group in relation to its subset supporting the grading. This leads to a result about rings graded by amenable groups and one about rings graded by supramenable groups, with the former yielding a new ring-theoretic characterization of amenability. The portion of the talk about Kropholler's problem is based on a paper that I wrote together with Johan Öinert and that was just published in October.

Towards the end of the lecture, I will discuss some earlier results about a property of rings that is dual to UGN, called the strong rank condition (SRC). Even more obscure than UGN, the SRC property was, I believe, first defined by T. Y. Lam in 1997 in his well-known book \(Lectures\) \(on\) \(Modules\) \(and\) \(Rings\). A ring \(R\) is said to satisfy \(SRC\) if, for every positive integer \(n\), there is no \(R\)-module monomorphism \(R^{n+1}\to R^n\). This property, too, happens to be closely related to amenability. The connection became clear in 2019 when Laurent Bartholdi (with a contribution by Dawid Kielak) proved that, for any group \(G\) and field \(k\), the group ring \(kG\) satisfies SRC if and only if \(G\) is amenable. Moreover, in another paper published that same year, Kropholler and I generalized Bartholdi's argument to certain kinds of group-graded rings.

The talk will also include a rather surprising example of a \(\mathbb{Z}\)-graded ring that fails to have UGN but whose base ring possesses UGN. In addition, I intend to mention two significant open questions, one pertaining to UGN, and the other to SRC.

The interpolation problem is one of the oldest problems in mathematics. In its most broad form it asks: when can a curve of a given type be passed through a given number of points? I'll survey work on the interpolation problem from Euclid to the modern day, ending with recent joint work of mine with Eric Larson.