Activities This Week

In geometric group theory, finitely generated groups are studied from the geometric properties of their Cayley graphs. This point of view sees the group as a metric space, and allows one to apply methods of metric geometry, such as the metric compactification (also called the horofunction compactification). When considering the canonical action of the group on its boundary in such compactifications, it turns out that finite orbits are closely related to virtual homomorphisms. In this talk we will go over definitions, examples, known results and open problems for current and future research.

I will begin with an introduction to algebraic K-theory, highlighting its connections to number theory and algebraic geometry, with a focus on descent phenomena. Next, I will provide an overview of chromatic homotopy theory and its relationship to K-theory. I will then discuss the redshift conjecture and several significant advances from recent years. Finally, I will present joint work with Carmeli, Schlank and Yanovski on the compatibility of K-theory with chromatic analogues of cyclotomic extensions. I will explain how this result contributed to the recent refutation of Ravenel’s long-standing telescope conjecture.