Dr. Robin J. Sroka, Mathematisches Institut

(Co)homological stability is a celebrated principle in algebraic topology that connects the (co)homology of the general linear groups of a ring to its algebraic K-theory. The work of this Emmy Noether Research Group examines subtle variants of this principle beyond the realm of algebraic topology, shedding new light on questions from algebra and representation theory, to geometry and dynamics. This is achieved in three interrelated research projects exploring new connections between (co)homological and K-theoretic invariants. In Project A, we unlock new insights about the (co)homology of arithmetic groups and the algebraic K-theory of number rings by analyzing the interplay between conjectures of Rognes and Church--Farb--Putman. In Project B, we develop a mechanism that leverages insights about the homology of an augmented algebra to reveal new facets of its Hochschild homology. This will pave the way for new applications of homological stability ideas to questions in algebra and representation theory. In Project C, we systematically deploy the full strength of (co)homological and K-theoretic techniques to illuminate the mathematical interface of the geometry of polytopes and dynamics, bringing forth an integrated view on recent advances relating Zakharevich's scissors congruence K-theory and Matui's conjectures on ample groupoids.