Algebra / Topology Seminar

Speaker: Peter Kropholler

Title: Potential applications of condensed mathematics to Galois Cohomology of profinite groups, TDLC groups, and maybe even Hilbert’s 3^rd problem

Abstract: The Clausen--Scholze theory of condensed mathematics promises abelian categories with good closure properties (under limits and colimits etc) and with enough compact projective objects. An abundance of projective objects is akin to a Holy Grail for anyone wishing to enjoy the benefits of using Farrell--Tate cohomology and so it seems likely that condensed mathematics will provide a framework for quickly attacking and solving a number of open problems. By turning to Galois cohomology and the cohomology of TDLC groups we do not have to jump in the deep end with condensation: instead we can dip a toe in the water. In this talk I will look at a number of conjectural cohomological vanishing theorems which are formulations of natural cohomlogical finiteness conjectures about TDLC and profinite cohomology and discuss the prospects of establishing these with the new technology. Hilbert’s 3^rd problem provides another environment where condensation could play a role thanks to an observation of Sullivan widely used by, for example, Dupont and Sah: this time the analogy might be better made with ‘diving in the deep end’. This work has been enhanced by discussions with Lukas Brantner, Ged Corob Cook, and Rob Kropholler.