Abstract: Given two Chow motives M and N satisfying Rost nilpotence, a natural question is whether their direct sum has the same property. Rather obviously this question is closely related to the famous and old Kothe conjecture. If this conjecture is true the answer to above question is yes. However it seems that most ring theorists believe Kothe's conjecture does not hold. This leads to studying stronger versions of Rost nilpotence, which (surprisingly?) hold in (almost?) all cases where usual Rost nilpotence is known, as I will explain in my talk.

Abstract: There are many results in topology showing that certain continuous maps from a space to itself have fixed points (most famously, Brouwer's fixed-point theorem). These results are often not accessible from the homotopy-theoretic point of view, since they usually depend on more than just the homotopy type of a given space, and many are also not constructive, making use of the law of excluded middle. After introducing the language of synthetic homotopy theory, we will see that studying fixed point properties in that setting provides a different, homotopy-invariant notion, and we will present (counter-)examples that can be obtained using constructive methods. For this talk, we also aim to avoid type-theoretic terminology, instead working from the topologist's perspective.