Guest lectures and seminars - Page 113

Erik Bølviken (University of Oslo) gives a lecture with the title: Where models meet reality - The Solvency II regulation of  European insurance

The Barratt nerve BSd X of the Kan subdivision Sd X of a simplicial set X \in sSet is a triangulation. The Barratt nerve is defined as taking the poset of non-degenerate simplices, thinking of it as a small category and then finally taking the nerve.Waldhausen, Jahren and Rognes (Piecewise linear manifolds and categories of simple maps) named this construction 'the improvement functor' because of the homotopical properties and because its target is non-singular simplicial sets. A simplicial set is said to be 'non-singular' if its non-degenerate simplices are embedded. There is a least drastic way of making a simplicial set non-singular called 'desingularization', which is a functor D:sSet -> nsSet that is left adjoint to the inclusion. The functor DSd^{^2} is the left Quillen functor of a Quillen equivalence where the model structure on sSet is the standard one where the weak equivalences are those that induce weak homotopy equivalences and the fibrations are the Kan fibrations. I will talk about the main steps of the proof that the natural map DSd X -> BX is an isomorphism for regular X. This implies that DSd^{^2} is a triangulation and that the improvement functor is less ad hoc than it may seem. Furthermore, I will explain how the result provides evidence that any cofibrant non-singular simplicial set is the nerve of some poset.   

Triangulated categories of motives over schemes are sort of the "universal derived categories" among various derived categories obtained by various cohomology theories like l-adic cohomology. Ayoub constructed them using the A1-homotopy equivalences and étale topology. I will introduce the construction of triangulated categories of motives over fs log schemes. Fs log schemes are kinds of "schemes with toroidal boundary," and A1-homotopy equivalences and étale topology are not enough to obtain all homotopy equivalences between fs log schemes. I will explain what extra homotopy equivalences and topologies are neeeded.