Classes in Zakharevich K-groups constructed from Quillen K-theory
Freitag, 22.10.21, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
(joint work with M. Groechenig)\n\nThe Grothendieck ring of varieties K0(Var) is defined a lot like K0\nin Quillen K-theory. However, vector bundles get replaced by varieties\nand instead of quotienting out exact sequences, we quotient out Z -> X\n<- U relations, where Z is a closed subvariety and U its open\ncomplement.\nThis ring plays a crucial role in motivic integration, as in the proof\nthat K-equivalent [that's yet another meaning of K...] varieties have\nthe same Hodge numbers.\n\n Even though things look very analogous, K0(Var) is not the K0 group\nof some abelian category (or Waldhausen etc). Usual K-theory\nfoundations do not apply. Zakharevich and Campbell established that\nthere is an analogous theoretical formalism nonetheless, so there are\nalso higher K-groups corresponding to K(Var). However, until recently,\nit was not known whether any such Kn(Var) for n>0 contains any\nnon-zero element beyond torsion classes. Some months ago, we managed to\ngive the first construction of such, indeed showing that for all odd\nn>=3 the group Kn(Var) is infinite-dimensional. To do this, we develop\ntwo new tools. Joint with M. Groechenig and A. Nanavaty, motivic\nrealizations give rise to maps out of K(Var), and (joint just with M.\nGroechenig) there is a kind of exponential map from Quillen K-theory to\nK(Var), allowing us to import Quillen K-theory classes to give rise to\nclasses living on abelian varieties in K(Var).
Algebraic independence and special functions
Freitag, 29.10.21, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
In this talk we are going to see that the solutions of many linear functional equations do not satisfy algebraic differential equations. As an application, we will see how it yields to the proof of the algebraic independence of some special functions.\n\nThis is a joint work with B. Adamczewski, C. Hardouin and M. Wibmer