In differential geometry, we know that an irrational torus is a pathological space rather than a smooth manifold. However, we can still do some kind of differential geometry by using a new categorical viewpoint towards geometry. In this talk, basic geometry, homotopy groups, smooth bundles, and some new work on tangent spaces for irrational torii will be introduced. If time permits, we will talk about differential forms, de Rham cohomology and foliation theory on irrational torii as well.

After recalling the definition of Chow motives I will present a proof of Rost nilpotence for surfaces and some three folds, and discuss some applications of this result.

Let k be a field and G be an algebraic group. If char k=0 Birger Iversen showed in 1976 that the pull-back Br(k)$\to$ Br (G) is an isomorphism if $G$ is simply connected. If fact, he proved this using topological methods for $k$ the field of complex numbers from which the general case follows by Galois cohomology. In my talk I will present one or two (if time permits) algebraic proofs of Iversen's result which show more: If $G$ is simply connected and $k$ arbitrary then the pull-back $n$-torsion Br(k)$\to$ $n$-torsion Br (G) is an isomorphism as long as $n$ is prime to the characteristic of $k$. If $k$ is not perfect of char $p$ I will show then that the pull-back $p$-torsion Br(k)$\to$ $p$-torsion Br(G) is not surjective for any semisimple isotropic connected linear algebraic group over $k$.