We plan to work through the small book "Topology from the Differential Viewpoint" by John Milnor, one of the Classics of the Princeton University Press and a mathematical jewel. It introduces the reader to some fundamental concepts of early topology, but instead of using algebraic and combinatorial tools it uses elementary methods from real analysis, also introducing some basic concepts of differential topology like differential manifolds, tangent bundles and vector fields. The topics center around the concept of degree of mappings between manifolds as developed by Brouwer in the beginning of the 20-th century, and the booklet leads up to some interesting and from the outset not so trivial results of algebraic topology. In particular the relationship between indices of vector fields and Euler characteristics and the classification of homotopy classes of maps from manifolds into spheres of the same dimension are dealt with in a beautiful way. The final talks will treat homotopy classes of maps from (n+k)-manifolds to k-dimensional spheres which leads to a geometric understanding of higher homotopy groups of spheres.

This seminar is intended for advanced bachelor students and beginning master students. For most of the talks a good understanding of Analysis of several variables will suffice. There will be one or two talks where a basic knowledge of singular homology is needed. This will be covered in the first half of Pavle Blagejovics lecture Topology 2.

We encourage all participants to give their presentation in English, but anyone who absolutely panics at the thought of giving her/his talk in English might use German. (No other language, please)