Why Hopf Lectures?
by Alfred Aeppli
"To me, it is a great idea to gather from time to time in memory of Heinz Hopf, at the Swiss Federal Institute of Technology in Zurich (ETH), for some special lectures on current mathematical research. We remember, Heinz Hopf (1894-1971) was active at the ETH as one of the finest professors from 1931 to 1965 (see History). He will always be an inspiration to many of us in mathematics and in mathematical education at the university level. He was a mind at work who thrived in the world of disciplined curiosity full of intellectual challenges. He was a mathematical thinker who taught us to think. Once he had arrived in Switzerland in 1931 he was happy to live in the community provided by the Swiss higher education system. He remained a loyal member of the ETH until he died in 1971. Heinz Hopf demonstrated doing mathematics in talks, lecture courses, discussions and in his publications. I remember the introductory linear algebra course in 1947-48, the impression Heinz Hopf made on us students was outstanding and unforgettable. A student, by attending a course given by Hopf, got a good answer to the question "What is mathematics?" The notions had to be clear, the intellectual tools of the highest caliber, the arguments compelling and impeccable. There never was any doubt what could be accepted as valid.
Heinz Hopf helped create first rate thesis work in a number of cases over the years, e.g. Stiefel, Eckmann, Gysin and Samelson were among his Ph.D. students. He became the initiator of a mathematical school in Switzerland cultivating the development of ideas - new at the time - in mathematical disciplines, which can be called "geometric" in a broad sense. They include general and algebraic topology, differential geometry, manifolds, structures on manifolds, topological groups, algebraic geometry.
For Hopf, a specific well defined question was usually the starting point of a project. One example is particularly illuminating. At the time the higher homotopy groups had been introduced (by Hurewicz) it was noticed that they are abelian - like the homology groups -, and "therefore" the question was raised: can the higher homotopy groups be used to prove anything new of interest? In response to this question, Hopf produced the Hopf fibration of the 3-sphere by circles, the Hopf invariant and the homotopy classification of the continuous maps from the 3- to the 2-sphere. In those days it wasn't even clear that there are non-trivial (i.e. essential or non-contractible) continuous maps from the 3- to the 2-sphere. Hopf's answer was one of the first results in "modern" homotopy theory. Many ramifications followed, including notions like homology groups of a discrete group, Eilenberg-MacLane spaces, Moore spaces, Postnikov towers, obstructions to cross sections in fiber bundles.
Another fundamental contribution is Hopf's theorem on the homology groups of compact group manifolds. This stimulated the development of Hopf algebras with their applications in many places, e.g. in the theory of cohomology operations. Global properties of manifolds were of great interest to Hopf, including existence and classification of various structures on manifolds. He had an extraordinary ability to see interesting phenomena, e.g. the cartesian product of the 3-sphere with the circle was first recognized by Hopf as an example of a non-kaehlerian complex manifold. In classical differential geometry, Hopf's nicest result is his characterization of the ordinary 2-sphere as the only compact surface of genus 0 in Euclidean space with constant mean curvature. Many more mathematical gems out of Hopf's workshop could be mentioned.
In conclusion: let us continue to do mathematics, and let us never forget those who did mathematics before us, among them Heinz Hopf."