The Bott periodicity theorem is of fundamental importance in many areas of mathematics, from algebraic topology to functional analysis. It appears unexpectedly in different guises and I would like to explain some of these as well as the influence it has had on the development of different fields. I will concentrate on two roles that periodicity plays. First, periodicity allows one to deloop cla...

Almost everyone in China knows this story, even elementary school pupils: Once upon a time, there was a weapon salesman claiming with confidence to one crowd that all his spears were capable of piercing every shield that existed in the world, then he turned around and told the other crowd mysteriously that all his shields were not penetrable. A bystander who heard both claims asked him, “How ab...

“Quantum holonomy” as defined by Berry and Simon, and based on the parallel transport of Bott and Chern, can be considerably extended. There is a natural “parallelity” W*dW = (dW)* W within the Hilbert--Schmidt operators W This defines parallelity and holonomy along curves of density operators e = WW*. There is an intrinsic nonlinearity in the parallel transport which dissolves for curves of pr...

We construct Bott-type Floer homology groups for the sym-plectic manifold (T S 1 ; can) and Bott-type Morse homology groups for the energy functional on the loop space of S 1. Both objects turn out to be isomorpic to the singular homology of the loop space of S 1. So far our objects depend on all choices involved, but the above isomorphism suggests further investigation to show independence of ...

We study the geometry and topology of Bott towers in the context of toric geometry. We show that any kth stage of a Bott tower is a smooth projective toric variety associated to a fan arising from a crosspolytope; conversely, we prove that any toric variety associated to a fan obtained from a crosspolytope actually gives rise to a Bott tower. The former leads us to a description of the tangent ...