It is well-known that a based space is of the weak homotopy type of a loop space iff it is a grouplike algebra over an $A_{\infty }$-operad. The classical model for such an operad consists of Stasheff’s associahedra. The present paper describes a similar recognition principle for free loop spaces. Let $𝒫$ be an operad, $M$ a $𝒫$-module and $U$ a $𝒫$-algebra. An $M$-trace over $U$ consists of a space $V$ and a module homomorphism $T:M\to {\text{End}}_{U,V}$ over the operad homomorphism $𝒫\to {\text{End}}_U$ given by the algebra structure on $U$. Let ${𝒞}_1$ be the little 1-cubes...

Summary of the three lectures. These notes are available electronically at http://www.ma.huji.ac.il/~drorbn/Talks/Srni-9901/notes.html.

Let $M$ be a $C^{\infty }$-manifold with a Riemannian conformal structure $C$. Given a regular curve $\gamma$ on $M$, the authors define a linear operator on the space of (differentiable) vector fields along $\gamma$, only depending on $C$, called the Fermi-Walker connection along $\gamma$. Then, the authors introduce the concept of Fermi-Walker parallel vector field along $\gamma$, proving that such vector fields set up a linear space isomorphic to the tangent space at a point of $\gamma$. This allows to consider the Fermi-Walker horizontal lift of...

The paper deals with extensions of the finite operator calculus of G.-C. Rota called $\psi$-extensions which give a framework for corresponding quantum group investigations. This also covers the instance of the well-known $q$-analogue of umbral calculus. The article also contains glossaries of the most important terms and notations used by Ward, Viskov, Markowsky and Roman on one side and the Rota-oriented notations on the other side.