Talk

A conjecture of Berger states that on a simply connected manifold $M$ in which all geodesics are closed, all geodesics have the same length. In this talk, we explain the proof of this conjecture for three dimensional manifolds. The proof is carried out in two parts.

A fundamental sequence of operads is that of the little n-cubes operads, originally due to Boardman-Vogt and May. It led to the notion of an E_n-operad, as one equivalent to the little n-cubes operad. There is no really structural definition of this notion, but since the introduction of the little n-cubes operads, many very different types of E-n-operads have been discovered. One of these is a quite elementary and seemingly rather central one:  the complete graphs operad, originally proposed by Berger and later modified in a paper by Brun, Fiedorwiecz, and Vogt.

It remains an open problem whether every fibered link in the 3-sphere is the binding of a braided open book; a positive answer to this question would prove a conjecture of Montesinos and Morton. While braided open books generalize braids, we introduce mutual arc presentations as a natural generalization of arc presentations of links in 3-manifolds. We show that if a fibered link admits a mutual arc presentation, then it is the binding of a braided open book.

The modular curve with $\Gamma_0(p)$-level structure has a
beautiful integral model over $\BZ_p$, and generalizations of this model
to any Shimura variety with $p$-parahoric level have been constructed in
the last 30 years. When passing from $\Gamma_0(p)$-level to
$\Gamma_1(p)$-level, the situation changes drastically. In the talk I
will explain a general method that potentially allows to deal with such
level structures. The method is based on the construction of torus