Colloquium Ical Atom Mailing List

The Mobius function is one of the most important arithmetic functions. There is a vague yet well known principle regarding its randomness properties called the “Mobius randomness law”. It basically states that the Mobius function should be orthogonal to any “structured” sequence. P. Sarnak suggested a far reaching conjecture as a possible formalization of this principle. He conjectured that “structured sequences” should correspond to sequences arising from deterministic dynamical systems. I will describe progress in recent years towards these conjectures building on major advances in ergodic theory, additive combinatorics, and analytic number theory.

In this talk, we explain how the classical Toda lattice model, one of the earliest examples of nonlinear completely integrable systems, can be used to demonstrate that certain configurations in the classical phase space are symplectic balls in disguise. No background in symplectic geometry is needed. The talk is based on joint work with Vinicius Ramos and Daniele Sepe.

A hyperkahler manifold is a compact holomorphically symplectic manifold of Kahler type. We are interested in hyperkahler manifolds of maximal holonomy, that is, ones which are not flat and not decomposed as a product after passing to s finite covering.

The group of automorphisms of such a manifold has a geometric interpretation: it is a fundamental group of a certain hyperbolic polyhedral space. I will explain how to interpret the boundary of this hyperbolic group as the boundary of the ample cone of the hyperkahler manifold. This allows us to use the fractal geometry of the limit sets of a hyperbolic action to obtain results of hyperkahler geometry.

One easy way of representing knot is via a knot diagram. However, inferring properties of the knot from its diagram and deciding when two diagrams represent the same knot are quite difficult problems. Surprisingly, when the diagram is sufficiently “twisty” then some structure starts to emerge. I will discuss two results of this nature: hyperbolicity of highly twisted knot diagrams and uniqueness of highly twisted plat diagrams.

Based on joint works with Yoav Moriah, Tali Pinsky and Jessica Purcell. All relevant notions will be explained in the talk.