Veranstaltungen

Toric contact manifolds provide an interesting class of contact manifolds. In this mini-​course we will introduce them, show how the ones with zero first Chern class can be determined by certain integral convex polytopes, called toric diagrams, and how to directly read relevant contact invariants from these toric diagrams. Plenty of hands-​on examples and some applications will be provided.

More information: https://math.ethz.ch/fim/activities/minicourses.htmlcall_made

The Strichartz estimate is one of fundamental tools in the study of nonlinear dispersive equations. This talk deals with (global- in-time) Strichartz estimates for Schrödinger equations with potentials decaying at infinity. The case when the potential decays sufficiently fast has been extensively studied in the last three decades. However, it has remained mostly unknown for slowly decaying potentials in which case the standard perturbation argument does not work. We instead employ several techniques from long-range scattering theory and microlocal/semiclassical analysis, and prove Strichartz estimates for a class of positive potentials decaying arbitrarily slowly. A typical example is the positive Coulomb potential in three space dimensions. As an application, we also obtain a modified scattering type result for the final state problem of the nonlinear Schrödinger equations with long-range nonlinearity and potential. This is partly joint work with Masaki Kawamoto (Ehime University).

The aim of this talk is to study the ergodicity of a class of partially hyperbolic flows constructed by taking unitary extensions to vector bundles of the geodesic flow over a closed negatively-curved manifold. We will show that, when the rank of the vector bundle is small compared to the dimension of the manifold, ergodicity can be completely characterized in geometric terms -- a certain connection admits no holonomy reduction. The proof combines classical hyperbolic dynamics, harmonic analysis, and, more surprisingly, real algebraic geometry (classification of polynomial maps between sphere). Joint work with Mihajlo Cekic.

Toric contact manifolds provide an interesting class of contact manifolds. In this mini-​course we will introduce them, show how the ones with zero first Chern class can be determined by certain integral convex polytopes, called toric diagrams, and how to directly read relevant contact invariants from these toric diagrams. Plenty of hands-​on examples and some applications will be provided.

More information: https://math.ethz.ch/fim/activities/minicourses.htmlcall_made

I will review the construction of polynomial solutions of KZ equations modulo an integer and the properties of the solutions, in particular, their p-adic limit.