Week 02.06.2025 – 08.06.2025

It follows from the Dirichlet theorem that every vector has `good' rational approximations. Singular vectors are the ones for which the Dirichlet theorem can be infinitely improved. An (obvious) example of singular vectors are the ones lying on rational hyperplanes. We will discuss the existence of totally irrational weighted singular vectors on manifolds, and also ones with high weighted uniform exponent. We will also mention some invariance of weighted uniform exponents in the case of manifolds. The talk is based on a joint work with Shreyasi Datta. 

If a set of massive objects collide in space and the fragments disperse, possibly forming black holes, then this process will emit gravitational waves. Computing the detailed gravitational wave-form associated with this process is a complicated problem, not only due to the non-linearity of gravity but also due to the fact that during the collision and subsequent fragmentation the objects could undergo complicated non-gravitational interactions. Nevertheless the classical soft graviton theorem determines the power  law fall-off of the wave-form at late and early times, including logarithmic corrections,  in terms of only the momenta of the incoming and outgoing objects without any reference to what transpired during the collision. I shall explain the results, briefly outline the derivation of these results and discuss possible generalizations and applications.