A meandric system of size n is the set of loops formed from two arc diagrams (non-crossing perfect matchings) on {1,⋯,2n}, one drawn above the real line and the other below the real line. Equivalently, a meandric system is a coupled collection of meanders of total size n. I will discuss a conjecture which describes the large-scale geometry of a uniformly sampled meandric system of size n in terms of Liouville quantum gravity (LQG) decorated by certain Schramm-Loewner evolution (SLE) type curves. I will then present several rigorous results which are consistent with this conjecture. In particular, a uniform meandric system admits macroscopic loops; and the half-plane version of the meandric system has no infinite paths. Based on joint work with Jacopo Borga and Ewain Gwynne.

In astrophysical fluid dynamics, stars are considered as isolated fluid masses subject to self-gravity. A classical model of a self-gravitating Newtonian star is given by the gravitational Euler-Poisson system, while a relativistic star is modeled by the Einstein-Euler system. I will review some recent progress on the local and global dynamics of Newtonian stars, and discuss mathematical constructions of gravitational collapse that show the existence of smooth initial data leading to finite time collapse, characterized by the blow-up of the star density. For Newtonian stars, dust-like collapse and self-similar collapse will be presented, and the relativistic analogue and formation of naked singularities for the Einstein-Euler system will be discussed.

TBA