Associated to any smooth projective curve C is its degree d Jacobian variety, parametrising isomorphism classes of degree d line bundles on C. Letting the curve vary as well, one is led to the universal Jacobian stack. This stack admits several compactifications over the stack of marked stable curves, depending on the choice of a stability condition. In this talk I will introduce these compactified universal Jacobians, and explain how their moduli spaces can be constructed using Geometric Invariant Theory (GIT). This talk is based on arXiv:2210.11457.

In this talk I will present several new stochastic representations for
solutions of the Navier-Stokes equations in a wall-bounded region,
in the spirit of mean field theory. These new representations are
obtained by using the duality of conditional laws associated with the Taylor diffusion family.
By using these representation, Monte-Carlo simulations for boundary fluid flows, including
boundary turbulence, may be implemented. Numerical experiments are given to demonstrate the usefulness of this approach.

It was recently shown by Aidekon and Da Silva how to construct a growth fragmentation process from a planar Brownian excursion. I will explain how this same growth fragmentation process arises in another setting: when one decorates a certain “critical Liouville quantum gravity random surface” with a conformal loop ensemble of parameter 4. This talk is based on joint work with Juhan Aru, Nina Holden and Xin Sun.