Cayley submanifolds in Spin(7) geometry are an analogue and generalisation of complex submanifolds in Kähler geometry. In this talk we provide a glimpse into calibrated geometry, which encompasses both of these, and how it ties into the study of manifolds of special holonomy. We then focus on the deformation theory of compact and conically singular Cayleys. Finally we explain how to remove conical singularities via a gluing construction.

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.