Minimal submanifolds are the critical points of the volume functional. If the second derivative of the volume is nonnegative, we say that such a minimal submanifold is stable.

After reviewing some basics of minimal submanifolds in a generic Riemannian manifold, I will give some motivations behind the Lawson--Simons conjecture, which claims that there are no stable minimal submanifolds in 1/4-pinched spheres. Finally, I will discuss my recent work with Giada Franz on the nonexistence of stable minimal submanifolds in conformal pinched spheres.

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.