Let \(\Sigma\) be a genus \(g\) surface with \(n\) punctures. We will define a variety that parametrizes \({\rm SL}_d\)-representations of \(\Sigma\) in which the loops around the punctures have fixed characteristic polynomial. We will discuss two properties, geometric irreducibility and smoothness. The proof of the former uses a method due to Liebeck-Shalev involving characters of the finite group \({\rm SL}_d(\mathbb{F}_q)\) and the Lang-Weil theorem from algebraic geometry. The proof of the second applies linear algebra to the differentials of the commutator and characteristic polynomial maps. Time permitting, we will define the action of the pure mapping class group of \(\Sigma\) on our variety and indicate how our two results are used in studying the orbits.

The classical Yamabe problem—solved through the work of Yamabe, Trudinger, Aubin, and Schoen—asserts that on any closed smooth connected Riemannian manifold $(M^n,g)$, $n\geq 3$, one can find a metric conformal to $g$ with constant scalar curvature. A fully nonlinear analogue replaces the scalar curvature by symmetric functions of the Schouten tensor. Traditionally, the existing theory has required the scalar curvature to have a fixed sign. In a recent work, we broaden the scope of fully nonlinear Yamabe problem by establishing optimal Liouville-type theorems, local gradient estimates, and new existence and compactness results. Our results allow conformal metrics with scalar curvature of varying signs. A crucial new ingredient in our proofs is our enhanced understanding of solution behavior near isolated singularities. I will also discuss extensions to manifolds with boundary, treating prescribed boundary mean curvature and the boundary curvature arising from the Chern–Gauss–Bonnet formula.

We prove local well-posedness as well as singularity formation for the g-SQG patch model on the plane (so on a domain without a boundary), with  $\alpha\in(0,\frac{1}{6}]$ and patches being allowed to touch each other. We do this by bypassing any auxiliary contour equations and tracking patch boundary curves directly instead of their parametrizations. In our results, which are sharp in terms of $\alpha$, the patch boundaries have $L^{2}$ curvatures and a singularity occurs when at least one of these $L^{2}$-norms blows up in finite time.