Course notes are here. Scribble notes for Lectures 1-2 are here. Scribble notes for Lectures 3-4 are here.

Disclaimer: Not everything may be covered.

We develop the spectral analysis of the Jacobi operator on the interfaces of soap-bubble clusters. By Plateau's laws, these always meet in threes at \(120^{\circ}\)-angles, and thus naturally interact via 3 linearly independent "conformal" boundary conditions (a mixture of Dirichlet and Robin). This gives rise to a self-adjoint operator, whose spectral properties determine the stability of the soap-bubbles -- whether an infinitesimal regular perturbation preserving volume to first order yields a non-negative second variation of area modulo the volume constraint. In essence, stability amounts to verifying a Poincaré-type inequality on soap-bubble clusters.

We verify that for all \(n \geq 3\) and \(2 \leq k \leq n+1\), the standard \(k\)-bubble clusters, conjectured to be minimizing total perimeter in \(\mathbb{R}^n\), \(\mathbb{S}^n\) and \(\mathbb{H}^n\), are indeed stable. In fact, stability holds for all Möbius-flat partitions, in which several cells are allowed to have infinite volume. In the Gaussian setting, any partition in \(\mathbb{G}^n\) \((n\geq 2)\) obeying Plateau's laws and whose interfaces are all flat, is stable. Our proof relies on a new conjugated Brascamp-Lieb inequality on partitions with conformally flat umbilical boundary, and the construction of a good conformally flattening boundary potential.

Joint work with Botong Xu.

A homogeneous, insulated object with a non-uniform initial temperature will eventually reach thermal equilibrium. Which point in the object takes the longest to reach this equilibrium? Intuition tells us that points "in the middle" of the material will generically reach this equilibrium temperature faster, and points "far from the middle" (i.e. at the boundary) should take more time to reach this temperature. The hot spots conjecture is a formalization of this intuition.

Rauch initially conjectured that points attaining the maximum temperature in the material would approach the boundary as time goes to infinity. Burdzy and Werner later disproved the conjecture for planar domains with holes. The general consensus, however, was that the conjecture should still hold for convex sets of all dimensions.

This talk will draw inspiration from a recurrent theme in convex analysis: most dimension-free results in convex analysis have a natural log-concave extension. We will motivate and construct the log-concave analog of the Hot Spots conjecture, and then disprove it. Using this log-concave construction, we will argue that the hot spots conjecture for convex sets is false in high enough dimensions.