A single soap bubble has a spherical shape since it minimizes its surface area subject to a fixed enclosed volume of air.  When two soap bubbles collide, they form a “double-bubble” composed of three spherical caps.  The double-bubble minimizes total surface area among all sets enclosing two fixed volumes.  This was proven mathematically in a landmark result by Hutchings-Morgan-Ritore-Ros and Reichardt using the calculus of variations in the early 2000s.  The analogous case of three or more Euclidean sets is considered difficult if not impossible.  However, if we replace Lebesgue measure in these problems with the Gaussian measure, then recent work of myself (for 3 sets) and of Milman-Neeman (for any number of sets) can actually solve these problems.  We also use the calculus of variations.  Time permitting, we will discuss an improvement to the Milman-Neeman result and applications to optimal clustering of data and to designing elections that are resilient to hacking.  http://arxiv.org/abs/1901.03934

The Allen-Cahn equation appears in the study of phase-transitions for a fluid with two-stable phases. It has been known from the work of Modica and Mortola that the level sets of the solution behave at large scales as minimal surfaces. This fact suggests that global solutions to the Allen-Cahn equation have the same rigidity properties as global minimal surfaces. In particular De Giorgi conjectured that the Bernstein theorem for minimal graphs is valid for the Allen-Cahn equation. I will discuss the history of this conjecture together with some of its nonlocal counterparts.

Differential equations deal with the same matters as children do: pictures in the plane. If a picture related to a differential equation remains (topologically) the same after the equation is slightly perturbed, this equation is structurally stable. If it is not, abrupt changes of the corresponding picture may occur under a small perturbation. These abrupt changes are the subject of the bifurcation theory. This talk gives a survey of the first three years of development of a new branch of the bifurcation theory: global bifurcations on the two sphere. Bifurcations in generic one-parameter families were classified; the answer appeared to be quite unexpected. An important and non-trivial question ”who bifurcates?” was answered. Natalya Goncharuk and the speaker defined a set called large bifurcation support; bifurcations that occur in a small neighborhood of this set determine the global bifurcations on the two-sphere. This result is a starting point for systematic classification of global bifurcations in two-parameter families. New examples of structurally unstable three-parameter families will be demonstrated. These are joint results of the speaker and his collaborators: N. Goncharuk, D. Filimonov, Yu. Kudryashov, N. Solodovnikov, I. Schurov and others. The talk will be addressed to a broad audience.

This is the fourth in a series of lectures on naive descriptive set theory based on an expository paper by Matt Foreman. We continue the discussion of universality properties of Polish spaces and subspaces of Polish spaces.