We review old and new properties of systems of linear partial differential equations with constant coefficients. We discuss solvability in different function classes, to observe very different solution spaces. We examine the existence of vector potentials in the different spaces, by which we mean systems Av=0 with the property v=Bu, where A and B are linear PDE operators with constant coefficients. Properties of the systems and their solutions are examined both from linear algebra and algebraic geometry angles. A special class of operators that are examined is that of constant rank operators, which are prevalent in the nonlinear analysis of compensated compactness theory. We will discuss some of the challenges of extending this theory to non-constant rank operators.

I will consider the analysis of probability distributions through their associated covariance operators from reproducing kernel Hilbert spaces. In this talk, I will show that the von Neumann entropy and relative entropy of these operators are intimately related to the usual notions of Shannon entropy and relative entropy, and share many of their properties. They come together with efficient estimation algorithms from various oracles on the probability distributions. I will also present how these new notions of relative entropy lead to new upper-bounds on log partition functions, that can be used together with convex optimization within variational inference methods, providing a new family of probabilistic inference methods (based on https://arxiv.org/pdf/2202.08545.pdf, see also https://francisbach.com/information-theory-with-kernel-methods/).

The first edition of Thompson’s famous book On Growth and Form appeared in 1917. It has subsequently been regarded as a foundational work in mathematical biology and a revolutionary contribution to the field of morphology. Most existing literature credits Thompson as a lone genius who produced the 793 pages of the 1917 edition and 1116 pages of the 1942 edition. Thompson’s correspondence presents a very different picture of this tome as one arising from extensive and ongoing – perhaps sometimes unwitting? – collaboration.

In space dimension 2 we present a finite element complex for the deformation operator acting on vectorfields and the linearized curvature operator acting on symmetric 2 by 2 matrices. We also present the tools that were used in the construction, namely the BGG diagram chase and the framework of finite element systems. For this general framework we can prove a de Rham theorem on cohomology groups in the flat case and a Bianchi identity in the case with curvature.