Why Mathematical Physics?

The 2022 Prizes from the International Mathematical Union featured two focusing on mathematical physics: a Fields Medal to Hugo Duminil-Copin, and the Carl Friedrich Gauss Prize for Applications of Mathematics to Elliott H. Lieb. Duminil-Copin (and his teacher, Stanislav Smirnov, 2010 Fields Medal) developed rigorous ways of thinking of a lattice at its critical local connection probability that leads to a path through the lattice (Fig. 2.1). They employed a discrete complex analysis. Then, they develop ingenious ways of describing that lattice at the critical point, ways amenable to rigorous mathematical proof.  In a lifetime of work, Lieb has employed mathematics as the usual physicists’ starting point to reveal the physics. For example, Lieb and collaborators (1964) showed how that Ising lattice might be thought of as a quantum field theory (with annihilation and creation operators). Themes are: Technique is physical. Rigor is revealing. Tricks are physical. And, mathematical physics often leads to deep mathematics.