Recent Progress in Mathematics

This volume brings together five chapters that examine open problems at the current frontier of mathematics. Each one traces how its topic developed, where it stands today, and where it may be heading. Four chapters are surveys, while the fifth draws more directly on original research. The fields covered include algebraic geometry, topology, partial differential equations, Riemannian geometry, and harmonic analysis.

Young-Hoon Kiem revisits enumerative geometry, looking at its work before string theory and the refinements that followed it, and then turns to recent progress in quantum singularity theory, Donaldson-Thomas theory for Calabi-Yau 4-folds, and Vafa-Witten invariants. Dongho Chae studies whether solutions to the three-dimensional incompressible Euler equations can blow up in finite time. He begins with Kato's classical local well-posedness theory and the Beale-Kato-Majda blow-up criterion, then reports on newer work on the analogous singularity question for the two-dimensional Boussinesq equations.

Simon Brendle describes recent advances that make it possible to classify every singularity model on a three-dimensional Riemannian manifold, and he offers a new proof of the classification of noncollapsed steady gradient Ricci solitons in dimension three. Hyeonbae Kang reviews the Neumann-Poincare operator, covering visibility and invisibility through polarization tensors, the decay of its eigenvalues and the localization of plasmons near surfaces, the effect of singular geometry on its essential spectrum, stress analysis, and the structure of the elastic version of the operator. Danny Calegari gives a concrete account of the shift locus as a complex built over a contractible building, describing its pieces through dynamically extended laminations and affine algebraic varieties with discriminant-like structure.

The book is intended for researchers working in these areas.