This book gathers written notes from three lecture series presented at the PIMS–IFDS–NSF Summer School on Optimal Transport, held at the University of Washington in June 2022. The summer school was the first major event organized by the Kantorovich Initiative, a nascent research consortium linking several universities in the Pacific Northwest and dedicated to advancing the mathematics of Monge–Kantorovich transport problems and their many applications. The mini-courses offered during the summer school—and the lecture notes collected here—are poised to have a substantial impact on the next generation of optimal transport researchers. The range of topics included in this volume reflects the remarkable breadth of contemporary research in the field.
 
Felix Otto and Lukas Koch’s contribution, based on Otto’s mini-course, presents a variational perspective on the regularity theory of optimal transport. Alfred Galichon and Antoine Jacquet’s notes, drawn from Galichon’s mini-course, explore the links between optimal transport and matching models in economics. Finally, Geoffrey Schiebinger’s notes survey his mini-course on applications of optimal transport in developmental biology.

Young-Heon Kim is a professor of mathematics at the University of British Columbia (UBC), Vancouver. His research interests are in theory and applications of optimal transport (OT), combining geometric, PDE, and probabilistic perspectives. His main results in the OT area, with his collaborators, include progress in martingale optimal transport theory, stochastic OT approach to the supercooled Stefan problem, pseudo-Riemannian geometry approach for regularity theory of optimal transport on geometric domains and general cost functions, analysis of Wasserstein barycenters on Riemannian manifolds, heat flow approach to Caffarell’s construction, as well as results in trajectory inference analysis for cell biology,  among others. Young-Heon obtained his Ph.D. in 2005 from Northwestern University and was a postdoctoral fellow at University of Toronto (2005–2008) and a visiting member at Institute for Advanced Study (2008/2009), before joining UBC.

Soumik Pal is a professor of mathematics at the University of Washington, Seattle, with adjunct appointments in the departments of applied mathematics and statistics. His research interests are in probability theory, especially in stochastic analysis and optimal transport. He did his Ph.D. from Columbia University in 2006. After spending two years at Cornell University as a visiting assistant professor, he joined University of Washington as a tenure-track in 2008. For his contributions, he was elected a fellow of the Institute of Mathematical Statistics (IMS) in 2025.

Brendan Pass is a professor in the Department of Mathematical and Statistical Sciences at the University of Alberta (Edmonton, Alberta, Canada). He works primarily on optimal transport, with a particular emphasis on multi-marginal problems, Wasserstein barycenters, unequal dimensional problems, and applications in economics