Course Code at the University of Toronto: MAT1502HF

Instructor: Professor Robert McCann, University of Toronto

Course Dates: September 11 - December 10, 2026
Mid-Semester Break: October 26-30, 2026

Lecture Time & Location:

• Thursdays, 4:00 PM - 5:30 PM (ET) at MP 137 (McLennan Physical Laboratories), University of Toronto
• Fridays, 11:00 AM - 12:30 PM (ET) at BA 6183 (Bahen Centre), Unviersity of Toronto

Format: In-Person Only

Course Description

Nonsmooth objects arise ubiquitously in nature, applications, and as limits of smooth objects. To analyze them requires ideas from geometry which apply outside the traditional framework of smooth manifolds. The rapidly developing areas of metric and metric-measure geometry provide an ideal setting for this, enabling powerful ideas to be simplified and generalized.

This course will be an introduction to these topics. It will expose how various notions of curvature (e.g. sectional and Ricci) can be estimated without a differentiable structure, using only distances and volumes, and the consequences which such estimates bring. Many important theorems from Riemannian geometry (volume growth estimates, spectral gaps, diameter bounds, factorization results,...) extend naturally to metric spaces with appropriate curvature bounds.These spaces also turn out to be the natural settings in which to analyze certain deterministic and probablistic dynamics modelling physical processes such as heat flow, chemical reactions, population spreading, and Markov chains. We touch on the consequences of bounds of sectional curvature type, before focusing on to the more subtle case where such bounds hold only in the Ricci (i.e. averaged) sense, a subject of intense and ongoing investigation.

We expose the necessary ideas from optimal transportation, which in particular allow more complicated spaces to be decomposed into collections of weighted one-dimensional metric spaces (`needles'). We discuss RCD(K;N) spaces and Gigli's geometric splitting theorem. Recalling that the Einstein field equation for gravity can be formulated in terms of Ricci curvature, if time and audience interests permit we shall explore how similar ideas can be used to formulate a nonsmooth theory of gravity still admitting key results such as the Hawking singularity theorem.

This course is offered in conjunction with the Fall 2026 Fields thematic semester on Optimal Transport in the Natural Sciences and Statistics.

Prerequisites: Some familiarity with measure theory is assumed. Student familiarity with Differential geometry (Riemannian or Lorentzian) can provide useful intuition but is by no means essential.

References:

• Ambrosio, Gigli and Savare: Gradient Flows in Metric Spaces and in the Space of Probability Measures. Birkhäuser 2005.
• Burago, Burago and Ivanov: A Course in Metric Geometry. AMS GSM #33, 2001.
• Alexander, Kapovitch, and Petrunin: An Invitation to Alexandrov Geometry Spring 2019.
• Cavalletti and Mondino, Optimal transport in Lorentzian synthetic spaces, synthetic timelike Ricci curvature lower bounds and applications, Camb. J. Math. 12 (2024) 417534.
• Gigli, The splitting theorem in non-smooth context arXiv:1302.5555.
• McCann, A synthetic null energy condition, Commun. Math. Phys. 405 (2024) 38:1-24.
• Villani: Optimal Transport: Old and New, Springer-Verlag 2009.