While Einstein’s theory of gravity is formulated in a smooth setting, the
celebrated singularity theorems of Hawking and Penrose describe many physical
situations in which this smoothness must eventually breakdown. In
positive-definite signature, there is a highly successful theory of metric and
metric-measure geometry which includes Riemannian manifolds as a special case,
but permits the extraction of nonsmooth limits under curvature and dimension
bounds analogous to the energy conditions in relativity: here sectional
curvature is reformulated through triangle comparison, while Ricci curvature
is reformulated using entropic convexity along geodesics of probability
measures.

This lecture explores recent progress in the development of an analogous
theory in Lorentzian signature, whose ultimate goal is to provide a nonsmooth
theory of gravity. We highlight how the null energy condition of Penrose
admits a nonsmooth formulation as a variable lower bound on timelike Ricci
curvature.

This talk is hybrid so will be available on zoom. To participate remotely,
please register via
zoom

Speaker Biography

Robert McCann is a professor of mathematics and Canada Research Chair in
Mathematics, Economics and Physics at the University of Toronto. He is a world
leader in the vibrant field of optimal transportation, and has played a
pioneering role in its rapid development since the mid 90’s. In particular, the
notion of displacement convexity, introduced in his PhD thesis, lies behind
many of the area’s myriad applications. His distinguished research record has
been recognized with many prestigious awards, including (among others) an
invited lecture at the 2014 International Congress of Mathematicians, election
to the Royal Society of Canada in 2014, the 2017 Jeffery-Williams prize of the
Canadian Mathematical Society and the 2023 W.T. and Idalia Reid Prize of the
Society for Industrial and Applied Mathematics.