The goal of this semester is to bring together a diversity of mathematical scientists with a range of experience and expertise, to pursue research on Nonsmooth Riemannian and Lorentzian Geometry.

Differential geometry is traditionally formulated on smooth manifolds,  such as the torus and the sphere.  But this is unsatisfactory from various points of view.  To analyze such geometries,  one must often take limits,  and smooth spaces can both approximate and be approximated by non-smooth spaces,  including piecewise linear manifolds such as polyhedral surfaces, but also more singular objects.  It is therefore desirable to formulate geometric notions which place smooth and nonsmooth spaces on an equal footing, and accommodate limiting processes including possible changes of topology and dimension (collapse).  

For spaces with sectional curvature bounds, such a framework was established in the mid-twentieth century based on triangle comparison techniques by the group of A. D. Alexandrov.  Despite decades of progress,  a number of fundamental questions about such geometries remain.  Moreover, in many settings, including Einstein's theory of gravity and the Hamilton-Perelman proof of the Poincaré and Thurston Geometrization conjectures,  it is Ricci rather than sectional curvature bounds that play a central role. Over the past fifteen years,  a theory of lower Ricci curvature bounds in measured geodesic spaces has blossomed, based on entropic convexity along geodesics in the space of probability measures,  and featuring seminal contributions by many scientists, since the pioneering work of Lott, Villani, and Sturm.

More recently,  the possibility of extending such ideas to the Lorentzian setting of general relativity has been raised using time-like triangle comparison on the one hand, and by using optimal transport of measures from past to future and entropic convexity properties to define time-like Ricci bounds and give a weak sense to the Einstein field equations, on the other.   This is especially enticing since general relativity, while set in a smooth four-dimensional spacetime,  often predicts that this smoothness must break down,  for example in blackhole interiors.  Such singularities are features rather than bugs in the theory; they predict for example that our universe originated in a big bang, yet many of their properties remain to be understood, such as the Penrose cosmic censorship conjecture.  Spaces satisfying synthetic time-like curvature bounds provide an exciting new framework and set of techniques for addressing such fundamental questions, both about the model and about the universe we live in.

The goals of this program include:

(i) to gather together leading researchers in non-smooth geometry and cognate areas (curvature flows,  mathematical general relativity, probability theory, discrete geometry, optimal transport), to have them identify key problems in the field and new applications on which significant progress is likely to be within reach; (ii) to provide a collaborative environment welcoming to minorities and women, which brings scientists together with trainees from different backgrounds to achieve substantial progress on some of these problems;  (iii) to expose new researchers to the promise of the field and its dazzling array of challenges,  while grounding them in its basic techniques; (iv) to provide an opportunity for sustained interactions and cross-fertilization between scientists working in different geographic regions and different disciplines;  (v) to make available to the wider community a series of broad interest talks and more specialized mini-courses.

Although the thematic program will spread from July to December of 2022,  it will also feature three weeklong workshops: 

Geometry of Spaces with Upper and Lower Curvature Bounds (12-16 September 2022).
Organized by Vitali Kapovitch (Toronto), Viktor Schroeder (Zurich), Catherine Searle (Wichita)

The study of metric spaces with upper and lower sectional curvature bounds  is a rich and well developed subject that goes back some 60+ years to the work of A.D. Alexandrov.  It has a wide variety of applications and connections with various areas of geometry such as geometric group theory, classical Riemannian geometry of manifolds with various sectional curvature bounds, integral geometry and the study of spaces with lower Ricci bounds.

The planned workshop has two main aims:

*    To bring together experts to report on recent progress and to establish new goals and connections between various areas of the field.
*    To familiarize young mathematicians  and graduate  students with the basics of the theory as well as recent developments. To this end we are planning to have two mini-courses: one on spaces with lower curvature bounds and one on spaces with upper curvature bounds.

The lecture series will be meant to review both the basics of the theory as well as recent developments. They should be of interest to all levels of researchers from graduate students to postdocs to specialists in the field. We also plan to have standalone talks.  

A particular goal is to bring together and familiarize scientists with each other's research: people working on various aspects of the theory of spaces with upper and lower curvature bounds as well as people who work on somewhat outside but closely related topics. Such topics have proved spectacularly successful with an extremely wide variety of applications from small cancellation groups, to relatively and semi-hyperbolic groups, automatic groups, Teichmüller theory, mapping class groups, automorphisms of free groups, random groups and virtually fibered conjecture for 3 manifolds.

The workshop will also aim to address structure theory of $\mathrm{CAT}(K)$ spaces, Alexandrov spaces of positive and nonnegative curvature with low cohomogeneity, relations to submetries and Riemannian foliations, analysis on Alexandrov spaces (both with upper and lower curvature bounds), relations between Alexandrov and integral geometry and harmonic maps to and from spaces with various curvature bounds and lastly, upper and lower sectional curvature bounds for Lorentzian spaces.

Mathematical Relativity, Scalar Curvature and Synthetic Lorentzian Geometry (3-7 October 2022)
Organized by Ghazal Geshnizjani (Waterloo and Perimeter), Robert McCann (Toronto), Richard Schoen (Irvine), Christina Sormani (Lehman College and CUNYGC)

The synthetic approach towards understanding Ricci curvature in the Riemannian setting has lead to groundbreaking results over the past quarter century.  Recent advances towards developing a synthetic version of sectional and Ricci curvature in the Lorentzian setting and of Scalar curvature in the Riemannian setting is highly promising and should lead to deeper understanding of Einstein's theory of gravity.  Identifying the relevant notions of convergence for spacetimes is another topic of current interest encompassed by this workshop.

Potential participants include mathematicians and physicists with diverse approaches to these questions  to foster communication and to forge a new and unifying perspective.  

Aspects of Ricci Curvature Bounds (14-18 November 2022)
Organized by Nicola Gigli (SISSA), Robert Haslhofer (Toronto),  Jan Maas (IST Austria),  Karl-Theodor Sturm (Bonn)

Structures having Ricci curvature bounded either from below and/or from above in some sense have been at the center of intense investigations in the last ten years, the reason being the deep effect at the geometric, analytic and probabilistic level that this sort of assumption carries. The study of such bounds and their geometric and analytic effects therefore attracts a constellation of mathematicians with a diverse set of skills and backgrounds. The conference aims at gathering specialists from different sectors to foster communication between various (sub)fields of research. We aim at covering the synthetic side of the story, the non-commutative one, the regularity of Ricci-limit spaces and geometric evolution problems.