Gromov, Hanke, Sormani, Yu's

Not Only Scalar Curvature

Upcoming talks

October 4, 2023

Aaron Naber: Ricci Curvature, Fundamental Group and the Milnor Conjecture (I)
(11:00 am NYC, 17:00 Paris)
It was conjectured by Milnor in 1968 that the fundamental group of a complete manifold with nonnegative Ricci curvature is finitely generated. In this talk we will discuss a counterexample, which provides an example $M^7$ with $\mathrm{Ric}>= 0$ such that $\pi_1(M)=Q/Z$ is infinitely generated.

There are several new points behind the result. The first is a new topological construction for building manifolds with infinitely generated fundamental groups, which can be interpreted as a smooth version of the fractal snowflake. The ability to build such a fractal structure will rely on a very twisted gluing mechanism. Thus the other new point is a careful analysis of the mapping class group $\pi_0(\mathrm{Diff}(S^3\times S^3))$ and its relationship to Ricci curvature. In particular, a key point will be to show that the action of $\pi_0(\mathrm{Diff}(S^3\times S^3))$ on the standard metric $g_{S^3\times S^3}$ lives in a path connected component of the space of metrics with $\mathrm{Ric}>0$.

Daniele Semola: Ricci Curvature, Fundamental Group and the Milnor Conjecture (II)
(12:15 pm NYC, 18:15 Paris)
The goal of this second talk will be to discuss more in detail some of
the main ideas involved in the construction of the counterexamples to
the Milnor conjecture.

We will review the topological construction and outline the key
geometric steps, with particular emphasis on those involving the
mapping class group of $S^3\times S^3$. Moreover, we will describe the
behavior of the asymptotic cones of the examples, in relationship with
the known restrictions.

October 18, 2023

Christian Bär:
(11:00 am NYC, 17:00 Paris)

Yipeng Wang:
(12:15 pm NYC, 18:15 Paris)

November 1, 2023

Marcus Khuri:
(11:00 am NYC, 17:00 Paris)

Brian Allen: On the Stability of Llarull's Theorem in Dimension Three
(12:15 pm NYC, 18:15 Paris)
Llarull's Theorem states that any Riemannian metric on the $n$-sphere which has scalar curvature greater than or equal to $n(n-1)$, and whose distance function is bounded below by the unit sphere's, is isometric to the unit sphere. Gromov later posed the Spherical Stability Problem, probing the flexibility of this fact, which we give a resolution of in dimension $3$. We show that a sequence of Riemannian $3$-spheres almost satisfying the hypotheses of Llarull's theorem with uniformly bounded Cheeger isoperimetric constant must approach the round $3$-sphere in the volume preserving Sormani-Wenger Intrinsic Flat sense. The argument is based on a proof of Llarull's Theorem due to Hirsch-Kazaras-Khuri-Zhang ( using spacetime harmonic functions and a characterization of Sormani-Wenger Intrinsic Flat convergence given by Allen-Perales-Sormani ( This is joint work with E. Bryden and D. Kazaras (

Previous talks

Seminar Info

Each session will consist of two lectures: an introductory colloquium and a related seminar on a recent advance in geometry. Not all the talks will concern scalar curvature but perhaps many will. The pair of talks will be followed by a discussion both in person and online. Everyone is welcome to join the session live (zoom) or watch the recorded videos of the talks later (IHES channel).

Zoom starts at 11:00 am NYC time:

click here to register and receive link

IHES Carmin TV Channel Info:


Misha Gromov, IHES

Bernhard Hanke, University of Augsburg

Christina Sormani, CUNY Graduate Center and Lehman College

Gouliang Yu, Texas A&M University