# Schedule of the Follow-up Workshop

## Monday, August 29

10:15 - 10:50 |
Registration & Welcome coffee |

10:50 - 11:00 |
Opening remarks |

11:00 - 11:45 |
Tapio Rajala: On density of Sobolev functions on Euclidean domains |

11:50 - 12:35 |
Nicolas Juillet: Examples in relation with a metric Ricci flow |

12:35 - 15:15 |
Lunch break |

15:15 - 16:00 |
Fernando Galaz-Garcia: Three-dimensional Alexandrov spaces with positive and non-negative curvature |

16:00 - 16:45 |
Tea and cake |

16:45 - 17:30 |
Fabio Cavalletti: Isoperimetric inequality under curvature dimension condition |

afterwards |
Reception |

## Tuesday, August 30

9:00 - 9:45 |
Shin-ichi Ohta: Nonlinear geometric analysis on Finsler manifolds: Some functional inequalities |

9:50 - 10:35 |
Matthias Liero: On entropy-transport problems and the Hellinger-Kantorovich distance |

10:35 - 11:00 |
Group photo and coffee break |

11:00 - 11:45 |
Ilaria Mondello: Geometric analysis on stratified spaces |

11:50 - 12:35 |
Raquel Perales: Convergence of Alexandrov Spaces |

12:35 - 15:15 |
Lunch break |

15:15 - 16:00 |
Informal talk / discussion |

16:00 - 16:45 |
Tea and cake |

16:45 - 17:30 |
Informal talk / discussion |

## Wednesday, August 31

9:00 - 9:45 |
Shouhei Honda: New stability results for sequences of metric measure spaces with uniform Ricci bounds from below |

9:50 - 10:35 |
Kohei Suzuki: Convergence of Brownian motions on RCD spaces |

10:35 - 11:00 |
Coffee break |

11:00 - 11:45 |
Emanuel Indrei: Quantitative logarithmic Sobolev inequalities |

11:50 - 12:35 |
Martin Kell: Sectional curvature-like conditions on metric spaces |

12:35 - 15:15 |
Lunch break |

15:15 - 16:00 |
Discussion |

16:00 - 16:45 |
Tea and cake |

16:45 - 17:30 |
Discussion |

## Thursday, September 1

9:30 - 10:15 |
Max Fathi: Ricci curvature and functional inequalities for interacting particle systems |

10:15 - 10:45 |
Coffee break |

10:45 - 11:30 |
Heikki Jylhä: L^{∞} estimates in optimal transport |

11:35 - 12:20 |
Jan Maas: Gradient flow structures for quantum systems with detailed balance |

12:20 - 15:15 |
Lunch break |

15:15 - 16:00 |
Gerardo Sosa: Automorphism groups of RCD(K,N)-spaces are Lie groups |

16:00 - 16:45 |
Tea and cake |

16:45 - 17:30 |
Discussion |

## Friday, September 2

9:30 - 10:15 |
Asma Hassannezhad: Eigenvalue problems in sub-Riemannian geometry |

10:15 - 10:45 |
Coffee break |

10:45 - 11:30 |
Asuka Takatsu: Curvature Dimension condition from the viewpoint of Information geometry |

11:35 - 12:20 |
Joe Neeman: rho-convexity and Ehrhard's inequality |

12:20 - |
Lunch break - end of workshop |

# Abstracts

## Fabio Cavalletti: Isoperimetric inequality under curvature dimension condition

We will review the one-dimensional localization method in the general framework of metric measure spaces and the recent proof of the isoperimetric inequality for essentially non-branching m.m.s. verifying the curvature dimension condition. We will also address rigidity and stability questions.

## Max Fathi: Ricci curvature and functional inequalities for interacting particle systems

I will present a few results on entropic Ricci curvature bounds, with applications to interacting particle systems. The notion was introduced by M. Erbar and J. Maas and independently by A. Mielke. These curvature bounds can be used to prove functional inequalities, such as spectral gap bounds and modified logarithmic Sobolev inequalities, which measure the rate of convergence to equilibrium for the underlying dynamic. This talk will focus on the case where curvature is nonnegative, but not necessarily strictly positive. Joint work with M. Erbar.

## Fernando Galaz-Garcia: Three-dimensional Alexandrov spaces with positive and non-negative curvature

I will discuss the topological classification of closed three-dimensional Alexandrov spaces with positive or non-negative curvature, both in the Alexandrov and CD(K,N) sense. This is joint work with Luis Guijarro, Michael Munn and Qintao Deng.

## Asma Hassannezhad: Eigenvalue problems in sub-Riemannian geometry

In this talk, we study upper bounds for eigenvalues of the sub-Laplacian on sub-Riemannian manifolds. Sub-Riemannian structures naturally occur in control theory, analysis of hypoelliptic operators, contact geometry and CR geometry. The sub-Laplacian is an intrinsic hypoelliptic operator in sub-Riemannian geometry. It is a deep relationship between its eigenvalues and the sub-Riemannian structure. I recall basic geometric properties of sub-Riemannian manifolds and discuss geometric bounds for eigenvalues of the sub-Laplacian. Further, I give examples on which eigenvalue upper bounds are independent of the geometry of the underlying manifold. This is joint work with Gerasim Kokarev.

## Shouhei Honda: New stability results for sequences of metric measure spaces with uniform Ricci bounds from below

In this talk we will give several stability results with respect to the measured Gromov-Hausdorff convergence under lower Ricci bounds, and applications. Applications include the continuity of Cheeger's isoperimetric constants and new almost suspension theorem. This is joint work with Luigi Ambrosio.

## Emanuel Indrei: Quantitative logarithmic Sobolev inequalities

We discuss the stability problem for the Gaussian logarithmic Sobolev inequality from two perspectives: the first is in the context of the Brenier map arising in mass transport theory; the second is based on Fourier analysis and involves the Beckner-Hirschmann entropic uncertainty principle.

## Nicolas Juillet: Examples in relation with a metric Ricci flow

Gigli and Mantegazza have observed how optimal transport and heat diffusion permit us to describe the Ricci flow (or at least its direction) uniquely from the metric aspects of Riemannian manifolds. The goal is to reformulate the Ricci flow so that it also makes sense for metric spaces.

I will present investigations and results obtained with Matthias Erbar that concerns some non-Riemannian limits of Riemannian manifolds, in particular the Euclidean cone and the Heisenberg group.

## Heikki Jylhä: L^{∞} estimates in optimal transport

It is well-known that for finite p the L^{p} transportation distances W_{p} metrize the weak convergence of probability measures (up to a convergence of p-th moments). However, the same result does not hold for the L^{∞} transportation distance W_{∞}. In light of this, we may ask whether convergence in W_{∞} can be characterized using weak convergence and some other conditions.

A natural approach to this problem is to study situations where W_{p} has lower bounds depending on W_{∞}, since we always have W_{p} ≤ W_{∞}. In a recent paper with Tapio Rajala we found a characterization for a situation where the distance from a reference measure in metric W_{p} is bounded from below by an increasing positive function of the corresponding distance in W_{∞}. Furthermore, we proved that a similar result holds on the level of optimal transport plans. As a corollary of our results we can characterize convergence in W_{∞} in the case where the measures are compactly supported.

## Martin Kell: Sectional curvature-like conditions on metric spaces

In this talk I present two concavity assumptions on the distance. The first one is the non-negative curvature analogue of Busemann’s non-positive curvature condition and resembles a sectional curvature-like condition comparable to the measure contraction property. It holds for certain non-Riemannian Finsler manifolds, but it is not clear whether it is compatible with Ohta’s Ricci curvature on Finsler manifolds. Whenever the n-dimensional Hausdorff measure is non-trivial then the measure contraction property holds and the space is a PI-space. Independent of this one always obtains a bi-Lipschitz splitting theorem. The second condition, called uniform smoothness, is dual to uniform convexity of the distance function and gives a convex exhaustion function, a first step towards a soul theorem.

## Matthias Liero: On entropy-transport problems and the Hellinger-Kantorovich distance

In this talk, we will present a general class of variational problems involving entropy-transport minimization with respect to a couple of given finite measures with possibly unequal total mass. These optimal entropy-transport problems can be regarded as a natural generalization of classical optimal transportation problems. With an appropriate choice of the entropy/cost functionals they provide a distance between measures that exhibits interesting geometric features. We call this distance Hellinger-Kantorovich distance as it can be seen as an interpolation between the Hellinger and the Kantorovich-Wasserstein distance. The link to the entropy-transport minimization problems relies on convex duality in a surprising way. Moreover, a dynamic Benamou-Brenier characterization also shows the role of these distances in dynamic processes involving creation or annihilation of masses. Finally, we will give a characterization of geodesic curves and of convex functionals.

This is joint work with Giuseppe Savaré and Alexander Mielke.

## Jan Maas: Gradient flow structures for quantum systems with detailed balance

We present a new class of transport metrics for density matrices, which can be viewed as non-commutative analogues of the 2-Wasserstein metric. With respect to these metrics, we show that dissipative quantum systems can be formulated as gradient flows for the von Neumann entropy under a detailed balance assumption. We also present geodesic convexity results for the von Neumann entropy in several interesting situations. These results rely on an intertwining approach for the semigroup combined with suitable matrix trace inequalities.

This is joint work with Eric Carlen.

## Ilaria Mondello: Geometric analysis on stratified spaces

In this talk we will give a brief introduction to the singular setting of stratified spaces, which generalize the notion of isolated conical singularity. We will present some geometric and analytic tools inspired by classical Riemannian geometry, in order to deduce a lower bound for the first non-zero eigenvalue of the Laplacian under the appropriate lower bound for the Ricci curvature.

## Joe Neeman: rho-convexity and Ehrhard's inequality

We say that a function of two real variables is rho-convex if its Hessian matrix, multiplied by rho on the off-diagonal, is positive semi-definite. This notion (and its generalization to functions of more than two variables) turns out to give simple proofs of various inequalities on Gaussian space, and in some cases also a characterization of equality and near-equality cases. We will present a new proof of Ehrhard's inequality using these methods.

## Shin-ichi Ohta: Nonlinear geometric analysis on Finsler manifolds: Some functional inequalities

A Finsler manifold is a manifold equipped with a (Minkowski) norm on each tangent space. Although the natural Laplacian is nonlinear for Finsler manifolds, one can establish the Bochner formula (O.-Sturm, 2014) and use it to develop the nonlinear analogue of the Gamma-calculus (of Bakry et al). In this talk, we treat applications to functional inequalities including dimensional versions of the Poincare(-Lichnerowicz) inequality, log-Sobolev inequality, and Sobolev inequality. This method gives us sharp estimates even for non-reversible metrics.

## Raquel Perales: Convergence of Alexandrov Spaces

We study sequences of integral current spaces (X_{j},d_{j},T_{j}) with no boundary such that (X_{j},d_{j}) are Alexandrov spaces with nonnegative curvature and diameter uniformly bounded from above and such that the integral current structure T_{j} has weight 1. We prove that for such sequences either they collapse or the Gromov-Hausdorff and Sormani-Wenger Intrinsic Flat limits agree. Joint work with Nan Li.

## Tapio Rajala: On density of Sobolev functions on Euclidean domains

I will present recent results on the density of the Sobolev space W^{1,q}(Ω) in W^{1,p}(Ω), when 1 ≤ p < q ≤ ∞ for domains Ω in the Euclidean space. Relating to this, I will also discuss removability of sets of measure zero for Sobolev functions and extension operators from w^{1,p}(Ω) to W^{1,p}(ℝ^{n}). This is joint work with P. Koskela and Y. Zhang.

## Gerardo Sosa: Automorphism groups of RCD(K,N)-spaces are Lie groups

## Kohei Suzuki: Convergence of Brownian motions on RCD spaces

We show that, under the RCD condition, the weak convergence of the laws of the Brownian motions is equivalent to the measured Gromov convergence (Gigli-Mondino-Savaré '13) of the underlying spaces.

## Asuka Takatsu: Curvature Dimension condition from the viewpoint of Information geometry

Information geometry is a geometry on the space of probability measures, which is completely different from Wasserstein geometry. In this talk, I generalize the Boltzmann entropy via Information geometry and revisit the curvature dimension condition.