Abstracts for Metric Measure Spaces and Ricci Curvature

Let $(M,g)$ be an $n$-dimensional complete Riemannian manifold. Let $x_0, y_0\in M$ and consider the loop space $P_{x_0,y_0}(M)=\{\gamma\in C([0,1]\to  M)~|~\gamma(0)=x_0, \gamma(1)=y_0\}$. Let $\nu^{\lambda}$ be the pinned measure defined by the transition probability $p(t/\lambda,x,y)$, where $p(t,x,y)$ denotes the heat kernel of the diffusion semigroup $e^{t\Delta/2}$.  Heuristically, we have
$$ d\nu^{\lambda}_{x_0,y_0}(\gamma)=\frac{1}{Z_{\lambda}} \exp\left(-\lambda E(\gamma)\right) d\gamma, $$

As the developments of the synthetic theories for lower bounds on Ricci curvature clearly illustrates, the connections between Eulerian notions (Gamma-calculus, Bakry-Emery theory etc.) and Lagrangian notions (Lott-Villani and Sturm theory) play an important role. In my lectures I will cover this topic, providing strong links between vector fields (derivations, in the metric measure setting) and solutions to the ODE. I will first cover the case of a single metric measure structure and then the case of measured Gromov-Hausdorff convergence.

The goal is to give a short overview of topics in symmetric transformations on metric measure spaces. We will address the questions of when is the group of symmetries of a m.m. space well-behaved? And of what can be concluded concerning the induced geometry of a space with symmetries?

We will introduce special types of couplings of Brownian motions on Riemannian manifolds.  We discuss the shy  couplings in which the motions do not meet in finite time and then the ones in which the distance between the motions is a given deterministic function.  On the model manifolds of constant curvature we show a complete characterization of such distance functions.  This is joint work with Mihai Pascu.