Alternatively have a look at the program.

## Existence and stability of flows associated to vector fields in metric measure spaces - part 1

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.

## Harnack Inequalities and Applications on Manifolds - part 1

We will introduce coupling by change of measures to establish Harnack inequalities and derivative formulas of heat semigroups for stochastic differential equations. In particular, equivalent inequalities are derived for curvature and second fundamental lower bounds on Riemannian manifolds with boundary.

## Asymptotics of spectral gaps on loop spaces

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

$$ d\nu^{\lambda}_{x_0,y_0}(\gamma)=\frac{1}{Z_{\lambda}} \exp\left(-\lambda E(\gamma)\right) d\gamma, $$

## Quasi-invariance of heat kernel and Wiener measures in infinite-dimensional hypoelliptic settings

The talk will concentrate on the recent progress in two different infinite-dimensional hypoelliptic settings. One is of infinite-dimensional Heisenberg groups equipped with a natural sub-Riemannian (hypoelliptic) metric, and another of the horizontal path space of a totally geodesic Riemannian foliation. In the first case the heat kernel measure satisfies a Cameron-Martin type quasi-invariance, while in the second case the horizontal Wiener measure is quasi-invariant with respect to the flows generated by suitable tangent processes.

## Existence and stability of flows associated to vector fields in metric measure spaces - part 2

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.

## Two short proofs of the measure rigidity theorem

In this talk I will show how the general existence of transport maps and the properties of interpolation measures can be used to show that any two reference measures with the measure contraction property must be mutually absolutely continuous.

## Topics in group actions on m.m. spaces

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?

## Path Integrals on Manifolds and their asymptotic Expansions

Path integrals, i.e. integrals where the integration domain is an infinite-dimensional space of paths, were introduced into physics by Richard Feynman and since then have been an important tool in thermodynamics, quantum mechanics and quantum field theory. Mathematically however, they are notoriously ill-defined, and there are several different approaches to turn path integrals into a rigorous mathematical concept.

## Deterministic couplings of Brownian motions on Riemannian manifolds

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.

## Random walks on ultra-metric spaces

We describe a class of non-local Dirichlet forms on ultra-metric spaces and obtain various upper and lower estimates of the corresponding heat kernels.

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