I will talk about optimal estimates for stochastic integrals
in the case when both rough paths and martingales play a role.

This is an ongoing joint work with Peter Friz (TU Berlin).

We derive generalized lower Ricci bounds in terms of signed measures. And we prove associated gradient estimates for the heat flow with Neumann boundary conditions on domains of metric measure spaces obtained through „convexification“ of the domains by means of subtle time changes. This improves upon previous results both in the case of non-convex domains and in the case of convex domains.
 

"We consider a spatial Lambda-Fleming-Viot process, a model in mathematical biology, with a randomly chosen (rough) selection field. We study the scaling limit of this process in different regimes. This leads to the analysis of semi-discrete approximations of singular SPDEs, in particular the Parabolic Anderson Model and allows to extend previous results to weakly nonlinear cases. The subject presented is based on joint works with Aleksander Klimek and Nicolas Perkowski."

In this talk I will introduce a continuous wetting model consisting of the law of a Brownian meander tilted by its local time at a positive level h, with h small. I will prove that this measure converges, as h tends to 0, to the same weak limit as for discrete critical wetting models. I will also discuss the corresponding gradient dynamics, which is expected to converge to a Bessel SPDE admitting the law of a reflecting Brownian motion as invariant measure. This is based on joint work with Jean-Dominique Deuschel and Tal Orenshtein.

We discuss the models of random geometry that are derived
from scaling limits of large graphs embedded in the sphere and
chosen uniformly at random in a suitable class. The case of
quadrangulations with a boundary leads to the so-called
Brownian disk, which has been studied in a number of recent works.
We present a new construction of the Brownian
disk from excursion theory for Brownian motion indexed
by the Brownian tree. We also explain how the structure
of connected components of the Brownian disk above a
given height gives rise to a remarkable connection with
growth-fragmentation processes.

We exhibit a new martingale coupling between two probability measures $\mu$ and $\nu$ in convex order on the real line. This coupling is explicit in terms of the integrals of the positive and negative parts of the difference between the quantile functions of $\mu$ and $\nu$. The integral of $|y-x|$ with respect to this coupling is smaller than twice the Wasserstein distance with index one between $\mu$ and $\nu$. When the comonotonous coupling between $\mu$ and $\nu$ is given by a map $T$, it minimizes the integral of $|y-T(x)|$ among all martingales coupling.

(joint work with William Margheriti)

Exceptional holonomy manifolds come with certain geometric data that include a Ricci flat metric. Finding examples is therefore very difficult. The task can be made more tractable by imposing symmetry.  The focus of this talk is the case of torus symmetry. For a particular rank of the torus, one gets a natural parameterisation of the orbit space in terms of so-called multi-moment maps. I will discuss aspects of the local and global geometry of these 'toric' exceptional holonomy manifolds. The talk is based on joint work with Andrew Swann.

In this talk we present p-order methods for unconstrained minimization of convex functions that are p-times differentiable with Hölder continuous p-th derivatives. We establish worst-case complexity bounds for methods with and without acceleration. Some of these methods are "universal", that is, they do not require prior knowledge of the constants that define the smoothness level of the objective function. A lower complexity bound for this problem class is also obtained. This is a joint work with Yurii Nesterov (Université Catholique de Louvain).