Lahars and huaicos are potent natural hazards that threaten lives and livelihoods. They comprise debris-laden fluid that flows rapidly down slopes, bulking up considerably as they progress. Owing to their rapid onset and the significant threat that they pose to communities and infrastructures, it is important to be able to predict their motion in order to assess quantitatively some of the impacts that they may cause. In this seminar I will present mathematical models of these flows and apply them to various natural settings, drawing on examples from Peru and the Philippines.  Along the way I will show some informative, idealised solutions, the susceptibility of these flows to roll wave instabilities, ways to prevent ill-posedness and how to include measured topography in the computation.

The presence of glaciers and ice sheets leaves a significant imprint
on Earth's surface.  We steadily improve our physical understanding of
the involved processes, from the erosion of kilometer-deep fjords in
crystalline bedrock to the broad ice-marginal deposition of sediments.
This talk will highlight observations of landforms, sedimentary deposits,
laboratory experiments, and models that aim to capture the interplay
between ice and substratum.  I show how the interplay may play a role in
the future evolution of the West Antarctic Ice Sheet in a warming climate.

I will review aspects of the theory of symmetry-protected topological phases, focusing on the case of one-dimensional quantum chains. Important concepts include the bulk-boundary correspondence, with bulk topological invariants leading to interesting boundary phenomena. I will discuss topological invariants and associated boundary phenomena in the case that the system is gapless and described at low energies by a conformal field theory. Based on work with Ruben Verresen, Ryan Thorngren and Frank Pollmann.

iii) identify a special class of correlated compressors based on the idea of random permutations, for which we coin the term PermK. The use of this technique results in the strict improvement on the previous MARINA rate. In the low Hessian variance regime, the improvement can be as large as √n, when d > n, and 1 + √d/n, when n<=d, where n is the number of workers and d is the number of parameters describing the model we are learning.

Wasserstein distance induces a natural Riemannian structure for the probabilities on the Euclidean space. This insight of classical transport theory is fundamental for tremendous applications in various fields of pure and applied mathematics. We believe that an appropriate probabilistic variant, the adapted Wasserstein distance $AW$, can play a similar role for the class $FP$ of filtered processes, i.e. stochastic processes together with a filtration. In contrast to other topologies for stochastic processes, probabilistic operations such as the Doob-decomposition, optimal stopping and stochastic control are continuous w.r.t. $AW$. We also show that $(FP, AW)$ is a geodesic space, isometric to a classical Wasserstein space, and that martingales form a closed geodesically convex subspace. Finally we consider computational aspects and provide a novel method based on the Sinkhorn algorithm.

The talk is based on articles with Daniel Bartl, Mathias Beiglböck and Stephan Eckstein.

In this talk we prove an analogue of the Brakke's $\epsilon$-regularity theorem for the parabolic Allen--Cahn equation. In particular, we show uniform $C^{2,\alpha}$ regularity for the transition layers converging to smooth mean curvature flows as $\epsilon\rightarrow 0$. A corresponding gap theorem for entire eternal solutions of the parabolic Allen--Cahn is also obtained. As an application of the regularity theorem, we give an affirmative answer to a question of Ilmanen that there is no cancellation in BV convergence in the mean convex setting.