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.

An $r$-uniform tight cycle of length $k>r$ is a hypergraph with vertices $v_1,\ldots,v_k$ and edges $\{v_i,v_{i+1},…,v_{i+r-1}\}$ (for all $i$), with the indices taken modulo $k$. Sós, and independently Verstraëte, asked the following question: how many edges can there be in an $n$-vertex $r$-uniform hypergraph if it contains no tight cycles of any length? In this talk I will review some known results, and present recent progress on this problem.