Week 28.10.2024 – 03.11.2024

A common challenge in using Markov chain for sampling from high-dimensional distributions is multimodality, where the chain may get trapped far from stationarity. However, this issue often applies only to worst-case initializations and can be mitigated by using high-entropy initializations, such as product or weakly correlated distributions. From such starting points, the dynamics can escape saddle points and spread mass correctly across dominant modes.

In this talk, I will discuss our results on convergence from high-entropy initializations for the random-cluster models on the complete graph. We focus on the Chayes–Machta or the Swendsen–Wang dynamics for the random-cluster model showing that these chains mix rapidly from specific product measures, even though they mix exponentially slowly from worst-case initializations. The analogous results hold for the Glauber dynamics on the Potts model. Our proofs involve approximating high-dimensional dynamics with 1-dimensional random processes and analyzing their escape from saddle points.

We will give a short overview of the Nielsen-Thurston Classification problem (classifying the homeomorphism type of surfaces) on finite-type surfaces and then move to infinite-type surface mentioning what is known and pointing out some difficulties. We will then discuss how to approximate, in the compact-open topology, a general self-homeomorphism of an infinite-type surface (joint with Y.Chandran) and potential definitions of pseudo-anosov mapping classes in the infinite-type setting (joint with F.Valdez).

A closed quantum system thermalizes in the sense of typicality, if any initial state will reach a suitable equilibrium subspace and stay there most of the time. For non-degenerate Hamiltonians, a sufficient condition for thermalization is the eigenstate thermalization hypothesis (ETH). Shiraishi and Tasaki recently proved the ETH for a perturbation of the Hamiltonian of free fermions on a one-dimensional lattice. The perturbation is needed to remove the high degeneracies of the Hamiltonian. We point out that also for degenerate Hamiltonians ETH implies thermalization. Additionally, we develop another strategy for proving thermalization by adding small generic perturbations. This is joint work with Stefan Teufel, Roderich Tumulka, and Cornelia Vogel.

By using a sharp quartic curvature pinching for the mean curvature flow in $\mathbb{S}^{n+m}$, $m\ge2$, we improve the quadratic curvature conditions. Through a blow-up argument, we establish both a codimension and a cylindrical estimate, which show that in regions of high curvature, the submanifold quantitatively becomes codimension one. In these regions, the submanifold is shown to be weakly convex and moves by translation or evolves is a self-shrinker. Additionally, a decay estimate ensures that the rescaled flow converges smoothly to a totally geodesic limit as time tends to infinity, without the need for iteration methods or integral estimates. Our approach relies on the preservation of the quartic pinching condition along the flow and gradient estimates that control the mean curvature in regions of high curvature.

Classical statistical mechanics describes the macroscopic properties of large numbers of particles. It has a hidden weakness: it assumes that the microscopic forces derive from a Hamiltonian. The same mathematical object then controls both the equations of motion, and the Boltzmann distribution. This is why quantities like pressure are not only time averages of forces (on a wall), but also thermodynamic state functions (which exist independently of any wall). Active matter systems are different. Their particles take energy out of the environment, and use it for dissipative self-propulsion, violating Hamiltonian dynamics. Examples include swimming micro-organisms, and synthetic colloids propelled by optical or chemical energy. The absence of a Hamiltonian-derived detailed balance principle requires a rebuild of statistical mechanics, with some surprising outcomes. For example: (i) the pressure of an active fluid on a wall is not a state function -- it depends on the type of wall\DSEMIC (ii) various interfacial phenomena, governed in equilibrium by a single surface tension, now involve different tensions, some of which can be negative. I will survey these among other surprises and, if time allows, say how they affect kinetic questions such as nucleation rates.

Talk 1 (15:00): Introduction to Graph-based Learning
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Matthew will give an introduction to graph-based learning, touching on variational problems on graphs, techniques from optimal transport, the calculus of variations, and large data limits.

Talk 2 (16:00): Discrete-To-Continuum Limits in Graph-Based Semi-Supervised Learning

Semi-supervised learning (SSL) is the problem of finding missing labels from a partially labelled data set. The heuristic one uses is that “similar feature vectors should have similar labels”. The notion of similarity between feature vectors explored in this talk comes from a graph-based geometry where an edge is placed between feature vectors that are closer than some connectivity radius. A natural variational solution to the SSL is to minimise a Dirichlet energy built from the graph topology. And a natural question is to ask what happens as the number of feature vectors goes to infinity? In this talk I will give results on the asymptotics of graph-based SSL using an optimal transport topology. The results will include a lower bound on the number of labels needed for consistency.