This week

Whether a localized microscopic defect will affect the macroscopic behaviour of a system is a fundamental question in statistical mechanics. For the Totally Asymmetric Simple Exclusion Process (TASEP) on $\mathbb{Z}$, this problem was originally posed by Janowsky and Lebowitz and became famous as the ``slow-bond” problem. If the wait time of jump for a particle at the origin is increased from an exponential with rate $1$ to that with rate $1-\epsilon$, is this effect detectable in the macroscopic current? Different groups of physicists, using a range of heuristics and numerical simulations, reached opposing conclusions on whether the critical value of $\epsilon$ is $0$. This was ultimately resolved rigorously in Basu-Sidoravicius-Sly which established that $\epsilon_c=0$. In this talk, we will study the effect of the current as $\epsilon$ tends to $0$ and in doing so explain why it was so challenging to predict on the basis of numerical simulations. In particular, we show that with the effect of the perturbation tends to 0 faster than any polynomial. Our proof focuses on the Last Passage Percolation formulation of TASEP. The talk is based on joint works with Allan Sly and Lingfu Zhang.

We consider the varifolds associated to a phase transition problem whose first variation of Allen-Cahn energy is Lp integrable with respect to the energy measure. We can see that the Dirichlet and potential part of the energy are almost equidistributed. After passing to the phase field limit, one can obtain an integer rectifiable varifold with bounded Lp mean curvature.

Modern machine learning models are typically overparameterised and
singular: many different trained models ~achieve the same optimal
loss. Singular learning theory (SLT) is an approach to statistical
learning theory for such models developed by Sumio Watanabe, rooted in
Bayesian statistics and singularity theory.

In the first introductory lecture, the main player will be the local learning coefficient and its applications. I will give and motivate its definition from several interlocked perspectives: free energy and model selection in Bayesian statistics, singularity theory, information theory and statistical
physics. We will see how the LLC is estimated in practice using
(stochastic) MCMC techniques. I will then present empirical results
showing how the LLC captures important structural aspects of training
and generalisation, both in toy models and in LLMs.


In the second lecture, I will explain some of the Bayesian statistics
and singularity theory that go into the main results of SLT. I will
also discuss various caveats involved when applying SLT to
developmental interpretability of realistic deep learning models. I
will sketch some of the current research on bridging those gaps
between theory and practice, including extensions of the LLC (refined
LLC, susceptibilities), SLT for other models of computations such as
noisy Turing machines, and a dynamical interpretation of the LLC in
terms of jet schemes. This may also be an opportunity to relate SLT
with some other strands of modern deep learning research. Finally,
time permitting, I will argue for an SLT-inspired perspective on the
role of generalisation in AI safety.