8 problems

Klartag's conjecture. The localisation method should generalise to the multidimensional setting by employing optimal transport of vector measures.

Let $\widehat\Pi_t$ be the normalized particle occupation measure of the catalytic branching random walk on $\mathbb Z^d$, and let $\lambda$ and $\lambda_0$ denote the ordinary and…

Nearly optimality conjecture. The sub-optimality of the results for $\thm{thm-main-poly}$ is an artefact of the proof, and the optimisation problem should be nearly optimal, up to…

Let $L$ be a localisation of groups, and let $G$ be a transfinitely nilpotent group. Here, localisation means an idempotent monad arising from the lifting-property factorisation de…

Let $L$ be a localisation of groups, that is, a functor equipped with the natural transformation and idempotence data described in the source. A group is nilpotent if it has a fini…

Let $cxi(z)=cxi(-z)$ with probability $p=p(|z|)$, where $p(n)cto1$ as $nctocinfty$, and fix a Pareto parameter $calpha>2$. Complete localisation is the property that the parabolic…

In the parabolic Anderson model with an i.i.d. random potential, complete localisation means that the solution is asymptotically concentrated at a single spatial site, while double…

Let $u(t,z)$ be the solution of the parabolic Anderson model with independent Weibull potential $\xi(z)$ of parameter $0<\gamma<2$, and let $U(t)=\sum_{z\in\mathbb{Z}^d}u(t,z)$. Co…