7 problems

Asymptotic-similarity conjecture. The random set $\Lambda$ achieves asymptotic similarity with the local maxima of the GEF.

Let $X$ be a Fano manifold with volume form $dV$, let $\beta_0<0$, and let $\beta$ range over the real numbers satisfying $\beta>\beta_0$. For each $N$, let $\mathcal{Z}_N(\beta)$…

Threshold-equality conjecture. The two invariants coincide:

Beta-convergence conjecture. If $X$ admits a Kähler–Einstein metric, then for every given $\beta<1$, $Z_{N_k,\beta}<\infty$ for sufficiently large $k$, and the empirical measures…

Canonical-current convergence conjecture. The currents $\omega_k$ converge to a Kähler–Einstein metric on $X$.

Let $N$ particles form a Coulomb gas at inverse temperature $\beta$, and consider observation lengthscales compared with the threshold $\beta^{-\frac12}N^{-\frac1{\mathsf d}}$. A p…

Let $X$ be a compact complex manifold, let $L$ be a line bundle, and let the singular weights and measures be as in the preceding theorem. The associated Bernstein–Markov property…