21 problems

Let $P$ be the spectral measure associated with the character $chi_\zw$ in the correspondence between characters and parameters…

Let $R$ be the radius of the restricted disc, let $(B_n)_{n\geq 0}$ be the active Bernoulli random variables, and condition on the set … Let $\\{|X_k|,k\in I\\}$ be the set of modu…

Let $K$ be a kernel on $\mathbb{R}$, let $\mathbb{P}_K$ be the associated point-process law, and let a discrete subset of $\mathbb{R}$ mean a subset with no finite accumulation poi…

Consider a determinantal point process on the real line with correlation kernel … Label its points by $\cdots<Z_{-1}<Z_0<Z_1<Z_2<\cdots$ so that $Z_{-1}<0\leq Z_0$. The sine-proces…

Let $\omega(x)$ be the continuous limit shape of a sequence of Young diagrams, and let the associated random Young tableaux have a limiting surface. Continuity conjecture. If…

A determinantal point process (DPP) assigns probabilities to subsets through a positive semidefinite ensemble matrix, and learning its likelihood means estimating the model paramet…

Let $E$ be the underlying space, let $\mathbb{P}$ be a point process on $E$, and let $\rho^R$ be an $L^1$-specialization. The pair $(\mathbb{P},\rho^R)$ is Giambelli compatible whe…

Optimal-measure claim. For any fixed $n\geq 1$, the optimal measure is $d\sigma=w^\lambda dx$ for some $\lambda\in(-1/2,\infty)$. The value of $\lambda$ may depend on $n$.

Universality conjecture. The coefficient $C_2$ has the form

Sharp variance asymptotics conjecture. The variance of the free-fermion linear statistic satisfies the two-sided bound above as $\hbar\to0$.

Simultaneous uniform interpolation conjecture. For $\mathbb{P}_{K_{\mathbb{D}}}$-almost every $X\in\operatorname{Conf}(\mathbb{D})$,

Let $A$ be the function defining the Zakharov–Shabat reflection coefficient, let … and let $A_s$ be the associated operator, with kernel $K_s=A_s^2$. Consider the Riemann–Hilbert p…

Let $X$ be the underlying space, let $\mathbb{F}$ be the scalar field, and let $K,Q:X^2\to\mathbb{F}$ be kernels. They are equivalent when, for every $r\geq 1$ and every…

A determinantal point process on a finite ground set is specified by a positive semidefinite kernel whose eigenvalues lie in $[0,1]$, and its maximum likelihood estimator is the es…

Partial-decoupling conjecture. The kernels of the partial decouplings of $Z$ are the only critical points of $\Phi_{L^*}$.

Let $\widehat\lambda^{\mathrm{med}}$ be the median-based estimator of the intensity $\lambda$ for a stationary determinantal point process, and let $W_n$ be the observation domain.…

Let $s_n=\sqrt{c_n}\,\mathrm{P}(N(\mathbf{X}\cap C_{n,0})=\lfloor\mathrm{Me}_Z\rfloor)$ be the density factor appearing in the asymptotic normality result for the jittered median-b…

Let $\mu$ be a determinantal random point field with a continuous kernel, and let $(\mathcal{E},\di)$ be the associated pre-Dirichlet form on $\Lm$. The notation $\Lm$ denotes the…

Interpolation conjecture. Given any countable dense set $H_0\subset H$, for almost every ${\mathscr X}$, the set

Let $\widehat{\rho}_N(t,x)$ denote the particle density of the drifted noncolliding Brownian motion with $N$ particles at time $t>0$, and let the width of its support be the length…

Uniform Hessian bound. There is a constant $C$, independent of $k$, such that