44 problems

Problem (Simon's Problem 1). Prove that for every $\nu\ge3$ there are $a<b$, with $b-a$ small, and an open interval $I\subset\mathbb R$ such that, almost surely, $H_\omega$ has pur…

Let $\mathcal H_6$ be a six-dimensional complex Hilbert space. For an integer $K$, let $\mathcal B_m=\{\psi_i^m:1\leq i\leq 6\}$, $1\leq m\leq K$, be orthonormal bases of…

Let $\tfrac12<\gamma<\tfrac32$, let $V:\mathbb R\to\mathbb R$ belong to $L^{\gamma+1/2}(\mathbb R)$, and set $V_-(x)=\max\{-V(x),0\}$. Write $E_j$ for the negative eigenvalues of t…

Consider a non-relativistic atom in the Born–Oppenheimer approximation: a point nucleus of positive integer charge $Z$ is fixed at the origin, and $N$ electrons interact through Co…

Prove that the spectrum of the Anderson model in dimension $\nu=2$ is dense pure point.

“Prove that Bose-Einstein condensation occurs in a continuum model of a weakly interacting Bose gas, or determine whether the long-held assertion fails.” The question does not spec…

For a finite-dimensional bipartite system, it suffices to consider equal local dimensions. Let $d\geq 3$, let $\mathcal H_d=\mathbb C^d$, and let $\rho$ be a density operator on…

Give a rigorous explicit solution of the fixed-frequency scalar-wave diffraction problem for two finite slits in the plane, including asymptotics of the solution. This is a broad r…

In 2002, P. Zhou analyzed the behavior of solutions of the Cauchy problem for perturbations of the defocusing NLS equation i ut + u{xx}- 2|u|^2 u - epsilon |u|^ell u =0, u(x,t=0) =…

The problem is an open-ended research program: develop numerical methods for solving Cauchy problems for nonlinear integrable equations by computing their scattering data and then…

The principal question is to determine the long-time behavior of a solution of the focusing nonlinear Schrödinger equation on the half-line, with prescribed initial profile $f$ and…

The research direction is to determine how broadly universality of computation times extends across numerical algorithms on random data. For input size $n$, let $X_n$ have input la…

In one spatial dimension, the KPZ equation is formally … where $\dot W$ is space-time white noise. Its Hopf–Cole solution is defined through the stochastic heat equation (SHE),…

Consider the Korteweg--de Vries (KdV) equation ut + u ux + u{xxx} =0 with initial data u(x, t=0) = u0(x),qquad xin mathbb{R}. In the 1970's, McKean and Trubowitz proved the remarka…

The research direction is to extend the Riemann–Hilbert/steepest-descent method to $n\times n$ Riemann–Hilbert problems, with the aim of treating generic situations in multi-matrix…

Given initial data $u_0(x)$, $x\geq 0$, smooth and compactly supported, and boundary data $u(0,t)=\sin(\omega t)$, $t\geq 0$, with $\omega\neq 0$, the question is: suppose $u(x,t)$…

The research direction is stated as: “What happens if the geometric distribution, say, is slightly perturbed? The challenge here is to develop an effective perturbation theory for…

The research direction is to understand the long-time behavior of perturbations of focusing cubic NLS near a multisoliton. Replacing the defocusing cubic term in the model equation…

For an integer $n\ge 2$, let $J(0)$ be an $n\times n$ tridiagonal TE1 matrix with diagonal entries $a_1(0),\ldots,a_n(0)$ and positive off-diagonal entries…

Let $(a_1,a_2,\ldots)$ be an i.i.d. sequence, and let $X$ denote an operation that produces a sequence $(X_1,X_2,\ldots)$ from it. The question is to characterize such an operation…

The question is to develop a microscopic model for the reported random-matrix behavior of car-parking spacings. The reported contrast is GUE statistics for cars parked on the right…

For focusing nonlinear Schrödinger (NLS), the question concerns whether the non-generic spectral behavior described below changes the long-time behavior of solutions, and, in the c…

The research direction is the one stated in the question: “The spectral problem with general data, both for NLS and also other related non-self-adjoint problems, however, is far fr…

Let $B$ be standard Brownian motion on $0, \infty)$ and interpret $B'$ as white noise. On $L^2((0, \infty))$, let the stochastic Airy operator with Dirichlet boundary condition at…

Let $M$ be a real $p\times n$ matrix whose columns are independent samples from the multivariate Gaussian distribution $N_p(\mu,\Sigma)$, where $\Sigma$ is positive definite. Cente…