18 problems

Mosco-convergence conjecture. For every fixed $\bar\rho\in\mathcal P_2^{\mathcal S}(\mathbb R^d)$,

Avoidance conjecture. If a compact connected component $C$ of $\operatorname{Crit}(f)$ consists of weakly strict saddle points, then $W^s_-(C)$ has measure zero. Moreover, if $M$ i…

The paper considers generalized gradient flows and convex optimization on entanglement polytopes in Hadamard manifolds. In particular, the preceding discussion uses unitary matrice…

Let $X \in \mathbb{R}^{n \times T}$ satisfy $C_X \succ 0$, and let $W(t)$ follow the continuous-time gradient-flow feedforward synaptic dynamics. Let $\mathcal{S}$ denote the stati…

Let $X \in \mathbb{R}^{n \times T}$ satisfy $C_X \succ 0$, and let $W(t)$ solve the continuous-time gradient-flow feedforward synaptic dynamics … Write $\mathcal{R}_{m,n}$ for the…

Let $\gamma(t)$ be a gradient flow line for an analytic function, and suppose that it has a limit point. Arnold--Thom gradient conjecture. The limit … exists. This is a stronger, fi…

Let $\rho_0\in\mathcal{P}_V$, let $S_r$ be the stated class of flows, and for $u\in S_r$ define the pushforward measure $\rho_t=u(t,\cdot)_{\#}\rho_0$. For $u$-regularized Stein F…

Let $u$ be one of the weak solutions to the sixth-order quantum-diffusion equation constructed in the proof of the existence theorem, let $u^0$ be its initial datum, and let…

Let $f$ be a real analytic function on an open set $U\subset\mathbb{R}^n$, and let $x(t)$ be a gradient flow line of $f$ with a limit point $x_0\in U$. Arnold–Thom conjecture. If…

Assume that $X X^T$ has full rank. Let $N$ be the depth, let $W(t)=W_N(t)\cdots W_1(t)$ denote the product along the gradient flow, and let $\mathcal{M}_k$ denote the rank-constrai…

Let $f$ be an analytic function and let $x(t)$ be a gradient flow line of $f$. Suppose that $x(t)$ has a limit point $x_{\infty}$ as $t\to\infty$. Stronger Arnold-Thom conjecture.…

Center-manifold conjecture. The PDE has an $n$-dimensional center manifold on which its flow is approximated by this ODE system, in the sense that the error terms of solutions are…

Let $\mathbb H$ be a finite-dimensional Euclidean space, and let $\phi:\mathbb H\to\mathbb R$ be a continuously differentiable Lipschitz function. A map…

Equal minimal subgradient norm conjecture. If

Gradient-flow global-maximization conjecture. For any data and prior data, up to a subset of $\mathcal{W}$ of measure zero, the flow $W(t)$ approximates a global maximum in the sta…

Unique-limit conjecture. The flow has a unique limit in the set of strictly convex functions.

Global existence and convergence conjecture. There is a smooth solution $A(t)$ for $t\in[0,\infty)$ to this flow for the Yang–Mills energy functional with respect to the $L^2$ Riem…

The diffusion equation can be viewed as a gradient flow in several distinct ways, including as the $L^2$-gradient flow of the Dirichlet integral, as the $H^{-1}$-gradient flow of t…