23 problems

Let $F$ be an integrand, and consider conditions involving only $F$ and a finite number of its derivatives. Morrey's conjecture. There is no such condition that is both necessary a…

Ball's sufficient conditions conjecture. Strict positivity of the second variation, together with suitable versions of quasiconvexity in the interior and at the boundary, should im…

Morrey's conjecture. Every rank-one convex function $\mathbf E:\mathbb R^{2\times2}\rightarrow\mathbb R$ is quasiconvex.

Let $H$ be a finitely presented freely indecomposable subgroup of a torsion-free word hyperbolic group $G$. The virtual normalizer of $H$ is the subgroup of elements of $G$ that no…

Quasiconvexity conjecture. The displayed function is quasiconvex at some matrix $A_0$. By homogeneity, quasiconvexity at $A_0$ implies quasiconvexity at $0$. This proposed strength…

Let $\mathcal{G}$ be a locally quasiconvex hyperbolic group. An immersion of finite directed height is an immersion $\psi:H\rightarrow F$ of graphs satisfying the finite directed-h…

Let $X$ be a compact $2$-complex with negative immersions. Write $\pi_1X$ for its fundamental group, and let local quasiconvexity and the finite-generation intersection property re…

Let $\mathscr F\colon \mathbb{R}^{2\times 2}\to\mathbb{R}$ be the local Burkholder functional, known to be rank-one convex. Let $\Omega$ be the domain, let…

Let $\Omega\subset\mathbb R^N$ be the domain, let $W$ be the integrand in … and suppose that $W(x,y,\cdot,\cdot)$ is separately curl Young quasiconvex. The local functional has the…

Let $W$ be the potential appearing in the energy, and let $Q(W(|\cdot|))$ denote its quasiconvex envelope on $\mathbb{R}^{N\times N}$. Let $W^{**}$ denote the convex envelope of th…

Let $H<G$ be a hyperbolic subgroup of a hyperbolic group $G$. Suppose the Cannon–Thurston map … exists, and suppose that $H$ is not quasiconvex in $G$. Non-quasiconvex subgroup con…

Let $B_p\colon \mathbb{R}^{2\times 2}\to\mathbb{R}$ be the Burkholder integrand, defined by … where $\lvert A\rvert$ denotes the operator norm. Iwaniec's conjecture. The integrand…

Conti–Müller–Ortiz conjecture. The two hulls coincide:

The case $d=3$ conjecture. Any non-polyconvex weak extremal $f\in {\cal C}_3$ is an extreme ray of ${\cal C}_3$. Moreover, if $f\in {\cal C}_3$ is a non-polyconvex extreme ray of…

Strongly quasiconvexity dichotomy. The strongly quasiconvex subsets of any non-relatively hyperbolic, hierarchically hyperbolic space are either hyperbolic or coarsely cover the en…

Assume that $H(p)=\varphi(|p|)$, where $\varphi:[0,\infty)\to\mathbb{R}$ satisfies (H7), and let … be the associated sequence, with $M_i=\varphi(s_{2i-1})$ and…

Let $f(\boldsymbol{\xi})=\boldsymbol{\xi}C\boldsymbol{\xi}^T$ be a quasiconvex quadratic form, where $\boldsymbol{\xi}\in\mathbb R^{3\times 3}$ and $C\in(\mathbb R^3)^4$ is a fourt…

Real-time generalized word problem conjecture. The language $\textsf{GWP}(G,H)$ is real-time for every quasiconvex subgroup $H$ of every hyperbolic group $G$.

Quasiconvexity conjecture. The complement of every $W^{1,p}$-extension domain with $1<p\le2$ is quasiconvex.

Relative quasiconvexity criterion. Then $H$ is relatively quasiconvex in $G$.

Let $\Psi$ be the determinant-based function on $M(2,2)$ defined in the source, and let $f:\mathbb{R}^2\to\mathbb{R}^2$ be smooth with compact support. Writing $Df$ for its Jacobia…

Let $d\geq 2$. For every subset $K\subset \mathbb{R}^d$, the main theorem asserts the existence of a connected set $\tilde\Gamma\subset\mathbb{R}^d$ containing $K$, with controlled…

Non-equivalence conjecture. If $m,n>1$ and $f$ is lower semicontinuous, then strong Morrey quasiconvexity and weak Morrey quasiconvexity are not equivalent.