8 problems

Fix a subspace dimension $d$ and distortion parameters $\alpha = \beta \in (0,1)$. A random sparse matrix $\bm{\Phi}: \mathbb{R}^n \to \mathbb{R}^k$ has at most $\zeta$ nonzero entr…

Let $\mathcal{A}$ be a unital C-algebra, let $0<\varepsilon<1$, and let $M,N\in\mathbb{N}$ and $\mathbf{x}_1,\dots,\mathbf{x}_M\in\mathcal{A}^N$. Modular Johnson–Lindenstrauss flat…

Nonlinear $p$-stable projection conjecture. With probability at least $1-N^{-(c-2)}$, for all $x,y\in P$, if $\lVert x-y\rVert_p=O(\epsilon^2)$ then

Low-dimensional special linear group conjecture. For every prime power $\mathfrak{q}$ there exist $D=D(\mathfrak{q})\geqslant1$ and positive constants $c=c(\mathfrak{q})$,…

Let $(\mathcal{M},d_\mathcal{M})$ be an $n$-point metric space. For $p,D\geqslant 1$, a bi-Lipschitz distortion-$D$ embedding of $\mathcal{M}$ into a metric space means an embeddin…

Let $a_1,a_2,\ldots,a_n,b_1,b_2,\ldots,b_n\in\mathbb{R}^n$ have Euclidean norm at most $1$, let $0<\varepsilon<1$, and set … where $C$ is the absolute constant from Theorem … , the…

Let $f(n,d,\varepsilon)$ denote the smallest $m$ such that every $n$-point subset of $\ell_2^d$ can be embedded into $\ell_2^m$ with distortion at most $1+\varepsilon$. Dimension-r…

Let $S$ be a midsurface, let $E^h(u^h)$ denote the elastic energy of a shell of thickness $h$, and suppose … For $i\geq 1$, write … when $i>1$, and let $\mathcal{V}_N$ denote the s…