7 problems

Let $D$ denote the Euclidean unit ball in $\mathbb{R}^n$, and let $M^*(A)$ denote the mean width functional used in the paper. For a set $S=\{x_i\}_{i=1}^{2^k}\subset\mathbb{R}^n$,…

Let $V$ be a $(K,n)$ regular set in $\mathcal{B}^N$, and let $\mathcal{N}(V,\eps)$ denote its covering number at scale $\eps$. Covering-number dependence conjecture. There exists a…

Let $S_1$ and $S_2$ be Lagrangian $n$-spheres in an $A_2^n$-configuration, intersecting transversely at a single point, and let $\tau_1$ and $\tau_2$ be symplectic Dehn twists alon…

Let $M$ be a compact manifold, let $m$ denote Lebesgue measure, and let $\operatorname{Diff}^{1}_{m}(M)$ be the space of $C^1$ Lebesgue-measure-preserving diffeomorphisms. A diffeo…

Generalized Sudakov minoration conjecture. There is a universal constant $C>0$ such that

Brownian-sheet small deviation conjecture. For $d\ge2$, as $\epsilon\downarrow0$,

Smooth small ball inequality conjecture. The displayed inequality should hold, with an implied constant depending only on $d$ and $\varphi$. This is a smooth analogue of the Haar s…