3 problems

Let $\mathbb{G}$ be an $m$-dimensional Carnot group with horizontal distribution $\Delta=\operatorname{span}\{X_1,\ldots,X_k\}$, let $I=[0,1]$, and set $H=L^2(I,\mathbb{R}^k)$. For…

Let $\mathbb{G}$ be a Carnot group with horizontal distribution $\Delta$, equipped with a left-invariant norm, and let $H=L^2(I,\mathbb{R}^k)$ be the control space for horizontal p…

Guth's conjecture. If $n\geq 4$, then there exists a map $f\in C^1(\mathbb S^{n+1},\mathbb S^n)$ such that