7 problems

Let nanowords be considered over an alphabet consisting of exactly one element, with homotopy defined by the nanoword homotopy relation. Stronger conjecture. Any two nanowords over…

Let $(M,g)$ be a compact Riemannian manifold, and consider the wave maps equation from the circle to $M$. Two states are homotopic when they lie in the same homotopy class as maps…

Monotonic spiral conjecture. There exists a continuous function $h:[0,1]\times[0,\infty)\to V$ satisfying $\lim_{t\to\infty}h(s,t)=0$ for all $s$, $h(0,t)=p(t)$, and, for linearly…

The spiral conjecture. Every relation-avoiding origin path $p$ is weakly homotopic to a relation-avoiding origin path $q$ for which there exist linearly independent vectors…

Let $C\in \mathfrak{C}_{p_0}$ be a normal curve, and let a minimum homotopy be a homotopy from $C$ to $p_0$ whose area is minimum. A $b$-move is a move in the homotopy that creates…

Let $M$ be a Riemannian surface and let $\gamma$ be a simple closed curve that is contractible through curves of length less than $L$. A monotone contraction covers $\gamma$ if…

Let $M$ be a Riemannian surface, let $\boldsymbol{\gamma}$ be a simple curve in $M$, and let $L>0$. Suppose that $\boldsymbol{\gamma}$ is homotopic to a curve with the opposite ori…