12 problems

Let $r(\alpha)$ be the integer that counts the number of rotations of the velocity vector along a curve $\alpha$. Curves with $r(\alpha)=m$ include the curves $\gamma_m$ in the fam…

Let $M^n$ be a closed $n$-manifold. A marked cubical decomposition of $M^n$ is a cubical decomposition together with the marking data used to distinguish its cubical cells, and two…

Filling homotopy conjecture. Regularly homotopic filling immersions of arbitrary surfaces are filling homotopic.

Let $X^n$ be a closed manifold and let $f:X^n\to\mathbb{B}^N$ be an immersion. Write $g=f^*g_{\mathbb{R}^N}$ for the induced metric, and let $\mathfrak{c}(f)$ denote its minimal no…

Let $X^4$ be a closed manifold and let $f:X^4\to\mathbb{B}^N$ be an immersion. Write $g=f^*g_{\mathbb{R}^N}$ for the induced metric, and let $\mathfrak{c}(f)$ denote its minimal no…

A compact $2$-complex $X$ has negative immersions when it satisfies the negative-immersion property defined in the source. An immersion is a map $\psi:H\rightarrow F$ between graph…

Let $(\gamma,H)$ be Bartnik boundary data on $S^2$, normalized by $\operatorname{area}_{\gamma}(S^2)=1$. A maximal static vacuum solution is an asymptotically flat static vacuum tr…

Let $(\gamma,H)$ with $H>0$ be the Bartnik boundary data of a locally flat 3-ball that is not an embedded ball in $\mathbb R^3$. Let ${\mathcal M}$ denote the space of Bartnik boun…

Let $F:\bar B\rightarrow\mathbb R^3$ be a smooth immersion of a 3-ball that is not an embedding, and equip the ball with the pulled-back Euclidean metric $F^*(g_{\mathrm{Eucl}})$.…

Smoothness and invertibility conjecture. For each $k>\frac{d}{2}+1$, the operator $\Psi^L$ depends smoothly on the immersion $f\in\operatorname{Imm}^k(M,N)$ and is invertible as a…

Classification conjecture. Every complete immersion $F\colon M^{n}\to\mathbf{R}^{n+1}$ with the central cross-cut property must be either a central cylinder or a tubular quadric.

Let $b=b(k)$ be the number of ones in the binary expansion of $k$, and assume $n+2\geq 2^{b(k)}$. Let $s=n-2b(k)+k+3$. Consider a skew-framed immersion … of a manifold $M^{b(k)-1}$…