19 problems

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…

A compact $2$-complex $X$ has nonpositive immersions if every compact, connected, collapsed combinatorial immersion $Y\rightarrow X$ has either trivial fundamental group or satisfi…

Let $X^m$ be a compact enlargeable manifold immersed in the unit ball $B^n(1)$, with $n>m$. Let $J_\nu$ be the Bessel function of order $\nu=\frac n2-1$, and let $j_\nu$ denote its…

Let $X$ be an $m$-manifold and let $S^N(1)$ be the unit sphere in Euclidean space. Assume $N\geq 100m^2$, and write $\operatorname{curv}^{\perp}$ for the normal curvature of an imm…

Let $X^m$ be a compact smooth manifold and let $B^N$ denote a Euclidean unit ball containing an immersion of $X$. Write $\operatorname{min.cirv}(X^m,B^N)$ for the infimum of the re…

Non-generic smooth immersion conjecture. The conclusion of Proposition 2 should hold without the genericity hypothesis: every Bing spine in $M$ parametrized by a smooth immersion o…

Smooth parametrization rigidity conjecture. Any Bing spine of $\times I$ that can be smoothly parametrized by an immersion of … .

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 $M^{n+1}$ be a space form of sectional curvature $-\ka^2$, for some $\ka\to 0$. Let $I$ be an interval containing $[-\ka,\ka]$ in its interior, and let $\sr F(M;I)$ denote the…

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…

Let $X$ be the source of a $\hat v$-generic immersion $\alpha:X\to A$, let $K_\alpha:\mathcal T(v)\to\mathcal F_{\leq 2}$ be the associated map, and let $\kappa^\ast$ be a generato…

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 $M^{b(k)-1}$…