17 problems

Let $X$ be a nilpotent complex algebraic variety such that $\pi_2(X)$ is a free abelian group of rank $r$. Assign to each holomorphic map $f:\mathbb{C}\mathbb{P}^1\to X$ a multiple…

Cohen–Jones–Segal's stability conjecture. The stability theorem should hold if and only if the evaluation map $E$ has the required rational homotopy property. The supplied statemen…

Let $Z$ be a smooth connected closed $n$-dimensional $K(\Gamma,1)$-manifold, and let $\operatorname{Diff}(Z)$ be its group of diffeomorphisms with the natural smooth topology. Farr…

Rational cohomology conjecture. The rational cohomology of $\mathscr{B}un_{G,X}^0$ is

Let $m$, $p$, and $n$ be integers, let $\mathcal{L}_{m\cdot p}^{n}$ be the space of long links with $m$ components of dimension $p$ in $\mathbb{R}^{n}$, and let…

Fix $m\geq 2$, $k\geq 1$, and $n\geq 3$. Let $\operatorname{Conf}(m,\mathbb R^n)$ be the configuration space of $m$ points in $\mathbb R^n$, and let … be the graphing map. Real gra…

Let $m\geq 2$ and $n\geq 3$, and consider the rational homotopy groups of the configuration space $\operatorname{Conf}(m,\mathbb R^n)$. Graphing extension conjecture. The injectivi…

Let $m\geq 2$, $n\geq 3$, and let $G$ denote the graphing map from the based loop space of the configuration space of $m$ points in $\mathbb R^n$ to the corresponding space of high…

The mod-$p^r$ hyperbolicity conjecture. Let $X$ be a finite simply-connected $CW$-complex. If $X$ is rationally hyperbolic, then it is mod-$p^r$ hyperbolic for all primes $p$ and p…

Let $X$ be a finite type space of finite Lusternik–Schnirelmann category. It is rationally hyperbolic when the dimension of $\pi_\ast(X)\otimes\mathbb{Q}$ is infinite, and let…

Let $X$ be a space and let $p$ be a prime. Consider the growth of the rational homotopy groups of $X$ and of the $p$-torsion in its homotopy groups. The rational–p-torsion growth c…

Let $X$ have dimension at most $n$, let $Y$ be simply connected, and let $f:X\to Y$ be nullhomotopic with Lipschitz constant at most $L$. Let … be a filtration of the indecomposabl…

Gromov's thickness conjecture. There is a homotopy between $f$ and $g$ of thickness $O(L^p)$, where $p$ depends only on the rational homotopy type of $Y$; perhaps $p$ can always be…

Goodwillie--Weiss truncation homotopy conjecture. If $d-\max\{m_i\mid 1\leq i\leq r\}>2$, then

Low-codimension rational homotopy splitting conjecture. For every degree $*\geq 0$ and every $d-\max\{m_i\mid 1\leq i\leq r\}>2$, the identity giving the rational homotopy splittin…

Let $D$ be the exceptional divisor of a resolution of an isolated rational hypersurface singularity of dimension at least $3$, and let $\Delta=\Delta(D)$ be its dual complex. Ratio…

Directional isoperimetric conjecture. There is such a $(k+1)$-chain $y$ for which, for every $(k+1)$-tuple $I$,