7 problems

Group-ring resolution conjecture. There is a function $f(n)$ such that every $n$-generated ideal in $\mathbb K[G]$ has a free resolution of length $r-1$.

Let $M$ be a closed Riemannian manifold whose fundamental group satisfies $(T_n)_{L^1}$, and suppose that $M$ is $(n-1)$-connected. The $(T_n)_{L^1}$ waist conjecture. Does the fam…

Let $M$ be an $n$-dimensional complete Riemannian manifold that is a $\operatorname{CAT}(\varkappa)$ space, and let $B_M(R)\subset M$ be a ball of radius $R$, with…

Let $K \subseteq \mathbb{R}^n$ be a centrally symmetric convex body, let $\ell \in \{1,\ldots,n\}$, let $\mu$ be a probability measure supported in $K$ with a log-concave density,…

Let $ell geq 1$ and let $0 < lambda_1 leq lambda_2 leq cdots leq lambda_n$. Let $Q = (0,lambda_1) times cdots times (0,lambda_n)$, and for a map $f:Q to \mathbb{R}^{ell}$ write…

The section develops convex-geometric waist inequalities for continuous maps, log-concave measures, and related volume estimates. Greater-generality conjecture. It should be possib…

Let $X$ be an $n$-dimensional Riemannian manifold with sectional curvature $K\geq 1$, and let $f:X\to\mathbb{R}^k$ be a continuous map. For a subset $Y\subset X$, write…