30 problems

Let $A\subseteq \mathrm{SO}(3,\mathbb{R})$ be open, and let $\mu$ denote normalized Haar measure on $\mathrm{SO}(3,\mathbb{R})$. For a unit vector $u\in\mathbb{R}^3$, define the an…

Let $\mathfrak{g}$ be a real compact Lie algebra, and replace the operators and relations in Theorem … holds for all real compact Lie algebras. The theorem for $\mathfrak{su}(2)$ e…

Inverse-linear conjecture. If $h$ has nonnegative curvature, then the inverse-linear path from $h_0$ to $h_1=h$ consists entirely of metrics with nonnegative curvature.

Let $G$ be a compact Lie group with a bi-invariant metric $h_0$ and Lie algebra $\mathfrak g$. For a left-invariant metric $h$ on $G$, let $\Phi$ be the unique $h_0$-self-adjoint p…

Let $X$ be a connected $p$-compact group with maximal torus $T_X$. Its maximal torus normalizer is represented by the loop space $NT_X$. Maximal torus normalizer conjecture. The gr…

Let $G$ be a compact connected Lie group, let $X$ be a $p$-complete space, and let $\mathcal{A}_p$ denote the mod-$p$ Steenrod algebra. Suppose that … as algebras over…

Let $G$ be a simply-connected compact Lie group with … where each $H_i$ is a simple Lie group. Product formula conjecture. The weak category, ordinary category, and strong category…

Let $K$ be a compact connected Lie group, and let $f$ and $h$ be complex-valued finite-type functions on $K$. Let $dk$ denote Haar measure on $K$. Mathieu's conjecture. If … for al…

Let $G$ be a compact Lie group and let $BG$ be its classifying space. Kono–Yagita's even-degree conjecture. The Brown–Peterson cohomology $BP^*(BG)$ is concentrated in even degrees…

Let $G$ be a compact Lie group, let $BG$ be its classifying space, and let $BP^*(BG)$ denote its Brown–Peterson cohomology ring. Kono–Yagita's nilpotence conjecture. The ring…

Adams's detection conjecture. For every compact connected Lie group $G$ and every odd prime $p$, the map $\Phi$ is injective. This strengthens Quillen's result that the kernel cont…

Benson's torus-generation conjecture. The bounded derived category $\mathcal{D}^b(C^*BG)$ is generated by the module $C^*BT$.

Benson's generation conjecture. The category $\mathcal{D}^b(C^*BG)$ is generated as a thick subcategory of $\mathcal{D}(C^*BG)$ by the objects $\operatorname{Hom}_{C_*G}(k,X)$ for…

Continuous Babai conjecture. There is an integer $m$ such that

Doubling minimiser conjecture. The subset $H_{\delta}$ minimises the doubling constant among measurable subsets $A$ satisfying

Breuillard–Green conjecture. The minimal value of the doubling constant for subsets of small measure should be about

Word-map nonconstancy conjecture. Every non-constant word defines a non-constant word map.

The optimal eigenfunction-bound conjecture. This estimate should hold for every $p>2+\frac{4}{d-2}$, with an $\varepsilon$-loss if $2\leq r\leq 4$. The paper establishes evidence f…

The optimal Strichartz-range conjecture. The estimate should hold on any compact Lie group for every $p>2+\frac{4}{d}$. The paper provides evidence for this assertion for class fun…

Let $G$ be a compact connected Lie group, and let $\mathcal M^G$ denote the space of left-invariant Riemannian metrics on $G$. For $g\in\mathcal M^G$, write $\lambda_1(G,g)$ for th…

Let $\Sigma$ be a finite-dimensional quantum control system on $\mathbb{C}^n$ of the form … where $A$ and the $B_i$ are skew-adjoint matrices, and normalize the drift by…

Let $G$ be a simple compact Lie group, let $k\geq 4$, and let $S$ be a symmetric set of size $k$ generating a dense subgroup of $G$. Let $\Delta^S$ denote the associated averaging…

Let $G$ be a sscc Lie group and let $S$ be a connected graph with $k$ vertices and $r$ edges. Write $d=\operatorname{rank}(G)$ and $l=\frac{\dim(G)-\operatorname{rank}(G)}{2}$, and…

Uniform volume-doubling conjecture. There is a constant $D(K)$ such that

Let $G$ be a compact Lie group equipped with a rational metric $g$. Let $d$ be the dimension of $G$ and $r$ the rank of $G$. Let $I\subset\mathbb{R}$ be a finite time interval. Com…