55 problems

Let $K$ be a compact connected Lie group with normalized Haar measure $dg$, and let $f$ and $g$ be complex-valued $K$-finite functions. Mathieu's conjecture. If … then … for all su…

Algebraic model conjecture. For each compact Lie group $G$, there is an abelian category $\mathcal{A}(G)$ and a Quillen equivalence

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 connected Lie group. For a left-invariant Riemannian metric $g$ on $G$, let $\lambda_1(G,g)$ denote the smallest positive eigenvalue of the Laplace–Beltrami op…

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…

Let $G$ be a compact Lie group, let $g_0$ be a bi-invariant metric on $G$, and consider the class of all Riemannian metrics on $G$. A metric is spectrally isolated if some neighbor…

Let $G$ be a connected compact non-abelian Lie group, and let $(g_1,\ldots,g_n)$ be an $n$-tuple of elements of $G$ with $n>1$ that generates a free group. A tuple is slightly defo…

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…

A 2-compact group is the 2-local homotopy-theoretic analogue of a compact Lie group; let $DI(4)$ denote the exotic 2-compact group constructed by Dwyer and Wilkerson, and let the 2…

Let $X$ be a connected $2$--compact group. A decomposition $X=X_1\times X_2$ is sought, where $X_1$ is the $2$--completion of a connected compact Lie group and $X_2$ is a product o…

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 $G$ be a compact simple group of rank $r$, let \textit{\text{\boldmathvarphi}} denote angular variables on a maximal Abelian torus, and let \textit{\text{\boldmathchi…

Let $G$ be a compact Lie group and let $\mathfrak{X}$ be a Zariski clopen subset of the space $\operatorname{Sub}(G)/G$ of conjugacy classes of subgroups. Write…

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…

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

Breuillard–Green conjecture. If $A$ has sufficiently small measure, then

Let $V$ be a real linear representation of a compact Lie group, with its isotropy stratification into strata. Real algebraicity conjecture. The closed strata are real algebraic sub…

Let $\rho\colon G\to\operatorname{GL}(V)$ be a linear representation of a compact Lie group $G$ on a real vector space $V$, and let $H\subset G$ represent an orbit type. Write…