20 problems

Localize all spaces at $3$. Let $B_2(27,35)$ denote the space occurring in the proposed spherical resolution, and let $\alpha_2$ be the element generating…

In the exceptional Hermitian symmetric case $G_{\Bbb R}=E\,VII$, one has $(\frak g,\frak k)=(\frak e_7,\frak e_6\oplus{\Bbb C})$ and rank $r=3$. Let $\Lambda$ be the set of partiti…

Let $G_2$ have basis $C_1,\ldots,C_{14}$, with subgroups $S$, $\Sigma$, and maximal torus $T$ generated respectively by $\{C_1,C_2,C_3\}$, $\{C_8,C_9,C_{10}\}$, and…

Supersymmetry conjecture. The supersymmetries are preserved under the breaking of $Spin(8)$ to $G2$, and this preservation can be understood as arising from the components of $F_4$…

Let $S^*$ denote the unit circle, and let $\tilde J_{G_2}$ be the weight function associated with the paper's Euler-angle parametrization of $G_2$. A function … is $\frac{1}{4}$-ad…

The twisted $E_8$-equivariant lift conjecture. The element $e_8$ lifts to

Consider the unipotent representation … of complex $E_7$, whose $K$-types have highest weights in the weight lattice but not in the root lattice and occur with multiplicity one. Mc…

Let $V=\mathrm E_{8,2}/\mathbb Z_2$, whose spin-$\frac32$ fields form the real representation $\mathbf{3875}$ of $\mathrm E_8$. Let $\star$ denote the $\mathrm E_8$-invariant commu…

Let $V=\mathrm E_{7,1}^2/\mathbb Z_2$, and consider the three $N{=}1$ structures constructed on $V$. The automorphism group is…

Let $V$ be an $N{=}1$ SVOA and let $V_{\mathrm{ev}}$ denote its even subalgebra. Consider the three exceptional even subalgebras $\mathrm E_{7,1}^2$, $\mathrm E_{7,2}$, and…

Let $\mathcal{O}_q(\mathbb{OP}^2)$ be the quantum Cayley plane and $\mathcal{O}_q(\mathbf{F})$ the quantum Freudenthal variety, each equipped with its Heckenberger--Kolb calculus.…

Huang's conjecture.

Kono's conjecture. For $i,j\in\mathtt{t}(G)$, there exists $k\in\mathtt{t}(G)$ satisfying

Generation conjecture for $BE_8$. The mod $2$ cohomology of $BE_8$ is generated by these two elements as an algebra over the mod $2$ Steenrod algebra.

Let $E_6$ be the compact simple Lie algebra, let $\operatorname{Aut}^+(E_6)$ denote its orientation-preserving automorphism group, and let $x_3,x_9,x_{11},x_{15},x_{17},x_{23}$ den…

Let $E_8$ be the simply connected compact exceptional Lie group, and let $\gamma$, $\gamma'$, $\sigma$, $\sigma'$ and $\upsilon_3$ be the commuting automorphism elements used to de…

Let $E_8$ be the compact exceptional Lie group, and let $\sigma'$ be the involution induced by an element of $Spin(8)\subset F_4\subset E_6\subset E_7\subset E_8$. For a subgroup…

Let $E_8$ be the exceptional Lie group, and let $BE_8$ be its classifying space. Write … for the adjoint representation, and let $w_{128}(\rho_8)$ denote its 128-th Stiefel–Whitney…

Let $PSL(2,37)$ denote the projective special linear group over the field with $37$ elements, and let $E_7$ denote a group of type $E_7$. Kostant's conjecture. The group…

Let $G$ range over the exceptional Lie groups. Deligne's conjecture. There is a one-parameter family of symmetric tensor categories $\mathcal{C}_t$ specializing to…