7 problems

Let $X$ be a $2d$-connected space, with $d\geq 1$, and let $M=\widetilde H^*(X;\mathbf F_2)$ be its reduced mod-$2$ cohomology. Assume that all cup products in $M$ are trivial, tha…

Let $D_n$ be the Dickson invariant algebra of $n$ variables over ${\mathbb F}_2$, viewed as a subalgebra of ${\mathbb F}_2[X_1,\ldots,X_n]$. A polynomial is hit if it lies in the i…

Characteristic-rank conjecture. The characteristic rank is

Anomalous-module freeness conjecture. This phenomenon should generalize to $\widetilde{\operatorname{Gr}}_5(2^t)$ and $\widetilde{\operatorname{Gr}}_4(n)$ with

Strong-form conjecture. As unstable algebras, one has

Weak-form conjecture. As graded algebras, one has

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…