17 problems

Let $P_n=\mathbb Z/2[u_1,\ldots,u_n]$ be the polynomial algebra with its unstable $\mathbb A$-module structure, where $\mathbb A$ is the 2-primary Steenrod algebra. Define the cohi…

Let $P=\mathbb F_2[t_1,\ldots,t_h]$ be the polynomial algebra on $h$ generators, regarded as a module over the mod-two Steenrod algebra $\mathcal A$, and let … be its degree-$n$ co…

Let $\mathcal{A}$ be the Steenrod algebra and let $\mathcal P_n$ be the polynomial algebra on $n$ degree-one generators over $\mathbb F_2$. For a degree $d$, let $\alpha(d)$ denote…

Let … Over the base field $\mathbb{C}$, let $M_n(\mathbb{C}):= H^{*,*}(BV_n/\mathbb{C};\mathbb{F}_2)$ and … where $A^{>0,*}$ consists of motivic Steenrod algebra elements of positi…

Let $\mathcal P_m$ be the polynomial algebra on $m$ generators over $\mathbb F_2$, let $Q^{\otimes m}$ denote its cohit module, and let $\mathsf{Param}$ be a parameter vector of de…

Let $P_h$ be the polynomial algebra on $h$ variables over $\mathbb F_p$, equipped with the action of the Steenrod algebra $\mathscr A_p$. For a positive integer $n$, write its $p$-…

Let $P_k=\mathbb F_2[x_1,\ldots,x_k]$ be the polynomial algebra, let $Q P_k$ denote its quotient by the image of the positive-degree Steenrod algebra, and let … be Kameko's squarin…

When $bb Z/2$ is regarded as a trivial $A$-module, the hit problem is equivalent to determining the cohits … as a graded vector space, or more generally as a graded module over the…

Coinvariant-dimension conjecture. Its dimension is

Let $k=\mathbb Z_2$, let $P_4=k[x_1,x_2,x_3,x_4]$, and write $Q^{\otimes 4}=k\otimes_A P_4$. For a positive integer $s$, Kameko's squaring operation is the $kGL_4(k)$-module epimor…

Coinvariant-dimension conjecture. This space is trivial if $s=2$ and has dimension $1$ if $s\neq 2$.

Let $\mathscr B_d(n)$ be the set of admissible monomials of degree $n$ in $\mathscr P_d$, let $\mathscr B_d(\omega)$ be the subset of weight $\omega$, and let $\overline{\Phi}$ be…

Admissible-monomial prediction. If $\omega$ is a weight vector, then

Let $M_n$ be the mod $2$ motivic cohomology algebra of the classifying space of the elementary abelian $2$-group of rank $n$, and let $QM_n^{d,*}$ denote its degree-$(d,*)$ quotien…

Let $P_k$ be the polynomial algebra on $k$ generators over $\mathbb{F}_2$, let $QP_k$ denote its indecomposable quotient under the Steenrod algebra, and write $(QP_k)_n$ for its de…

Let $\mathcal{A}$ be the mod-$2$ Steenrod algebra, and let a monomial of total degree $d$ have exactly $r$ odd exponents. For an integer $m$, let $\alpha(m)$ be the number of nonze…

Let $\widetilde{\mathbf C}$ be the quotient of $\mathbb{R}[\mathbf{x}]$ by the subspace of hit-polynomials for the operators $\widetilde D_k$, and let…