62 problems

Odd-primary transfer conjecture. The transfer homomorphism $Tr_h^{\mathscr A_p}(\mathbb F_p)$ is one-to-one for any odd prime $p$ and $1\leq h\leq 4$. This extends the known low-ra…

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…

Fix a finite prime field $\mathbb{F}_p$ and let $\mathcal{A}_p$ denote the Steenrod algebra. For a topological space $X$, write $H^*(X)$ for its cohomology with coefficients in…

Kuhn's strong non-realization conjecture. If the mod $p$ cohomology of $X$ belongs to $athcal{U}_n$ for some integer $n$, then it is locally finite.

Let $X$ be a simply connected polyGEM of finite type at $2$, and let $\widetilde{\operatorname{H}}^*X$ denote its reduced cohomology with coefficients in $\mathbb{F}_2$. Grodal's c…

Let $p$ be a prime and let ${\EuScript A}$ be the mod $p$ Steenrod algebra. A module over ${\EuScript A}$ is topologically realizable if it is isomorphic to the mod $p$ cohomology…

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…

Let $G$ be the group acting in the hyperreal setting, let $C_2$ be its indicated subgroup, and let $E_{2+\sigma}$ and $BBU_{\mathbb R}$ denote the spaces appearing in the source. W…

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 $\mathcal B_m$ be its symmetric subalgebra, and let $f$ be a monomial in $\mathcal P_m$. Deno…

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 $p$-…

Detection conjecture. The nonzero element $t$ is detected by the sixth algebraic transfer $Tr_6(\mathbb F_2)$.

Let $\mathcal A$ be the mod-two Steenrod algebra, let $s\geqslant0$, and set … Let $P$ denote the Adams periodicity operator, and let $h_i$, $g_i$, $f_i$, $c_i$, $D_2$ and the disp…

Let $p$ be an odd prime, let $\mathcal A_p$ be the mod-$p$ Steenrod algebra, and let $Tr_h^{\mathcal A_p}$ denote the odd-primary algebraic transfer from the $GL_h(\mathbb F_p)$-co…

Let $M$ be an unstable $\mathcal A$-module, let $\mathscr R_h$ be the $h$-th Singer functor, and consider the algebraic approximation … Here positive stem means positive difference…

Nilpotence conjecture for $b_{11}$. The class $b_{11}$ is nilpotent in

Filtration submodule and annihilation conjecture. For each $1\leq k<\min(m,n)$, $\mathcal{F}_{n,k}$ is an $\mathcal{A}$-submodule and a ${\mathcal{D}}_n$-submodule of…

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…

Let $\mathscr A_2$ be the mod-$2$ Steenrod algebra, let $n_d$ be the internal degree sequence used in the paper, and let $Tr_5^{\mathscr A_2}$ denote the fifth algebraic transfer.…

Let $\mathscr A_2$ be the mod-$2$ Steenrod algebra and let $Tr_5^{\mathscr A_2}$ denote the fifth algebraic transfer from the $GL_5$-coinvariants of primitives to…

Let $\mathbb P_{n_d}^{\otimes 5}$ denote the degree-$n_d$ part of the polynomial algebra on five generators over $\mathbb Z_2$, let…

Let $n\geq 1$, let $\epsilon$ be $0$ or $1$, and let $l\geq 0$. Set $m=2^{n+1}-\epsilon+2l$, and let $k_n(\operatorname{Gr}_d({\mathbb R}^m))$ denote the dimension of…