5 problems

Ault–Fiedorowicz conjecture. For every $k$-algebra $A$, there is an isomorphism

Let $k$ be a commutative ring, let $k[t]$ be the polynomial algebra, and for $m\geq 0$ let $HS_1(k[t])_{t^m}$ denote its $t^m$-layered symmetric homology, obtained from the decompo…

Let $k$ be a commutative ground ring and let $n$ be a nonnegative integer. Consider the truncated polynomial algebra $k[t]/(t^n)$ and its degree-one symmetric homology…

Let $Sym_*^{(p)}$ denote the $p$th symmetric chain complex, and let $H_*(Sym_*^{(p)})$ be its homology. Free abelianity conjecture. The group $H_*(Sym_*^{(p)})$ is free abelian for…

Let $k[t_1,t_2,\dots,t_n]$ be the polynomial ring in $n$ indeterminates over $k$, and let $C_\infty$ denote the monad associated to an $E_\infty$ operad. Write $S^i$ for the $i$-sp…