64 problems

Parity conjecture. If $(1-\epsilon)\alpha$ is non-zero, then $w$ is even.

Let ${\mathbf{Spt}}^\Sigma_{{\mathbb P}^1}(k)$ be the category of symmetric $\mathbb{P}^1$-spectra over $k$, and let ${\operatorname{{\mathcal S}{\mathcal H}}}(k)$ be its homotopy…

Let $k$ be the base field, let ${\operatorname{{\mathcal S}{\mathcal H}}}_{S^1}(k)$ be the homotopy category of $S^1$-spectra, and let $\Sigma^\infty_{{\mathbb P}^1}$ and…

Let $k$ be the base field and let ${\operatorname{{\mathcal S}{\mathcal H}}}_{S^1}(k)$ be the Morel–Voevodsky homotopy category of $S^1$-spectra. For an integer $d$, let…

Morel's stable $\mathbb{A}^1$-connectivity conjecture. Under these hypotheses, $L_{\operatorname{mot}}E$ is $n$-connective.

Let $S=\operatorname{Spec}\mathbf Z$. For an $R$-linear MW-module $M$ over $S$, write $\big(\mathrm{FT}^{R}\big)$ for the property that, for every $S$-scheme $X$ of finite type, al…

Uniqueness conjecture for relative motivic spheres. If $X$ is $\mathbb{A}^1$-homotopic to $\mathbb{A}^2_S\backslash\{0\}$, then $X$ is isomorphic to $\mathbb{A}^2_S\backslash\{0\}$…

Morel's conjecture.

Let $F$ be a field of low cohomological dimension. Consider the motivic Adams spectral sequence based on $\operatorname{BPGL}\langle 2 \rangle$, and let $n$ be a positive integer.…

Let $f/g:\mathbb{P}^1\to\mathbb{P}^1$ be a pointed rational function with vanishing locus $D=\{r_1,\ldots,r_n\}\subset\mathbb{A}^1_k$. Let $m_i(x)\in k[x]$ denote the monic minimal…

Let $F$ be a field, and let $\mathrm{SH}_F$ denote the motivic stable homotopy category over $F$. Let $\mathrm{MGL}$ be the algebraic cobordism spectrum, let $a_i$ be generators of…

Let $F$ be a field. Write $\mathcal{A}^{\star,\star}_{F}$ for the algebra of motivic Steenrod operations and $H\mathbb{F}_{p,F}^{\star,\star}H\mathbb{F}_{p,F}$ for the correspondin…

Let $k$ be the base field, let $\mathcal{X}=(X,D)$ be an object of $\category{mSm}$, and let $q\geq 0$. Write $\mathcal{X}^{\circ}$ for the complement of the boundary,…

Let $S\in\mathrm{Rg}$ be a base ring, let $\mathrm{MS}_{\mathrm{Nis}}(S)$ denote the Nisnevich motivic stable category over $S$, and let $\mathbf{MGL}$ be the algebraic cobordism s…

Let $S\in\mathrm{Rg}$ be a base ring, and let $\mathrm{SH}(S)$ denote the stable motivic homotopy category over $S$. Let $\mathbf{MGL}\in\mathrm{SH}(S)$ be the algebraic cobordism…

Let $k$ be a flexible field and let $F/k$ be a finitely generated field extension. The isotropic Milnor K-ring of $F/k$, denoted $\operatorname{\mathrm{k}}^M_*(F/k)$, is the quotie…

Improved Adams-filtration bound. Then $x$ has Adams filtration at least $3n$.

Mixed-differential conjecture. There exists a $\tau^{2^n}$-periodic differential

Let $S$ be a derived scheme. Write $\mathrm{H}\mathbb Q$ for the rational motivic cohomology spectrum over $S$, and let $\mathbbm{1}_{\mathbb Q}^+$ denote the positive eigenspace i…

Connective-cover $p$-completion conjecture. If $X$ is $p$-complete, then $\tau_{\geq 1}X$ is also $p$-complete.

Let $\mathcal{C}$ be the realization target, let $L$ be its Tate object, and let $\nu(L^{\otimes *})$ be the filtered object induced by the slice tower. Write…

Let $X$ be a topologically contractible smooth affine complex variety, and let $x$ be a chosen closed basepoint. Asok–Østvær's conjecture. The pointed variety $(X,x)$ is stably…

Let $BPGL\langle m\rangle$ be the truncated motivic Brown–Peterson spectrum, and let its homological slice spectral sequence (HSSS) have differentials $d_r$. The differentials disp…

Let $S$ be a noetherian excellent finite dimensional base scheme of characteristic $0$. The Ayoub conjecture. The heart of the perverse homotopy t-structure on $\mathrm{DM}(S)$ is…

Let $S$ be a scheme. Write $\Kthlog(S)$ for its logarithmic $K$-theory spectrum and $\Kth(S)$ for its algebraic $K$-theory spectrum. Logarithmic K-theory comparison conjecture. For…