15 problems

Positive 2-curvature conjecture. $M$ admits a Riemannian metric of positive $2$-curvature if and only if

Global ultracommutative structure conjecture. Equivariant $\operatorname{TMF}$ admits the structure of a global ultracommutative ring.

A sigma orientation assigns, for each compact Lie group $G$ and each virtual representation $V\in \mathbf{R}\mathrm{O}(G)$ equipped with a string structure, an equivalence of $G$-e…

Let $\mathrm{tmf}$ be the connective topological modular forms spectrum, let $\overline{\mathrm{tmf}}$ denote its cofiber in the unit decomposition, and let…

Let $Mj_3$ be the $E_\infty$-ring spectrum constructed by Baker, equipped with the map $Mj_3\to \mathrm{tmf}$. Baker's conjecture. The induced map on homotopy groups … is surjectiv…

In the $\mathrm{tmf}$-Adams spectral sequence for $Z$, let $\epsilon_j,\bar\epsilon_j\in\{0,1\}$ and let $m\gg0$. Differentials conjecture, part 2. The $v_2$-families … support dif…

Let $x_i=h_{i,0}^2$ in the $\mathrm{tmf}$-Adams spectral sequence for $Z$. Hidden-extension conjecture. There are hidden extensions … This is suggested by comparing the known…

Let $x_j=h_{j,0}^2$. In the $\mathrm{tmf}$-Adams spectral sequence for $Z$, use indices $i\geq3$, $l\geq i+3$, $m<2^{i+1}-1$, $k_j\geq0$, and $\epsilon_j,\bar\epsilon_j\in\{0,1\}$.…

Let $E_r$ denote the pages of the tmf-resolution spectral sequence for $Z$, whose $E_2$-page satisfies $\prescript{\mathrm{tmf}}{}{E}_2^{s,t}(Z)=0$ for $s>(t-s+12)/11$. Vanishing-l…

$\mathcal{E}_\infty$ Thom spectrum conjecture. There is an $\mathcal{E}_\infty$ morphism $Mj\to\mathrm{tmf}_1(3)$ which factors through $Mj/\!/w_4,w_8,w_{12}$, and the induced morp…

Epimorphism conjecture. The $\mathcal{E}_\infty$ orientation $Mj_3\to\mathrm{tmf}$ induces a ring epimorphism

Epimorphism conjecture. A $2$-local characteristic morphism

Characteristic morphism conjecture. Each of the three displayed maps is a characteristic morphism.

Loop-space TMF to twisted K-theory conjecture. There is a map

Membrane partition function conjecture. The membrane partition function takes values in twisted $tmf$.