32 problems

Let $X_1$ and $X_2$ be smooth, spin, closed, oriented four-manifolds, and let … be their generalized fiber sum, also smooth, spin, closed, and oriented. Let $d_1$ and $d_2$ be thei…

The group $cpi_d\operatorname{TMF}$ is the degree-$d$ part of topological modular forms, and $cpi_0$ denotes connected components of the space of theories. A two-dimensional…

For every positive integer $n$, let $\operatorname{TMF}_0(n)$ denote the associated spectrum of topological modular forms and let $\operatorname{MF}_0(n)_0$ denote the degree-zero…

Stolz–Teichner conjecture. The degree-$n$ topological modular forms of $X$ are

Let $\mathit{tmf}$ denote the connective topological modular forms spectrum, let $\Sigma_3$ be the symmetric group on three letters, and let $\mathit{tmf}^{t\Sigma_3}$ denote its T…

Let $p$ be a prime and let $\ell$ be a generator of $\mathbb{Z}_p^\times$ when $p$ is odd, or of $\mathbb{Z}_2^\times/\{\pm 1\}$ when $p=2$. Write $Q(\ell)_{K(2)}$ for the $K(2)$-l…

Let $X$ be the target of a 3-dimensional $\mathcal{N}=1$ supersymmetric sigma model, and let $X_+$ denote $X$ with a disjoint basepoint. Sigma-model boundary-condition conjecture.…

Let $b$ be a symmetric bilinear form and let $\theta_b$ be the holomorphic section of the complex Looijenga line bundle…

Let $Q\colon\mathbb{Z}^d\to\mathbb{Z}$ be positive definite and unimodular, let $b$ be the corresponding even bilinear form, and let $\Theta_Q=\Theta_b$ be its theta function. Let…

Let $\mathcal{Z}$ denote the invariants of closed 3- and 4-manifolds and the proposed maps associated to cobordisms, valued in modules over $\operatorname{TMF}$. TMF-valued TQFT co…

Let $\operatorname{Bil}\colon \operatorname{Lat}^{\operatorname{op}}\to\operatorname{AbGroup}\subset\mathrm{PicGrpds}$ be the functor of bilinear forms, and let…

Let $G$ be a compact Lie group acting on a manifold $X$, let $X{/{\!/}^{\nabla}}G$ denote the quotient stack encoding the action with connection data, and let…

Let $A_{-2c}$ be the kernel of … For $c=32,31/2,15,14$, the kernel is respectively isomorphic to $\mathbb{Z}/3,\mathbb{Z}/2,\mathbb{Z}/2,\mathbb{Z}/2$. The kernel-generator conject…

Let $\mathbb{V}$ be a unitary self-dual VOSA of central charge $c=n/2$, equivariant under a finite group $G$ with anomaly…

Let $\mathbb{V}$ be a unitary self-dual VOSA of central charge $c=n/2$ containing the affine Lie VOA $\mathfrak{g}_k$, and let $G$ be the corresponding compact simply-connected gro…

Let $(M_{8\ell+2},g_M^{\mathrm{spin}})$ be a closed differential spin manifold, and let $g_M^{\mathrm{string}}$ be a lift of its spin structure to a differential string structure.…

The VOSA-to-$\operatorname{TMF}$ class conjecture. The VOSA should determine a class

The twisted $E_8$-equivariant lift conjecture. The element $e_8$ lifts to

Let $V$ be a rational vertex operator algebra with central charge $c=24n$, where $n$ is an integer, and suppose that $V$ has only one irreducible module. Let the character of $V$ b…

Let $X$ be a parameter space, and let $\nu\oplus k\in (I_\mathbb{Z}\Omega^\text{spin})^4(X)$ be the anomaly of a 2d spin $\mathcal{N}=(0,1)$ supersymmetric theory. Twisted Segal–St…

Let $G$ be a symmetry group, let $\nu$ be the gravitational anomaly, and let $k$ be the 't Hooft anomaly, with $\nu\oplus k\in (I_\mathbb{Z}\Omega^\text{spin})^4(BG)$. For a simple…

Let $R$ be an $\mathbf{E}_1$-ring spectrum and write $A=\pi_0 R$. Suppose that there is a class $D\in \pi_{-d}R$ witnessing the Anderson self-duality of $R$. Let $F\colon R\to R$ b…

Let $\nu\in\mathrm{TMF}^{-3}(\mathrm{pt})$ be the class corresponding to $[\overline{\operatorname{Fer}(3)}]$, let $\{24\Delta\}\in\mathrm{TMF}^{-24}(\mathrm{pt})$ be the class cor…

Let $\SQFT^\bullet$ denote the spectrum whose spaces classify compact unitary $\mathcal N=(0,1)$ superconformal field theories, graded by anomaly. Let $\mathrm{TMF}^\bullet$ denote…

Structured ring spectrum conjecture. The global sections satisfy