36 problems

Let $W_0$ and $W_1$ be closed, connected, oriented $3$-manifolds. A rational homology ribbon cobordism from $W_0$ to $W_1$ is a ribbon cobordism $W$ from $W_0$ to $W_1$ such that t…

Let $\tau$ be a collection of singularity types, let $X_\tau$ be the classifying space for $\tau$-cobordisms, let $\mathcal K_\tau$ be the Kazarian space, let $\mathcal K^+_\tau$ b…

Universality conjecture. The formal group ${\frak F}(x\otimes 1,1\otimes x)$ is the universal object in the category of formal groups over a (topological) Hopf algebra.

Seiberg's deformation conjecture. Any quantum field theories with the same symmetry $G$ and the same anomaly ${\mathbf{Z}}_{d+1}$ can be deformed to each other by adding degrees of…

Let $M$ be a closed surface, and let $I_k$ denote its Friedlander–Nadirashvili invariant associated with the $k$-th Laplace eigenvalue. Friedlander–Nadirashvili cobordism-invarianc…

Let nanowords be considered over an alphabet consisting of exactly one element, with homotopy defined by the nanoword homotopy relation. Stronger conjecture. Any two nanowords over…

Let $M^n$ be a closed $n$-manifold. A marked cubical decomposition of $M^n$ is a cubical decomposition together with the marking data used to distinguish its cubical cells, and two…

Let $\overline{A_2}(l+1)$ and $\overline{\Sigma}^2(l)$ be the singularity loci associated with the relevant maps, and consider appropriate resolutions of these loci. Cobordism conj…

Let $l$ be an integer, and let $\overline{A_2}(l+1)$ and $\overline{\Sigma}^2(l)$ denote the corresponding singularity loci. Their Segre–Stiefel–Whitney classes are defined in the…

Atiyah–Hirzebruch's conjecture. These integrality relations give all the integral relations for Chern numbers.

Linear lower-bound conjecture. The minimal number satisfies

Glazyrin–Pak unit-triangle conjecture. There are two unit triangles $\Delta_1,\Delta_2\subset\mathbb{E}$ which are not cobordant.

Glazyrin–Pak cobordism conjecture. For every integral curve $\gamma$, there is a unit rhombus $\rho$ and a dome over $\gamma\cup\rho$.

Let $\mathcal{C}_0$ be the decoration data for an $E_1$-monoidal $(\infty,n-1)$-category $\mathscr{F}_{n-1}(\mathcal{C}_0)$ of singular manifolds in $\mathbb{R}^n$, framed in codim…

Let the Thom–Pontryagin construction produce a topological space representing cobordism classes of decorated foams, and let Quillen's K-theory space be the space representing the h…

Let $Y$ be the spectral cover in the logarithmic cotangent bundle $T^{\vee}X(\log {\rm Wob})$, let ${\rm Wob}$ be the wobbly locus, and suppose a logarithmic Higgs sheaf is equippe…

Cobordism conjecture. This cobordism is oriented if $M$ and $N$ are oriented, the codimension $k$ is odd, and $r$ is even.

The invariant $\Delta_a$ is defined for a $3$-manifold together with a suitable label $a$, and a $\operatorname{Spin}^c$ homology cobordism is a homology cobordism preserving the r…

Norm-one conjecture. The partition functions of these manifolds satisfy

Cobrordism-invariant conjecture. For invertible topological orders in branch-independent bosonic systems, the topological invariant $Z^\text{top}(M)$ is a cobordism invariant.

Let $k$ be the relative normalisation in the cobordism group with Spin$^c$ structure and gauged $U(1)_{(0)}$ symmetry, denoted by…

A physically consistent configuration in quantum gravity may carry a cobordism charge if it represents a class in a cobordism group; a non-vanishing cobordism group would therefore…

Ribbon rational homology cobordism conjecture. If there exists a ribbon $\mathbb{Q}$-homology cobordism from $Y_0$ to $Y_1$ and one from $Y_1$ to $Y_0$, then $Y_0$ and $Y_1$ are di…

A topological quantum field theory is a quantum field theory whose observables depend only on topological bordism data, and an invertible phase is an invertible field theory, equiv…

Assuming Hypothesis H, M-brane charge quantization is discussed through Cobordism cohomology theories, including quaternionic and unitary Cobordism; write $M\!f$ for such an associ…