18 problems

Vanishing Einstein functional conjecture. Suitably regular two-dimensional spectral triples will have the Einstein functional identically vanishing.

Let $M$ be a stably causal Lorentzian manifold whose sectional curvatures are much less than $1$, and let $\gamma$ be a timelike curve from $s$ to $t$ in $M$ whose radius of curvat…

Let $X_\Gamma$ be a derived $\Gamma$-space, let $\mathcal D_{\mathrm{coh}}(X_\Gamma)$ be its coherent derived category, and let $\mathsf{Phys}_\Gamma$ be the category of $\Gamma$-p…

For an $M$-band system, let the quantum states be represented by points in the complex projective space $\mathbb{C}P^{M-1}$, and let the strong quantum isoperimetric inequality (QI…

An extended phase space is a Poisson manifold functorially associated with a Poisson covariant system, and it has the structure of a classical principal bundle. A crossed product i…

A quantum projective space has a quantum metric $g$ and a connection whose curvature defines an appropriately defined Ricci tensor. The Einstein condition conjecture. The Ricci ten…

Connectedness and over-completeness conjecture. For every irreducible matrix configuration, ${\cal M}$ is connected, and the quasi-coherent states are over-complete, meaning

Let $V^1$ be the unit-volume Liouville random measure defined from the Liouville measure by normalizing its total mass to $1$, under the integrability assumptions stated in the sur…

Let $coperatorname{Vol}$ denote the total mass of the Liouville measure, and consider the corresponding unit-volume Liouville measure on a sphere-like manifold. Sheffield's unit ar…

Dolbeault–Dirac spectral triple conjecture. The point spectrum of $D_{\overline{\partial}}$ has finite multiplicity and tends to infinity. Consequently,

The ${\cal B}_1$ basis diagonalizes curvature-related Wilson loop operators around edges, while the ${\cal B}_2$ basis encodes spatial geometric data such as areas and dihedral ang…

For a spherical simplex $\sigma$, let $\sigma'$ be a dual simplex whose lengths are determined by the dihedral angles of $\sigma$. The normalized Wilson loops around triangles are…

The construction uses a Hilbert space and operators for a $(3+1)$-dimensional theory of flat connections with curvature defects, obtained from the Hilbert space and operators of a…

Let $Z(N,k)$ be the partition function and let the modified grand potential $J(\mu,k)$ be defined by … Let $\mu_{\mathrm{eff}}$ denote the effective chemical potential, and let…

In a framework where boundary quantum states encode geometric data, entanglement entropy is associated with the quantum geometry of the boundary and spacetime. Bianchi's conjecture…

Let auxiliary tetrahedra be scissor-congruent when they are related by the scissor-congruence symmetry identified in the semiclassical description of the volume spectrum. The assoc…

In the pre-geometric setting, quantum spacetime is a sum over quantum histories of superposed pre-geometric spin networks. Emergence conjecture. Continuous spacetime emerges as a c…

Let $P$ be an arbitrary polygon, and suppose its classical area has a quantum interpretation analogous to the geometric formula for the parallelogram $P(p_1,p_2)$, in which the qua…