31 problems

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

A spectral triple for a commutative geometry satisfies the order-one condition only up to compact operators, as occurs in certain strictly noncommutative $q$-deformed examples. Sta…

Let $SU_q(2)_\sigma$ be the fixed-point subalgebra associated with the twisted spectral triple described above, and let $u$ be the generator of $K_1(SU_q(2))$ represented by the sp…

Let $\cA=C^\infty(\mathbb{T}^2)\rtimes\mathbb{R}$, with generators $u_1,u_2$ for the torus part and $v_t$ for the crossed-product action, and let $D$ be the Dirac-type operator def…

Let $\cA=C^\infty(\mathbb{T}^2)\rtimes\mathbb{R}$, with subalgebra $C^\infty(\mathbb{T}^2)$ and generators $V_t$ for the crossed-product action. Consider the spectral triple on…

Let $\cA=C^\infty(\mathbb{T}^2)\rtimes\mathbb{R}$, and let $(\cA,L^2(\mathbb{T}^2,E),\tilde{Q})$ be the spectral triple described in the paper. Denote by $\Omega_{\tilde{Q}}(\cA)$…

Let $\mathcal{A}_1=\mathcal{D}_{L^2}(\mathbb{R}^{2N})$ be the algebra of Moyal multipliers acting on the relevant Hilbert space, let $\mathcal{B}(\mathbb{R}^{2N})$ denote the propo…

Fix real numbers $C$, $a$, and $b$. Consider quantum Riemannian $1$-spaces with measure whose associated Hilbert space comes from a unital involutive algebra and a state, with non-…

Non-vanishing torsion conjecture. Such spectral triples have non-vanishing torsion.

Compact-resolvent conjecture. The asymptotic behaviour of $D_{\overline{\partial}}$ satisfies the compact-resolvent condition for all irreducible quantum flag manifolds.

Roșu's conjecture. The operator $D_{\overline{\partial}}$ allows one to construct a spectral triple for all irreducible quantum flag manifolds.

Exponential growth conjecture. The eigenvalues of the Dolbeault–Dirac operators of all irreducible quantum flag manifolds have exponential growth.

Let $M$ be a compact Riemannian manifold, let $H_\Lambda$ be the truncated Hilbert space, and let $\mathbb{P}(H_\Lambda)_{\epsilon_\Lambda^2}$ denote the localized state space at s…

Let $(C^(M)_\Lambda,H_\Lambda,D_\Lambda)$ be a truncated commutative spectral triple, let $\mathbb{P}(H_\Lambda)$ be the projective space of $H_\Lambda$, and let $d_\Lambda$ be the…

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

Let $\mathcal{O}_q(G/L_S)$ be an irreducible quantum flag manifold of weak Gelfand type, and let $(Omega^{(bullet,bullet)},kappa)$ be its covariant Kähler structure, unique up to r…

Let $K_H$ be the harmonic Sierpinski gasket equipped with its geodesic metric, and let the direct sum spectral triple be the one constructed from the constituent curves. The associ…

Let $\mathbb{C}_q[G/L_S]$ be an irreducible quantum flag manifold, and let $\Omega^{(0,\bullet)}$ be the subcomplex of its Heckenberger--Kolb calculus. Denote by…

The paper constructs spectral triples for algebras associated with one-dimensional subshifts, including a spectral triple on the noncommutative crossed-product algebra generated by…

Admissible-operator conjecture. The class of admissible Dirac operators should contain all operators proposed in [DS] and their fluctuations.

The operators use a representation $c0$ of $algA_S$ on a Hilbert space $caH$, a positive operator $Theta$, a Dirac-type operator $DD$, an automorphism $beta$, and a functional…

Categorification conjecture. The category of finite even $S^o$-real spectral triples over $\mathbb{C}$ and spectral triple morphisms is equivalent to the category of finite-dimensi…

Spectral-dimension and spectral-measure conjecture. The spectral dimension satisfies

Equality conjecture. The equality $d(p,q)=\widetilde d(p,q)$ should hold for every pair of points $p,q$ on a large class of globally hyperbolic Lorentzian spin manifolds.

Continuum spectral-triple conjecture. The Kasparov module