21 problems

Constant-term spectral-action conjecture. The constant term of the spectral action of $\mathcal{D}_A$ on the noncommutative $n$-torus is proportional to the constant term of the sp…

An $n$-dimensional noncommutative torus is the algebra generated by unitary elements $U_1,\ldots,U_n$ satisfying … where $e(t)=e^{2\pi i t}$ and $\theta$ is an antisymmetric matrix…

Let $A_\theta$ be the noncommutative two-torus, and let the loci $\Theta_\tau$ and ${\cal H}_{r,\tau}$ be as defined above, with $r_0$ the unique number of the form $\pm\theta+n$ f…

Standard-bundle generation conjecture. Every holomorphic bundle on $T_{\theta}$ is a successive extension of standard holomorphic bundles.

Let $G$ be a Galois group and let $\mathscr{A}_{RM}^{2n}$ be a suitable even-dimensional noncommutative torus associated with $G$. Let $\sigma_n$ be an $n$-dimensional representati…

Generic noncommutative torus kernel conjecture. We conjecture that

Let $\mathcal{S}(\mathcal{D}_A,\Lambda,\phi)$ denote the spectral action for a selfadjoint one-form $A$ on the noncommutative torus, and let…

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

Let $A_\theta$ be the algebra of the noncommutative $2$-torus, let $J A_\theta J^*$ be its opposite-algebra representation, and let $D$ be an admissible Dirac operator on the nonco…

Higher-dimensional noncommutative-torus conjecture. The connection between quasi-standard $t$-structures and noncommutative tori extends to the higher-dimensional case.

Holomorphic-bundle heart conjecture. There is a natural $t$-structure associated with every point of $\widetilde{{\bf LG}({\mathbb R})}$ whose heart is equivalent to the category o…

Let $L_{CM}={\mathbb Z}+\tau_D{\mathbb Z}$ be a lattice with complex multiplication by the ring of integers in the imaginary quadratic field ${\mathbb Q}(\sqrt{-D})$. Let…

Let $p$ be a prime with $p=3\pmod 4$, let ${\cal E}(p)={\cal E}_{CM}^{(-p,1)}$ be the associated $\mathbb Q$-curve, and let $h_K$ be the class number of the relevant imaginary quad…

Let ${\cal E}_{CM}^{(-D,f)}$ be an elliptic curve with complex multiplication by the order of conductor $f$ in the imaginary quadratic field, and let ${\cal A}_{RM}^{(D,f)}$ be the…

Consider the spectral action on a noncommutative torus with perturbation … Let $\Theta$ be the deformation matrix, and let the associated Yang--Mills term be…

Let $\mathcal{D}$ be the Dirac operator of a noncommutative torus and let $\mathcal{D}_A$ be its fluctuation by a selfadjoint one-form $A$. Let $\Theta$ be the deformation matrix,…

Let $\mathcal{D}_A$ denote the fluctuated Dirac operator on a noncommutative torus, and let $\mathcal{D}+A$ denote the corresponding operator on the commutative torus. Spectral-act…

Let $A_\theta$ be a noncommutative torus with smooth subalgebra $A_\theta^\infty$, and let $\varphi_A$ be the $*$-automorphism defined by … where…

Let $E_{CM}$ be an elliptic curve with complex multiplication, let ${\theta}_{E_{CM}}$ be the real quadratic irrational associated with $E_{CM}$, and let…

Consider a noncommutative $n$-torus with Dirac operator $\mathcal{D}$ and a fluctuated Dirac operator $\mathcal{D}_A$. Let $\Theta$ denote the skew-symmetric deformation matrix, an…

Let $\mathcal{D}$ be the Dirac operator on a noncommutative $n$-torus, let $A$ be a selfadjoint one-form, and write $\mathcal{D}_A$ for the corresponding fluctuated Dirac operator.…