12 problems

Maximal-class conjecture. In dimensions $n>8$, the maximally symmetric rank $3$ distribution with $6$-dimensional square is of maximal class.

Let $\mathfrak{g}$ be the Lie algebra generated by the raising and lowering operators of the periodic Fredkin chain with $N$ sites. Let $\mathrm{S}^z$ be the third component of the…

Anti-adjoint representation conjecture. There exist coefficients $\gamma_k$ such that

Let $\Sigma^\pm$ be the raising and lowering operators for the periodic Fredkin chain, let $C$ be the cyclic shift operator, and let $\mathrm{H}$ be its Hamiltonian. An eigenstate…

Total-spin Cartan decomposition conjecture. The operator $\mathrm{S}^z$ can be written as

Type $C_N$ symmetry-algebra conjecture. The operators $\Sigma^\pm$ generate the algebra of the Serre relations with rank $k=N$ and Cartan matrix

Raising and lowering operator conjecture. There exist raising and lowering operators $\Sigma^\pm$ satisfying

The fractional quantum Hall effect has an area-preserving-diffeomorphism symmetry whose quantum realization involves the Girvin-Macdonald-Platzman algebra. The Magnetoroton is a sp…

Abelian subalgebra dimension conjecture. The dimension satisfies

Algebraicity conjecture. All symmetry operators $K_{ij}$ are algebraic differential operators in the variables $\rho_{ij}$, with coefficients linear in the $\rho_{ij}$.

Let $M$ be a real-analytic connected holomorphically nondegenerate CR-hypersurface of CR-dimension $n$, and let $\mathfrak{s}(M)$ denote its symmetry algebra. Beloshapka's conjectu…

Let a scalar linear second-order ordinary differential equation be complexified, and let the resulting system of two real second-order ordinary differential equations be linear and…