10 problems

Faithfulness conjecture. The homomorphism $\kappa$ is faithful, that is, injective.

Let $S({\bf C}^{N})^{\infty}_{\mathrm{AP}}$ denote the asymptotically periodic sequences and let $S({\bf C}^{N})^{\infty}_{\mathrm{EP}}$ denote the eventually periodic sequences. W…

Property $T$ integrality conjecture. The representation $\varrho$ is integral.

Simpson's motivic conjecture. The representation $\varrho\otimes k$ is a direct summand over $k$ of the monodromy representation of a motive over $X$, that is, it comes from geomet…

Let $Y$ be a rational homology $3$-sphere. Call $Y$ $SU(2)$-abelian if every representation $\pi_1(Y)\to SU(2)$ has abelian image. Zhang's conjecture. If $Y$ is $SU(2)$-abelian, th…

Let $|0\rangle_p$ and $|0\rangle_q$ be the vacuum states in the Hilbert spaces $L^2_p({\cal F})$ and $L^2_q({\cal F})$, respectively, associated with the Sobolev parameters $p$ and…

Simpson's conjecture. Every such representation has integral image. This connects rigidity of local systems on projective varieties with arithmeticity; the source presents it as Si…

Let $K$ be a two-bridge knot, and let $\Phi_K(2,y)$ be its Riley polynomial specialized at meridian trace $2$. The knot signature is denoted by $\sigma(K)$. Riley's conjecture. The…

Rank monotonicity conjecture. The obtained representation of $C_f$ has rank no less than the rank of the initial representation of $C_g$.

Let $\Gamma$ be an infinite sofic group, let $\Sigma$ be a sofic approximation, and let $V$ be a finite-dimensional representation of $\Gamma$. The finite-dimensional vanishing con…