20 problems

Let $F/k$ be the field extension in the source, let $G$ be the field automorphism group, and let $\mathcal I_G$ be the category of homotopy-invariant smooth $G$-representations. Fo…

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

Let $H$ be a Hilbert space and let $H_1,H_2,\ldots,H_n$ be subspaces of $H$, forming a system of subspaces $S=(H;H_1,H_2,\ldots,H_n)$. For each subspace, consider the orthogonal pr…

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…

Simple-arc sufficiency conjecture. There is a series of finite-dimensional representations

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. The SU(2)-abelian L-space conjecture. Every…

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 $p$ be a prime, and let $\beta_{3,p}$ denote the Burau representation of the braid group $B_3$ modulo $p$. Faithfulness conjecture. The representation $\beta_{3,p}$ is faithful…

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…

Let $2\leq m\leq M$, let $\Gamma$ be a uniform lattice in $\operatorname{Isom}(\operatorname{\mathbb{H}}^m_{\operatorname{\mathbb{C}}})$, and let … be a convex co-compact represent…

Deroin–Tholozan conjecture. In each Hitchin fiber, the Hitchin section maximizes the translation length.

Kaplansky's conjecture. If $Q_1$ and $Q_2$ have perfectly identical representations over ${\mathbb Z}$, then they are equivalent over ${\mathbb Z}$, or are constant multiples of re…

Abundance conjecture. The Perron–Frobenius measure is potentially just one of many probability measures that could give faithful representations of the Cuntz–Krieger $C^*$-algebra…

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

Finite representability conjecture. The Clifford algebra of any given form is finitely representable.

Maximal-dimensional locus conjecture. For any $K$, the maximum-dimensional part of the Zariski closure of $U_K$, respectively that of $V_K$, is a complex curve.

Let $M$ be a compact 3-manifold and let $\gamma\in\pi_1(M)-\{1\}$. Finite-ring separation conjecture. There exists a finite commutative ring $R$ with identity and a homomorphism ……

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…