12 problems

Let $H_{b}^{\alpha}(X,GL(2,\mathbb R))$ denote the class of the cocycles considered in the source. Viana's continuity conjecture. Lyapunov exponents are always continuous among…

Let $\mathfrak A$ be an associative algebra with a trace, and let $D_1,\dots,D_{2n}$ be derivations satisfying conditions (i)--(iii) from Section 2.4. Let $\Sigma_{N,2n-1}$ denote…

Amenable cocycle closure conjecture. One has

Lipschitz right-inverse conjecture. There exists $L=L(\lambda,C)$ and a not necessarily linear function $S_{\alpha,A}:X\to X$ such that

Measurable cocycle quotient conjecture. For any measurable map $\omega\mapsto \rho_\omega$ from $\Omega$ to $Z^1_{<m}(G,X,S^1)$, there exists a set of positive measure…

Let $t\in\mathbb{N}$. A cocyclic Hadamard matrix is a Hadamard matrix constructed from a cocycle over a group. Cocyclic Hadamard conjecture. There exists a cocyclic Hadamard matrix…

Let $c^2_p(\beta,\gamma)$ and $c^2_q(\beta,\gamma)$ be the conjectural non-commutative Hilbert modular $2$-cocycles associated with points $p$ and $q$. Let $b_{pq}^1$ be the interv…

Let $c^2_p(\beta,\gamma)$ denote the non-commutative Hilbert modular symbol associated with a point $p$ and group elements $\beta,\gamma$. Let $c^1$ denote the associated non-commu…

Let $c^1_{p_1,p_2;p_3}$ denote the non-commutative Hilbert modular symbol associated with points $p_1,p_2,p_3$. Non-commutative Hilbert modular symbol 1-cocycle conjecture. The sym…

A Hadamard matrix is an $n\times n$ matrix with entries in $\{1,-1\}$ whose rows are pairwise orthogonal. Given a group $G$, a cocyclic matrix is a matrix $M_f=(f(g,h))$ arising fr…

Let $(\alpha,A)$ be an analytic cocycle, and let $A_\epsilon$ denote its complexified cocycle at imaginary shift $\epsilon$. Assume that $L(\alpha,A_\epsilon)=0$ for $a<\epsilon<b$…

Let $\Gamma$ be a group generated by $g_{1},\dots,g_{n}$ and assume that $L(\Gamma)$ can be embedded into $R^{\omega}$. Let $V\subset\ell^{2}(\Gamma)^{n}$ be the subspace … Let…