11 problems

Let $f(x)\in\mathbb{Z}[x]$ be a monic expanding polynomial. A polynomial has the Height Reducing Property (HRP) if, writing $|f(0)|=c$, there is a polynomial $g(x)\in\mathbb{Z}[x]$…

Separation conjecture. If g\text{in \hspace{-.8em}/}\langle\alpha,\beta\rangle, then $\alpha$ and $\beta$ lie in the same connected component of

Let … and define … Thus, $\Omega_2$ is the set of double zeros in $(0,1)$ of power series with coefficients in $\{-1,0,1\}$. Infinitely many components conjecture. The set…

Let $R$ be a real closed field. For a smooth algebraic variety $X$ over $R$, write $X^{\operatorname{h}}$ for its associated space of real points. A generalized connected open subs…

Let $\mathcal{P}(a_1,\dots,a_m\mid -b_1\mid\dots\mid -b_n)$ be a nonempty stratum of residueless meromorphic differentials on genus-zero curves, with zeros of orders…

Let $\mathcal{C}$ be a tuple of semisimple conjugacy classes in $\operatorname{GL}_n\langle\sigma\rangle$, not necessarily generic, and suppose that the character variety…

Let $G$ be a group, and let $\mathfrak{c}$ denote the cardinality of the continuum. A subgroup of $G$ is unconditionally closed if it is closed in every Hausdorff group topology on…

Let $A$ be an expanding integer matrix whose characteristic polynomial is … and let $\mathcal{D}=\{0,1,\dots,m\}v$ with $m\geq1$ and $v\in\mathbb{R}^2$ such that $\{v,Av\}$ is line…

Asok–Dorant's conjecture. If $k$ is an infinite field and $X$ is a smooth $\mathbb A^1$-connected scheme, then $U$ is also $\mathbb A^1$-connected. This conjecture was answered aff…

Let $A$ be an expanding integer matrix, and let the height reducing property (HRP) denote the property of its characteristic polynomial used to determine connectedness of the self-…

Let $A\in M_2(\mathbb{Z})$ be expanding with $|\det(A)|=3$, let $f$ be its characteristic polynomial, and let ${\mathcal{D}}=\{0,1,b\}v$ with $b>1$ and $v\in\mathbb{R}^2$ such that…