8 problems

Dual-number manifold conjecture. In analogy with the Newlander–Nirenberg theorem, $TM$ should already be a manifold defined over the ring

Let ${\cal H}_T(\mathbb{Q}_{\varepsilon})$ be the category of Hodge-Tate structures over $\overline{\mathbb{Q}}_{\varepsilon}$, and let…

Let $F$ be an algebraically closed field, let $F_{\varepsilon}=F[\varepsilon]/(\varepsilon^2)$, and let ${\cal Q}_{\bullet}(F_{\varepsilon})$ be the Lie coalgebra formed from the n…

Let $F_{\varepsilon}=F[\varepsilon]/(\varepsilon^2)$, and let ${\cal Q}_{\bullet}(F_{\varepsilon})$ be the Lie coalgebra combining the non-Euclidean and Euclidean scissor-congruenc…

Let $F$ be a field, let $F_{\varepsilon}=F[\varepsilon]/(\varepsilon^2)$, and let ${\mathbb Q}_{\varepsilon}$-mod be the tensor category whose objects are…

Let $A=A_s+A_d\epsilon$ be an $n\times n$ dual number matrix, where $A_s$ is its standard part, $A_d$ is its dual part, and $\rho(A_s)$ denotes the spectral radius of $A_s$. The ze…

Let $\mathbb{DC}$ denote the dual complex numbers and $\mathbb{DR}$ the dual real numbers, with $\epsilon^2=0$. A rank-$k$ dual complex matrix $\bm{A}_k$ is a dual complex matrix w…

Let $D$ be the algebra of dual numbers, let $\mathscr{D}$ be the associated 2-category of projective bimodules, and for each positive integer $k$ let $\mathcal{J}_k$ denote the two…