17 problems

Ordered-group projective-plane conjecture. An ordered abelian group cannot locally trace define an infinite projective plane.

A limit group is a finitely generated fully residually free group. Let $F^{\mathbb{Z}[t]}$ be Lyndon's free exponential group. Myasnikov–Remeslennikov embedding conjecture. Every l…

Non-orderability conjecture for higher-rank lattices. Lattices in simple Lie groups of rank at least two have no left invariant total order.

Division-algebra embedding conjecture. If $G$ is left-orderable, then ${\mathbb Z}G$ embeds in a division algebra.

Zero-divisor conjecture. The group ring

Miller–Starchenko conjecture. Every $\mathcal I$-definable one-dimensional torsion-free ordered group is $\mathcal I$-definably isomorphic to either $(I,0,+,<)$ or…

Non-interpretability conjecture. The structure $(\mathbb{Z};+,C_\alpha)$ does not interpret $(\mathbb{Z};+,C_\beta)$, and conversely.

Invariant-subsemigroup separation conjecture. For every $P\subseteq C^+$ such that $P$ and $C^+\setminus P$ are closed under concatenations and cyclic shifts, there exists an invar…

ETOG representation conjecture. Every bi-positional prefix-independent winning condition coincides on periodic sequences with some energy condition over a totally ordered group.

Transfer conjecture. Many assertions that hold for o-minimal expansions of ordered fields should also hold for definably complete locally o-minimal expansions of ordered groups hav…

Positive-relator quotient conjecture.

Ordered-group decomposition conjecture. There is an o-minimal expansion ${\mathscr R^{\square}}$ of an ordered abelian group such that

Ordered-abelian-group conjecture. An ordered abelian group cannot trace define an infinite field.

Irrational cyclic-order conjecture. The structure $(a0;+,C_{a0})$ does not interpret $(a0;+,C_{b2})$, and conversely.

Let $E$ be a real Hilbert space with $\dim(E)=2$, let $h$ be a unit vector in $E$, and let $f:\mathbb{R}_{\ge0}\to\mathbb{R}_{\ge0}$ satisfy $f(d)=0$ if and only if $d=0$. Define ……

Let $X$ be a totally ordered Abelian group with respect to addition, and let $F:X^n\to X$ be an associative, idempotent, monotone function. The function $F$ is reducible if there e…

Let $\nu:F^*\rightarrow\mathbb{Z}^n$ be a maximal valuation, and let $\mathcal{O}=\{f\in\mathscr{F}:\nu(f)\geq 0\}$ with maximal ideal $\mathcal{m}=\{f\in\mathscr{F}:\nu(f)>0\}$ in…