68 problems

Stable-field conjecture. Every infinite stable field has no separable extensions; equivalently, every infinite stable field is separably closed.

A higher-spin field of spin $s$ is a field represented by a symmetric $s$-tensor, and the free theory has abelian gauge symmetries. For $s=3$, the higher-spin tower conjecture. all…

Let $K$ be a field whose theory is supersimple. A field is pseudo-algebraically closed ($PAC$) if every geometrically integral variety defined over it has a rational point. Supersi…

A spin equation is associated with a quasi-homogeneous polynomial; in the simplest $A_{r-1}$ case it has the form … A marked point with trivial orbifold structure is called a Ramon…

Let $K/k$ be a nontrivial extension of fields. Write $\operatorname{Aut}(K/k)$ for the group of automorphisms of $K$ fixing $k$ pointwise, acting on $K- k$. Finite-orbit conjecture…

Let $\mathcal{M}$ be a manifold with boundary $\partial\mathcal{M}$, and let $B_m$ denote the supergravity field appearing in the boundary action. The action may contain terms prop…

Let $c_j$ be the weights in the metric-affine action, and let Minkowski space be equipped with connections of the form … Here $R(x)$ denotes the curvature of the connection, and th…

For a field, the Kaplansky radical consists of the nonzero elements represented by every norm quadratic form in two variables. Kijima–Nishi conjecture. For quadratic extensions, th…

Let $\mathcal{G}$ be a graph with fields $\phi_i,\tilde\phi_i$ describing branching trees, and let $\mathfrak{r}$ and $\mathfrak{t}$ be graph points. Write…

A field is called stable if its first-order theory is stable, simple if its theory is simple, separably closed if it has no proper finite separable algebraic extension, and pseudo-…

Mal'tsev's conjecture. The class of multiplicative groups of fields cannot be characterized elementarily, that is, by first-order formulas.

Let $K$ be a number field. A triplet $(a,b,c)$ is called compositum feasible over $K$ when it arises from field extensions whose compositum has the specified degree $c$ and whose i…

Let $n\in\mathbb{N}$. A field $\mathbb{K}$ is an $n$-FSEANI if the parametric family of systems of equations given by (EE) has a non-zero solution in $\mathbb{K}$ for every choice…

Trial projection conjecture. The solution of the first equation of the master system is the map $\pi_{\mathrm{trial}}$ defined by

Lüroth-type conjecture. There is a unique field extension $L|k$ in $F$ such that

Let spacetime have dimension $D$, and consider different $(D-1)$-dimensional foliations with leaves on which the covariant Hamilton–Jacobi functions are integrated. A foliation and…

Consider a field theory with solitonic solutions, and distinguish the appearance of an inner substructure in a soliton from the existence of a larger number of normal modes around…

Let $\alpha$ be as in the source, and let an invariant set mean a set closed under addition and multiplication. Countable invariant decomposition conjecture. If the proposed constr…

Let a finite decomposition of the positive elements of a field be given, and regard its pieces as ${\mathbb Q}$-convex sets. A separating hyperplane is a hyperplane separating piec…

Let $C\mathrm{XF}$ be the theory of fields excluding the curve $C$ over the base field $K_0$, and let $M$ be a model of $C\mathrm{XF}$. The property $\mathrm{SOP}_3$ is the strict…

Let $F$ be a field and let $X$ be an algebraic closure of $F$, or let $F$ be an ordered field and let $X$ be its real closure. In either case, quantifier elimination is classically…

The $\barphi^6$ model has antikink-kink ($\bar{\rm K}{}\rm K$) configurations whose collisions can exhibit resonant scattering, even though the individual kinks support no bound st…

Let $F$ be a field, and write $F^\times$ for its multiplicative group. A group is indecomposable if it cannot be expressed as a nontrivial direct product of groups. Nonexistence co…

Pure-field NIP$_n$ conjecture. For every $n\geqslant 2$, no strictly NIP$_n$ pure fields exist; equivalently, a pure field is NIP$_n$ if and only if it is NIP.

For a Turing degree $x$, let $\mathds{R}_{x}$ denote the set of real numbers of degree at most $x$, and for Turing degrees $x$ and $y$, let $x\oplus y$ be their join and…