27 problems

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…

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…

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

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 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.

Let $\mathscr{D}\subseteq\mathbb{R}$ be the set of left-Diophantine numbers. Field conjecture. The set $\mathscr{D}$ is a field. This is a weaker consequence of the algebraicity co…

Strict containment conjecture. The class of weakly locally finite division rings strictly contains the class of Stewart's division rings.

Consider pure 4-dimensional $\mathcal N=1$ supergravity on $\mathbb R^4$. Let $\mathbb C^2$ carry its standard holomorphic symplectic structure, and write…

Consider 5-dimensional $\mathcal N=1$ supergravity on $\mathbb R^5$. Let $\mathbb C^2$ carry its canonical holomorphic symplectic structure, and let Poisson BF theory be defined on…

Let $D$ be a non-commutative division ring algebraic over its center. A division subring is centrally finite if it has finite dimension over its center. Mahdavi-Hezavehi's weak Kur…

Let $D$ be a division ring with center $F$. A subfield $K$ of $D$ is maximal if it is maximal among subfields of $D$. Maximal-subfields algebraicity conjecture. If every maximal su…

Let $K\subset L$ be a field extension. The linear matching conjecture. If $K\subset L$ possesses the local linear matching property, then it possesses the linear matching property.…

Prime-degree linear acyclic matching conjecture. There are infinitely many primes $p$ for which there exists a field extension $K\subset L$ with $[L:K]=p$ such that $K\subset L$ do…

Four-term sum-product conjecture. There are $a,b,c,d\in F$ such that

Product and ascending base-change conjecture. (1) The compositum of two $l$-$i$-modular extensions is $l$-$i$-modular: if $K/k$ and $L/k$ are $l$-$i$-modular, then $K(L)/k$ is $l$-…

Let $\mathbb{F}\subset\mathbb{K}$ be an extension of fields, and let $S$ be a zero-nonzero pattern that is spectrally arbitrary over $\mathbb{F}$. Field-extension conjecture. Then…

Let $F$ be an infinite field, and write $F^{\times}$ for its multiplicative group. A group is decomposable if it is isomorphic to a direct sum of two nontrivial subgroups. Decompos…

Let $K$ be a superrosy field and let $v$ be a non-trivial valuation on $K$. Its value group is the ordered abelian group of values, and its residue field is the quotient field indu…

Let $K$ be a field such that for every finite extension $L$ of $K$ and every natural number $n>0$, the index $[L^*:(L^*)^n]$ is finite. If $\operatorname{char}(K)=p>0$ and…

Let $K$ be an infinite field with NIP such that, for every finite extension $L$ of $K$ and every natural number $n>0$, the index $[L^*:(L^*)^n]$ is finite. The Hasson–Shelah defina…

A field is PRC (pseudo real closed) if it satisfies the PRC property; its absolute Galois group is small if it has only finitely many closed subgroups of each finite index. The sup…

A field is NIP if its theory has the non-independence property. The superrosy NIP field conjecture. Every infinite superrosy field with NIP is either algebraically closed or real c…

A supersimple field is a field whose theory is supersimple; a field is PAC if every absolutely irreducible variety over it has a rational point, and its absolute Galois group is sm…

Bottom theorem conjecture. For almost all \text{\boldmathsigma}\in\operatorname{Gal}(K)^e, the field K_s(\text{\boldmathsigma}) is a finite separable extension of no proper…