22 problems

High-dimensional realizability conjecture. The known standard non-negativity conditions on $g_2$ should be sufficient to ensure the existence of point processes in all space dimens…

Let $\mathrm{Part}_n$ denote the proposed cubical assembly analogue of $\mathrm{Part}(-)$ that makes the first $n+1$ layers of a cubical assembly into partitioned assemblies, and l…

Let $A$ be the partial combinatory algebra underlying the assembly category, and let cubical assemblies be the cubical analogue of assemblies. The constructions of $W$-types with r…

Let $A$ be the partial combinatory algebra underlying the assembly category, and let $\mathrm{p}\mathbbm{Grpd}(\operatorname{Asm}(A))$ be the category of partitioned groupoid assem…

Let $A$ be the partial combinatory algebra underlying the assembly category, and let $\mathrm{p}\mathbbm{Grpd}(\operatorname{Asm}(A))$ denote the category of partitioned groupoid a…

Let $A$ be the partial combinatory algebra underlying the assembly category, and let $\mathbbm{Grpd}(\operatorname{Asm}(A))$ denote its category of groupoid assemblies. A reflectiv…

Let $\mathcal{L}\subset\mathbb{RP}^2$ be an essential pseudoline arrangement, and let $M$ be its matroid. Let $Q$ be the Hirzebruch quadratic form of $M$, and let the semistable co…

Canonical realizations conjecture. The set of realizations of $M$ over $\mathbb{E}$ canonically injects into the set of $\hat{H}\mathbb{E}$-module structures on $\hat{H}M$.

Integer-scaling criterion. If $\{\frac{\lambda_1}{\alpha_1},\ldots,\frac{\lambda_n}{\alpha_n}\}$ is $C_{\mathbb{Q},3}$-realizable, then it is $C_{\mathbb{Q},3}$-realizable if and o…

Kleene's relative consistency conjecture. Formalizing function realizability and the model-theoretic consistency proof for $\mathcal{FIM}$ should yield a metamathematical proof tha…

Kleene's realizability conjecture. Every closed theorem of $\mathrm{HA}$ is realizable.

Counterpart conjecture. Some of these local operators have counterparts in the toposes

Let $(\mathscr{A},{\preccurlyeq},{\to})$ be an implicative structure, and interpret closed $\lambda$-terms and their types in a polymorphic type system with binary intersections. A…

Realizable fiber representation conjecture. Every locally realizable COM is a fiber of a realizable OM.

Realizability conjecture. For an arbitrary sign pattern ${\bar \sigma}$, the only type of pairs $(pos,neg)$ which can be non-realizable has either $pos$ or $neg$ vanishing. Equival…

The paper considers interactive realizers, continuous functions on states of knowledge that specify further guesses for extending the knowledge and reactions to the discovery of ne…

Non-realizability conjecture. There is no arrangement in $\mathbb{C}^3$ with the same incidence as $\mathcal{A}$.

Dominance and synthetic-domain-theory conjecture. A subcollection of the $\fe0^1_1$-sets forms a dominance in ${\cal E\!f\!f}_{\cal J}$, and this topos has a model of Synthetic Dom…

Pitts' indexing conjecture. Pitts' local operator gives a neat indexing of the hyperarithmetical functions, which could be fruitful in developing recursion theory with hyperarithme…

Pitts' indexing conjecture. Pitts' local operator gives a useful indexing of the hyperarithmetical functions.

Higher-identity realizability conjecture. Replacing realizers of identity proofs by higher identity proofs witnessing that this square commutes up to equivalence is the appropriate…

Consider methods for proving non-realizability of triangulated surfaces, including oriented matroid methods and the method proposed by Timmreck. Oriented-matroid strength conjectur…